UPPSALA DISSERTATIONS IN MATHEMATICS

89

Department of MathematicsUppsala University

UPPSALA 2015

Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and

Several Complex Variables

Håkan Persson

Dissertation presented at Uppsala University to be publicly examined in Polhemssalen,Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 5 June 2015 at 10:15 for thedegree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Professor Evgeny Poletsky (Department of Mathematics, Syracuse University, NY,USA).

AbstractPersson, H. 2015. Studies of the Boundary Behaviour of Functions Related to PartialDifferential Equations and Several Complex Variables. Uppsala Dissertations in Mathematics89. 52 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-506-2458-8.

This thesis consists of a comprehensive summary and six scientific papers dealing with theboundary behaviour of functions related to parabolic partial differential equations and severalcomplex variables.

Paper I concerns solutions to non-linear parabolic equations of linear growth. The mainresults include a backward Harnack inequality, and the Hölder continuity up to the boundary ofquotients of non-negative solutions vanishing on the lateral boundary of an NTA cylinder. It isalso shown that the Riesz measure associated with such solutions has the doubling property.

Paper II is concerned with solutions to linear degenerate parabolic equations, where thedegeneracy is controlled by a weight in the Muckenhoupt class 1+2/n. Two main results are thatnon-negative solutions which vanish continuously on the lateral boundary of an NTA cylindersatisfy a backward Harnack inequality and that the quotient of two such functions is Höldercontinuous up to the boundary. Another result is that the parabolic measure associated to suchequations has the doubling property.

In Paper III, it is shown that a bounded pseudoconvex domain whose boundary is α-Hölderfor each 0<α<1, is hyperconvex. Global estimates of the exhaustion function are given.

In Paper IV, it is shown that on the closure of a domain whose boundary locally is the graphof a continuous function, all plurisubharmonic functions with continuous boundary values canbe uniformly approximated by smooth plurisubharmonic functions defined in neighbourhoodsof the closure of the domain.

Paper V studies Poletsky’s notion of plurisubharmonicity on compact sets. It is shown thata function is plurisubharmonic on a given compact set if, and only if, it can be pointwiseapproximated by a decreasing sequence of smooth plurisubharmonic functions defined inneighbourhoods of the set.

Paper VI introduces the notion of a P-hyperconvex domain. It is shown that in such adomain, both the Dirichlet problem with respect to functions plurisubharmonic on the closureof the domain, and the problem of approximation by smooth plurisubharmoinc functionsin neighbourhoods of the closure of the domain have satisfactory answers in terms ofplurisubharmonicity on the boundary.

Keywords: uniformly parabolic equations, non-linear parabolic equations, linear growth,Lipschitz domain, NTA-domain, Riesz measure, boundary behavior, boundary Harnack,degenerate parabolic, parabolic measure, plurisubharmonic functions, continuous boundary,hyperconvexity, bounded exhaustion function, Hölder for all exponents, log-lipschitz,boundary regularity, approximation, Mergelyan type approximation, plurisubharmonicfunctions on compacts, Jensen measures, monotone convergence, plurisubharmonic extension,plurisubharmonic boundary values

Håkan Persson, Department of Mathematics, Analysis and Probability Theory, Box 480,Uppsala University, SE-75106 Uppsala, Sweden.

© Håkan Persson 2015

ISSN 1401-2049ISBN 978-91-506-2458-8urn:nbn:se:uu:diva-251325 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-251325)

List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I K. Nyström, H. Persson, O. Sande, Boundary estimates fornon-negative solutions to non-linear parabolic equations. Calc. Var.Partial Differential Equations. Advance online publication.doi:10.1007/s00526-014-0808-8.

II K. Nyström, H. Persson, O. Sande, Boundary estimates for solutions tolinear degenerate parabolic equations. Preprint, 2014.

III B. Avelin, L. Hed, H. Persson, A note on the hyperconvexity ofpseudoconvex domains beyond Lipschitz regularity. Preprint, 2014.

IV B. Avelin, L. Hed, Persson, Approximation of plurisubharmonicfunctions. Preprint, 2014.

V R. Czyz, L. Hed, H. Persson, Plurisubharmonic functions on compactsets. Ann. Polon. Math. 106 (2012), 133–144.

VI L. Hed, H. Persson, Plurisubharmonic approximation and boundaryvalues of plurisubharmonic functions. J. Math. Anal. Appl. 413(2014), no. 2, 700–714.

Reprints were made with permission from the publishers.

Contents

Part I: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 The Interior Regularity of the Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Subharmonic Functions and the Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . 151.4 The Boundary Harnack Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Non-tangentially Accessible Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6 Approximation of Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.7 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Part II: On the Appended Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Parabolic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Several Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Notation

Part IRn . . . . . . . . . . . . . . . real Euclidean n-space;Cn . . . . . . . . . . . . . . . complex Euclidean n-space;Ω . . . . . . . . . . . . . . . . a domain in Rn or Cn;E . . . . . . . . . . . . . . . . the closure of the set E;E ⊂⊂Ω . . . . . . . . . . denotes that E ⊂Ω;∂E . . . . . . . . . . . . . . . the topological boundary of the set E;d(x,E) . . . . . . . . . . . the Euclidean distance from x ∈ Rn to the set E ⊂ Rn;B(x,r) . . . . . . . . . . . . the Euclidean ball with center x and radius r;βn . . . . . . . . . . . . . . . . the volume of the unit ball in Rn;dV . . . . . . . . . . . . . . . the n-dimensional Lebesgue measure;dσ . . . . . . . . . . . . . . . the (n−1)-dimensional surface measure;|x| . . . . . . . . . . . . . . . the Euclidean norm of x, when x ∈ Rn;|E| . . . . . . . . . . . . . . . the Lebsegue measure of E, when E is a set;C(E) . . . . . . . . . . . . . the set of continuous functions on the set E;C∞

0 (E) . . . . . . . . . . . . the set of compactly supported, smooth functions on E;fflE f dV . . . . . . . . . . . the integral avarage of f over E, i.e., 1

|E|´

E f dV ;oscE( f ) . . . . . . . . . . the oscillation of f , i.e., supE f − infE f ;∂

∂ν. . . . . . . . . . . . . . . the outward normal directional derivative;

∇ ·F . . . . . . . . . . . . . the divergence of the vector field F ;c = c(a1, . . . ,a j) . . . denotes that the constant c depends on

the parameters a1, . . . ,a j.

Part IIΩT . . . . . . . . . . . . . . . Ω× (0,T );ST . . . . . . . . . . . . . . . ∂Ω× [0,T ];Cr(x0, t0) . . . . . . . . . B(r,x0)× (t0− r2, t0 + r2);C0,α . . . . . . . . . . . . . . Hölder continuous with exponent α;X . . . . . . . . . . . . . . . . a compact set in Cn.

7

Part I:Background

1. Harmonic Functions

1.1 IntroductionDefinition 1. A function u ∈C2(Ω) is said to be harmonic if

∆u =n

∑j=1

∂ 2u∂x2

j= 0, (1.1.1)

throughout the domain Ω⊂ Rn. 1

The equation (1.1.1) is known as the Laplace equation, and is one of themost fundamental partial differential equations (PDEs), both from a theoret-ical point of view, and from the point of view of applications. It has beenthe subject of intense studies for at least a couple of centuries, and to thisday, there are still unresolved questions related to the Laplace equation andharmonic functions.

One could say that the Laplace equation serves as a benchmark for muchresearch in partial differential equations. By studying the Laplace equation,one might get hints of what results could be expected to hold for more generalequations. And if you manage to prove a result for a more general equationwhich contradicts your intuition from the Laplace equation, it either meansthat you have made a mistake, or that you are on to something really important.The aim of this chapter is to provide such a benchmark, to which we can relatethe results of the thesis.

1.2 The Interior Regularity of the Laplace EquationAt an intuitive level, it is helpful to think about harmonic functions as mod-elling a state of equilibrium, such as a steady state temperature distributionor the density of a chemical concentration in equilibrium. Mathematically,this intuition is expressed by the following theorem, known as the mean valueformula.

1Throughout this thesis, Ω will always denote a domain in Rn (or in Cn ∼= R2n). For a list ofother notational conventions, see page 7.

11

Theorem 2. Suppose that u is a continuous function in Ω. Then u is harmonicif and only if

u(x) =

B(x,r)u(y)dV (y), (1.2.1)

for all balls B(x,r)⊂Ω.

The mean value formula is actually a special case of the more general the-orem.

Theorem 3 (The Poisson integral formula). Suppose that u ∈ C(B(x0,r)

)is

harmonic in B(x0,r), and let βn denote the volume of the unit ball. Then

u(x) =r2−|x− x0|2

nβnr

ˆ∂B(x0,r)

u(y)|x− y|n

dσ(y), ∀x ∈ B(x0,r). (1.2.2)

Furthermore, given a function u∈C(∂B(x0,r)

), the function defined by (1.2.2)

is harmonic and

limy→x

u(y) = u(x), ∀x ∈ ∂B(x0,r).

Proof. We prove the theorem in the case n ≥ 3. The case n = 2 is provedsimilarly. It is enough to prove the theorem with r = 1 and x0 = 0. The fulltheorem then follows from a change of variables.

We start with assuming that u is harmonic, and want to prove the formula(1.2.2). For this, let

G(x,y) =1

nβn

1

|y− x|n−2 −1∣∣∣|x|y− x|x|

∣∣∣n−2

. (1.2.3)

Note that ∣∣∣∣|x|y− x|x|

∣∣∣∣= |x− y| , for |y|= 1, (1.2.4)

so G(x,y) = 0 for |y| = 1. Furthermore, straightforward differentiation of(1.2.3) shows that

1−|x|2

nβn

ˆ∂B(0,1)

u(y)|x− y|n

dσ(y) =ˆ

∂B(0,1)u(y)

∂

∂νG(x,y)dσ(y).

12

Letting ∆y denote the Laplace operator with respect to y, it follows fromGreen’s theorem and the fact that ∆u = 0 that for each ε > 0,ˆ

∂B(0,1)u(y)

∂

∂νG(x,y)dσ(y) =

ˆB(0,1)\B(x,ε)

u(y)∆yG(x,y)dV (y) (1.2.5)

+

ˆ∂B(0,1)

G(x,y)∂

∂νu(y)dσ(y) (1.2.6)

+

ˆ∂B(x,ε)

G(x,y)∂

∂νu(y)dσ(y) (1.2.7)

−ˆ

∂B(x,ε)u(y)

∂

∂νG(x,y)dσ(y). (1.2.8)

As was noted above, it follows from (1.2.4) that (1.2.6) is zero. By differen-tiating, we see that ∆yG(x,y) = 0 for x 6= y, so the term (1.2.5) also vanishes.Letting ε→ 0, it follows from elementary estimates that the term (1.2.7) tendsto zero and that the term (1.2.8) tends to u(x).

To show that the function defined by (1.2.2) is harmonic, one only has todifferentiate under the integral sign. The statement about the boundary valuesof the function follows from elementary estimates.

Proof of Theorem 2. If u is harmonic, it follows from the Poisson formula andthe coarea formula thatˆ

B(x0,r)u(y)dV (y) =

ˆ r

0

ˆ∂B(x0,ρ)

u(y)dσ(y)dρ = βnrnu(x0). (1.2.9)

For the reverse implication, assume that B(x0,r)⊂Ω and let v be the function

v(x) =r2−|x− x0|2

nβnr

ˆ∂B(x0,r)

u(y)|x− y|n

dσ(y).

Since v is harmonic, it suffices to prove that u = v. To prove that v≥ u, assumethat v(x)−u(x)< 0 is a minimum of the function v−u. Since v is harmonic

v(x)−u(x) =

B(x,ρ)v(y)−u(y)dV (y),

for all ρ such that B(x,ρ) ⊂ B(x0,r). Since v− u is continuous and vanisheson the boundary, this is a contradiction.

A similar argument shows that u≥ v.

Another important corollary is the following.

Theorem 4. Harmonic functions are smooth.

Proof. Differentiate under the integral sign in the Poisson integral formula.

13

The intuition that a harmonic function describes an equilibrium is also sup-ported by the following theorem, known as the Harnack theorem.

Theorem 5 (The Harnack theorem). Suppose that u is a non-negative har-monic function in the ball B

(x0,(1+ ε)r

), for some ε > 0. Then there is a

constant C =C(n,ε) such that

supx∈B(x0,r)

u(x)≤C infx∈B(x0,r)

u(x).

Proof. Suppose that x,x′ ∈ B(x0,r). If |x− x′| ≤ εr/4, it follows that

u(x) =4n

βn(εr)n

ˆ

B(x,εr/4)

u(y)dV (y)≤ 4n

βn(εr)n

ˆ

B(x′,εr)

u(y)dV (y) = 4nu(x′).

Iterating this estimate in a chain of pairwise intersecting balls of radius εr/4,we arrive at the estimate

u(x)≤ 44n/εu(x′),

for any x,x′ ∈ B(x0,r).

It is important to note that whereas the mean value formula characterisesharmonic functions, versions of the Harnack theorem hold for a wide variety ofpartial differential equations. This makes the Harnack theorem one of the mostfundamental tools in the study of partial differential equations. As an exampleof the strength of this theorem we use it to prove two important theorems forharmonic functions.

Theorem 6 (The maximum principle). Suppose that u is harmonic in Ω. If itattains its maximum in an interior point, it is constant.

Proof. Suppose that x0 ∈Ω is such that u(x0) = supΩ u and let r = d(x0,∂Ω).Applying the Harnack theorem on the non-negative harmonic function v(x) =u(x0)−u(x) we can deduce that u(x) = u(x0) in the ball B(x0,r/2). Using thesame argument on points y ∈ ∂B(x0,r/2) it now follows from induction that uis constant throughout Ω.

Theorem 7. Suppose that u j is a decreasing sequence of harmonic functionson B(0,4r). Then the following statements are equivalent:

i) There exists a point x ∈ B(0,r) such that lim j u j(x)>−∞;ii) the functions u j converge uniformly to a harmonic function on B(0,r).

Proof. It follows from Harnack’s theorem that there is a constant C such thatwhenever j < k:

u j(y)−uk(y)≤C(u j(x)−uk(x)

),

14

for all y ∈ B(0,r). Since the right-hand side of the above inequality tends tozero as j,k→ ∞, it follows that u j is a uniform Cauchy sequence on B(0,r).This shows that u j uniformly converges to a continuous function u. It fol-lows from Theorem 3 that u is harmonic.

1.3 Subharmonic Functions and the Dirichlet ProblemWhen studying the boundary behaviour of harmonic functions, one of the firstproblems one might consider is the Dirichlet problem, that is, the problem offinding a harmonic function with prescribed boundary values. Formally, thiscan be stated as follows.

Definition 8. Suppose that f ∈ C(∂Ω) is continuous. To solve the Dirichletproblem is to find a harmonic function u in Ω such that

limy→x

u(y) = f (x), ∀x ∈ ∂Ω.

It is important to note that it follows from the maximum principle (Theorem6) that if a solution to the Dirichlet problem exists, it is unique.

The attentive reader, may have noticed that we already studied the Dirichletproblem in Section 1.2, namely in Theorem 3. That is, the solution to theDirichlet problem on B(x0,r) is given by

u(x) =r2−|x− x0|2

nβnr

ˆ∂B(x0,r)

f (y)|x− y|n

dσ(y).

This formula relies directly on the symmetric structure of balls. For domainswith more complex structure, more work has to be done to establish the exis-tence of a solution of the Dirichlet problem. One of the most powerful meth-ods to do this is Perron’s method, which localises the Dirichlet problem to thegeometry of the boundary. It relies on the notion of sub- and superharmonicfunctions.

Definition 9. An upper semicontinous function u on Ω is called subharmonicif it satisfies the following property:

Whenever B(x,r)⊂Ω and h∈C(B(x,r)

)is harmonic in B(x,r) and satisfies

u≤ h on ∂B(x,r), then it follows that u≤ h on B(x,r).A function v if called superharmonic if −v is subharmonic.

It can be proved that an upper semicontinuous function u (not identically−∞) is subharmonic if and only if it satisfies ∆u ≥ 0 in the sense of distribu-tions, see for example [Hör07, p. 148].

Since harmonic functions can be thought of as models of equilibria,(smooth) subharmonic functions can be thought of as modelling the density

15

of a quantity tending to equilibrium from below, for example temperature un-der heating. This is made more clear from the following analogue of Theorem2.

Theorem 10. Suppose that u is an upper semicontinuous function in Ω. Thenu is subharmonic if and only if

u(x)≤

B(x,r)u(y)dV (y),

for all B(x,r)⊂Ω.

Proof. This can be proved using the same argument as in the proof of Theorem2.

The connection between subharmonic functions and the solution of theDirichlet problem is given by the following theorem.

Theorem 11. A necessary and sufficient condition for the Dirichlet problemon Ω to be solvable is that there for every point x0 ∈ ∂Ω exists a r > 0 and anegative subharmonic function ψ on Ω such that

limx→x0

ψ(x) = 0, (1.3.1)

but

limx→y

ψ(x)< 0, (1.3.2)

for any other point y ∈ ∂Ω.

A function like the function ψ in the theorem is called a barrier at x0. It iseasy to show that Ω has a barrier at x0 ∈ ∂Ω if, and only if, Ω∩B(x0,r) hasa barrier at x0. A more deep result, known as Bouligand’s lemma, is that thecondition (1.3.2) is superfluous.

For the proof of the theorem we need the following important property ofsuprema of subharmonic functions.

Lemma 12. Suppose that A is a set of subharmonic functions on Ω. If thefunction

u(x) = supϕ(x) : ϕ ∈ A,is locally bounded from above, then the function

u∗(x) = limsupy→x

u(y)

is subharmonic.

16

Proof. If ϕ ∈ A and B(x,r)⊂⊂Ω, it is clear that

ϕ(x)≤

B(x,r)u∗(y)dV (y).

Taking the supremum of the left-hand side we get that

u(x)≤

B(x,r)u∗(y)dV (y).

Now pick x j → x such that lim j→∞ u(x j) = u∗(x). It is clear that for j largeenough, B(x j,r)⊂⊂Ω. Hence, by Fatou’s lemma,

u∗(x)≤ limj→∞

B(x j,r)

u∗(y)dV (y) (1.3.3)

≤

B(x0,r)u∗(y)dV (y). (1.3.4)

It now follows from Theorem 10 that u∗ is subharmonic.

Proof of Theorem 11. If the Dirichlet problem can be solved, a barrier canbe constructed by solving the Dirichlet problem with boundary data f (x) =−|x− x0|2.

For the reverse implication, construct the function

u(x) = sup

ϕ(x) : ϕ is subharmonic in Ω,

limsupΩ3y→x0

ϕ(y)≤ f (x0) for all x0 ∈ ∂Ω.. (1.3.5)

Since it follows from Lemma 12 that the upper semicontinuous regularizationu∗ is subharmonic, it follows from the construction of u that u ≤ u∗ ≤ u, andconsequently, u is subharmonic.

For a ball B(x,r)⊂⊂Ω, let

uB(x) =

r2−|x−x|2

nβnr

ˆ∂B(x,r)

u(y)|x− y|n

dσ(y), if x ∈ B(x,r);

u(x) , if x ∈Ω\B(x,r).(1.3.6)

The function uB is subharmonic, and u≤ uB. But because of the constructionof u, this means that u = uB, and thus u is harmonic in B(x,r). Since B(x,r)was arbitrary, this means that u is harmonic.

It remains to show that liminfΩ3x→x0 u(x) ≥ f (x0). To this end, considerfor ε > 0 the function

v(x) = f (x0)+Cψ(x)− ε. (1.3.7)

17

It is clearly subharmonic and by choosing C > 0 large enough, it follows from(1.3.2) that v≤ u. Since

liminfΩ3x→x0

v(x) = f (x0)− ε,

we are finished.

A domain Ω on which the Dirichlet problem can be solved, is called regular.The next theorem exemplifies how barriers can be used to exploit geometricinformation of the boundary to deduce the regularity of the domain.

Theorem 13. Suppose that Ω satisfies an outer ball condition, that is, thatthere for each x0 ∈ ∂Ω exists a ball B(x0,r) such that B(x0,r)∩ Ω = x0.Then Ω is regular.

Note that this implies that all smooth domains are regular.

Proof. The function ψ defined by

ψ(x) =

log r

|x−x0|, if n = 2;

r2−n−|x− x0|2−n , if n≥ 3;

is a barrier at x0.

Questions related to the regularity of domains are studied in Paper III andPaper VI.

We end this theorem with a theorem connecting the study of boundary be-haviour of harmonic functions to measure theory.

Theorem 14. Suppose that Ω is bounded and regular. For each x ∈ Ω, thereexists a unique regular Borel measure dωx = dωΩ

x such that for each f ∈C(∂Ω):

u(x) =ˆ

∂Ω

f (y)dωx(y),

where u is the solution to the Dirichlet problem on Ω with boundary values f .The measure dωx is called the harmonic measure (with respect to Ω).

Proof. Let u f denote the solution to the Dirichlet problem with boundary dataf . For each x∈Ω, define the linear functional f 7→ u f (x), acting on C(∂Ω). Bythe maximum principle, this functional is bounded, and hence the conclusionof the theorem follows from Riesz’s representation theorem.

Note that for a fixed Borel set E ⊂ ∂Ω, the harmonic measure ωx(E) is aharmonic function of the variable x. Formally, it can be regarded as the solu-tion of a generalised Dirichlet problem with boundary data χE , the indicator

18

function of E. This can be formalised by approximating χE with continuousfunctions.

It is also interesting to note that Theorem 3 gives an explicit expression forωΩ

x in case Ω is a ball. In most cases, no such explicit formula can be known.

1.4 The Boundary Harnack PrincipleIn the previous section, we studied the existence of a harmonic function uwith continuous boundary values f . This means that when approaching theboundary, u eventually becomes close to f . One might now wonder if twoharmonic functions sharing the same boundary values, tend to the boundaryvalue at the same rate when approaching the boundary. Intuitively, we expectthis to be true, but it is important to realise that a priori, we can only concludethat the difference of the solutions is close to zero. The relevant “dimensionfree” statement is however that the quotient of the solutions stays boundedwhen approaching the boundary. Question of this type is the theme of thecurrent section.

All studies of the boundary behaviour of functions is intimately connectedto the geometry of the boundary in question, but to emphasise the ideas, wehere consider the most elementary situation: a flat boundary. We use the fol-lowing notation

B+r =

(x1, . . . ,xn) ∈ B(0,r) : xn > 0

,

S+r =(x1, . . . ,xn) ∈ ∂B(0,r) : xn > 0

,

Σr = B(0,r)∩(x1, . . . ,xn) : xn = 0,

and we define the following interior reference points:

Ar = (0, . . . ,0,r) ∈ Rn. (1.4.1)

A theorem of the following type is called a Carleson estimate. It is often thestarting point for quantitative studies of boundary behaviour. See Figure 1.1for an illustration of the geometric setting.

Theorem 15. Suppose that u is a positive harmonic function in B+r , vanishing

continuously on Σr. Then there is a constant C =C(n) such that

supB+

ρ

u≤Cu(Aρ), ∀ρ <r2.

Proof. Extend u harmonically to Br using Schwarz reflection, that is u(x) =−u(x) for xn ≤ 0, where x denotes the reflection

(x1, . . . ,xn) 7→ (x1, . . . ,−xn).

19

Aρ

r ρ

B+r

B+ρ

Figure 1.1. The geometric setting of Theorem 15.

Now for x ∈ B+ρ , it follows from Theorem 3 that

u(x) =4ρ2−|x|2

2nβnρ

ˆ∂B(0,2ρ)

u(y)|x− y|n

dσ(y) (1.4.2)

≤ 4n 43

3ρ2

2nβnρ

ˆ∂B(0,2ρ)

u(y)∣∣Aρ − y∣∣n dσ(y) =Cu(Aρ). (1.4.3)

It is important to note that the constant C is universal and scale-invariant(that is, the estimate holds with the same C for all functions satisfying theassumptions of the theorem, and the constant is independent of r).

In the above proof we used the explicit formula for the harmonic measureon the ball (the Poisson integral formula). When studying domains with lesssymmetry, or when studying more general PDEs, such explicit formulae donot exist. Instead, one has to make do with estimates of the harmonic measure.The following is an example of the most fundamental of such estimates.

Theorem 16 (Doubling property). Let ωx denote the harmonic measure withrespect to B+

r . There is a constant C =C(n) such that when x ∈ B+r/2,

ωx(B(Ar,2ρ)∩S+r

)≤Cωx

(B(Ar,ρ)∩S+r

), ∀ρ > 0.

20

Proof. Using Schwarz reflection and the Poisson integral as in the proof ofTheorem 15, we see that there exists a constant C =C(n) such that

ωx(B(Ar,2ρ)∩S+r

)=

r2−|x−Ar|2

nβnr

(ˆB(Ar,2ρ)∩S+r

1|x− y|n

dσ(y)

)

≤Cr2−|x−Ar|2

nβnr

(ˆB(Ar,ρ)∩S+r

1|x− y|n

dσ(y)

)=Cωx

(B(Ar,ρ)∩S+r

).

The next theorem shows an application of the doubling property of the har-monic measure.

Theorem 17. Suppose that u and v are positive harmonic functions in B+r ,

vanishing continuously on Σr. Then there is a constant C =C(n) such that thefollowing holds for all ρ ≤ r/2:

1C

u(Aρ)

v(Aρ)≤ u(x)

v(x)≤C

u(Aρ)

v(Aρ), ∀x ∈ B+

ρ/2.

Proof. It follows from Theorem 15 that there exists a constant C1 such that

u(x)≤C1u(Aρ

), ∀x ∈ B+

ρ , (1.4.4)

and it follows from the Harnack inequality that there exists a constant C2 suchthat

v(Aρ)≤C2v(x), ∀x ∈ B(Aρ ,ρ/2

). (1.4.5)

Letting ωx denote the harmonic measure with respect to B+ρ , it now follows

from the maximum principle that

u(x)≤C1u(Aρ

)ωx(S+ρ ), ∀x ∈ B+

ρ , (1.4.6)

and

v(x)≥ 1C2

v(Aρ)ωx

(S+ρ ∩B

(Aρ ,ρ/2

)), ∀x ∈ B+

ρ . (1.4.7)

The right inequality now follows from (1.4.6), (1.4.7) and Theorem 16. Theleft inequality is proved analogously.

By a clever application of the previous theorem, we can show that not onlyis the quotient u/v bounded, it is Hölder continuous. A theorem of this type issometimes called a boundary Harnack estimate.

21

Theorem 18. Suppose that u and v are positive harmonic functions in B+r ,

vanishing continuously on Σr. Then there are constants C = C(n) and α =α(n) such that ∣∣∣∣u(x)v(x)

− u(y)v(y)

∣∣∣∣≤C |x− y|α , ∀x,y ∈ B+r/2.

Proof. Since u and v are smooth and nonzero in B+r , we only have to consider

y ∈ Σr/2. The theorem will follow by induction if we can prove that

oscB(y,ρ/2)

uv≤ θ osc

B(y,ρ)

uv, (1.4.8)

for some θ < 1 and ρ < r/2. Letting

w1(x) = u(x)− v(x) infB(y,ρ)

(u/v),

w2(x) = v(x) supB(y,ρ)

(u/v)−u(x),

we see that w1(x)+w2(x) = v(x)oscB(y,ρ)(u/v), and since both w1 and w2 arepositive, at least one of w1 or w2 is greater than 1

2 v(x)oscB(y,ρ)(u/v). Let usassume it is w1; the case when it is w2 is analysed analogously. By Theorem17 and this assumption on w1, there is a constant C such that

infB(y,ρ/2)

w1

v≥ 1

Cw1(Aρ)

v(Aρ)≥ 1

2Cosc

B(y,ρ)

uv, (1.4.9)

so by the construction of w1,

oscB(y,ρ/2)

uv= osc

B(y,ρ/2)

w1

v≤ sup

B(y,ρ)

w1

v− 1

2Cosc

B(y,ρ)

uv

(1.4.10)

=

(1− 1

2C

)osc

B(y,ρ)

uv. (1.4.11)

Theorems of the type considered in this section are studied in Paper I andPaper II.

1.5 Non-tangentially Accessible DomainsThe results of the previous section hold true in much more general domains,with boundaries far from flat. In this section, we describe a wide class ofdomains which is tailor-made for quantitative boundary estimates, namely the

22

Ω

r

x0

r/M

Ar(x0)

Figure 1.2. An interior corkscrew point

class of non-tangentially accessible (NTA) domains due to Jerison and Kenig[JK82]. We also give an example of how one goes about making boundaryestimates in an NTA domain.

Fundamental for the definition of an NTA domain is the notion of a non-tangential ball. A ball B(x,r)⊂Ω is called M-non-tangential (with respect toΩ) if

rM≤ d(B(x,r),∂Ω

)≤Mr.

A collection B jkj=1 of M-nontangential balls with the property that x ∈ B1

and y ∈ Bk and B j ∩B j+1 6= /0, for j ∈ 1, . . . ,k− 1, is called an M-Harnackchain of length k from x to y. It follows from the Harnack theorem (Theorem5) that if u is a positive harmonic function in Ω, and there exists an M-Harnackchain of length k from x to y, then

u(x)≤Cku(y),

where C = C(n,M) is the constant appearing in the statement of the Harnacktheorem. We are now ready to state the definition of an NTA domain.

Definition 19. A bounded domain Ω is called non-tangentially accessible(NTA) if there exist M ≥ 2 and r0 > 0 such that the following conditions aresatisfied:

23

Figure 1.3. The boundaries of the above domains are quasicircles, and hence thedomains are NTA.

i) corkscrew condition: for any x0 ∈ ∂Ω, and r : 0 < r < r0, there existsAr(x0) ∈Ω satisfying

rM

< |Ar(x0)− x0|< r and d(Ar(x0),∂Ω)>rM,

ii) Rn \Ω satisfies the corkscrew condition,iii) Harnack chain condition: if x1,x2 ∈ Ω, and d(x j,∂Ω) ≥ ε > 0 and|x1− x2| ≤ Cε for some C, then there exists an M-Harnack chain oflength k from x1 to x2, where k may depend on C, but not on ε .

This definition merits some comments. First, we note that the point Ar(x0)of condition i) takes on the role of the reference point Ar in (1.4.1); see Figure1.2 for an illustration. Secondly, condition ii) assures us that Ω is regular (seeTheorem 23 below). Finally, the discussion preceding the definition shows thatcondition iii) implies that the values of a positive harmonic function at pointsnear the boundary can be scale-invariantly compared. We will later give anexample of what this means, but first we want to gain some insight into whichdomains are NTA, and which are not.

The model example of an NTA domain is a Lipschitz domain, and it is fairlystraightforward to show that this is the case. More generally, Jerison and Kenig[JK82] have showed that any domain whose boundary is of the Zygmund-Λ∗class (see [Zyg02] for the definition of this class) is NTA. As functions inthe Zygmund-Λ∗ class can be nowhere differentiable, this means that NTAdomains can have very rough boundaries. Jerison and Kenig have also shownthat domains in R2 bounded by quasicircles are NTA [JK82, Theorem 2.7].This means that for example the domain bounded by the closed von Kochcurve in R2, and domains bounded by certain Julia sets in R2 are NTA, seeFigure 1.3 and [CG93].

Figure 1.4 shows schematic examples of domains that are not NTA. Theleftmost domain fails to satisfy condition i), whereas the domain in the middlefails to satisfy condition ii). The rightmost domain fails to satisfy conditioniii); note however that the solid body in R3 resulting from rotating the domainin space does satisfy condition iii).

24

Figure 1.4. Examples of domains that are not NTA

To illustrate how the conditions of the definition of an NTA domain areused to prove boundary estimates, we prove the following weak version of theCarleson estimate.

Lemma 20. Suppose that Ω is an NTA domain with constants r0 and M andthat u is positive and harmonic in Ω. Then there are constants C = C(n,M)and γ = γ(n,M) such that if x0 ∈ ∂Ω and r < r0,

u(x)≤C(

rd(x,∂Ω)

)γ

u(Ar(x0)

),

for all x ∈Ω∩B(x0,r).

Proof. Let ρ = d(x,∂Ω), and let y ∈ ∂Ω be a point realising this distance. Bycondition i) of Definition 19, there is point Aρ(y) satisfying

d(Aρ(y),∂Ω)≥ ρ

M,

and ∣∣Aρ(y)− x∣∣≤ 2ρ.

By condition iii) of Definition 19, there exists an M-Harnack chain of lengthk = k(M) from x to Aρ(y) in Ω. By the discussion preceding Definition 19, itfollows that

u(x)≤Cku(Aρ(y)

),

where C = C(M,n) is the constant appearing in the statement of the Harnacktheorem. As long as 2 jρ < r0, we can iterate this estimate to get

u(x)≤Cku(Aρ(y)

)≤C2ku

(A2ρ(y)

)≤ . . .≤C( j+1)ku

(A2 jρ(y)

).

Now let j be such that 2 jρ ≤ r ≤ 2 j+1ρ . Again using condition iii) of Defini-tion 19 and the Harnack theorem we see that u

(A2 jρ(y)

)≤Cu

(Ar(x0)

). The

conclusion of the theorem now follows from the choice of j.

The geometry of NTA domains plays a central role in Paper I and Paper II.

25

1.6 Approximation of Harmonic FunctionsIn Section 1.4 we saw how powerful it can be to be to extend a harmonicfunction u ∈ C(Ω) across the boundary, ∂Ω. In situations when this methodis unavailable, it can sometimes be enough to approximate u by harmonicfunctions defined in neighbourhoods of Ω. The following theorem gives anexample when this is possible.

Theorem 21. Suppose that there is a δ > 0 such that for x ∈ ∂Ω and r > 0,

∣∣B(x,r)\ Ω∣∣

|B(x,r)|≥ δ .

If u ∈C(Ω) is harmonic in Ω, then u can be uniformly approximated by func-tions harmonic in neighbourhoods of Ω.

To simplify the proof of the theorem we isolate the main idea behind theproof to the following lemma.

Lemma 22. Suppose that Ω is as in the statement of Theorem 21 and supposethat u is a positive continuous subharmonic function defined in some neigh-bourhood of Ω. Then there is a function v ∈C(Ω) that is harmonic in Ω andcoincides with u on ∂Ω and v can be uniformly approximated by functionsharmonic in neighbourhoods of Ω.

Proof. Let Ω j be a sequence of smooth domains such that Ω j ⊂ Ω j+1 and⋂∞j=0 Ω j = Ω. Suppose also that u is defined on Ω0. Let

v j(x) = supϕ(x) : ϕ is subharmonic in Ω0 and ϕ ≤ u in Ω0 \Ω j.

It then follows by construction that v j(x)≥ v j+1(x)≥ u≥ 0 in Ω0 and v j = u inΩ0 \Ω j. Arguing as in the proof of Theorem 11, we see that v j is harmonic inΩ j. Moreover, since Ω j is regular, v j is continuous and hence subharmonic.Let

v(x) = limj→∞

v j(x).

It follows from Theorem 7 that v is harmonic on Ω, and by construction v≥ uon ∂Ω. It remains to show that v ≤ u on ∂Ω. To do this, we note that itfollows from Theorem 10 that v(x) is subharmonic in Ω0, and by constructionv(x) = u(x) for x ∈Ω0 \ Ω.

26

Since v is positive, it follows from Jensen’s inequality and Theorem 10 thatvk is subharmonic for each k ∈ N. For x ∈ ∂Ω it thus follows that

v(x)k ≤ 1βnrn

ˆB(x,r)∩Ω

v(y)kdV (y)+1

βnrn

ˆB(x,r)\Ω

v(y)kdV (y)

≤ |B(x,r)∩ Ω||B(x,r)|

supB(x,r)∩Ω

v(y)k +|B(x,r)\ Ω||B(x,r)|

supB(x,r)\Ω

u(y)

≤ (1−δ ) supB(x,r)

v(y)k + supB(x,r)

u(y)k,

whence

v(x)k ≤ (1−δ ) limsupy→x

v(y)k + limsupy→x

u(y)k

= (1−δ )v(x)k +u(x)k.

This means that v(x)≤ δ− 1

k u(x) and letting k→∞ we see that v(x)≤ u(x).

Proof of Theorem 21. By Weierstraß’s theorem, there are smooth functions u jdefined in Rn such that u j → u uniformly on ∂Ω. By picking a large enoughC and writing

u j(x) =(

u j(x)+C(|x|2 +1

))−C(|x|2 +1

),

we see that in any neighbourhood of Ω, u j can be written as the difference oftwo smooth, positive subharmonic functions. By Lemma 22, there is a har-monic function h j ∈C(Ω) coinciding with u j on ∂Ω, and this function h j canbe uniformly approximated by harmonic functions defined in neighbourhoodsof Ω. It follows from Theorem 7 that h j converge uniformly to a harmonicfunction h in Ω. Since h j→ u uniformly on ∂Ω, it follows from the maximumprinciple that h = u on Ω.

Studying the proof of Theorem 21, we realise that we actually also haveproved the following theorem.

Theorem 23. Suppose that Ω is as in Theorem 21. Then Ω is regular.

Proof. Let f be the boundary data of the Dirichlet problem. Proceed as inthe proof of Theorem 21, but let u j approximate f instead of u. The functionh = lim

j→∞h j solves the Dirichlet problem.

Theorems of the type considered in this section are studied in Paper IV,Paper V and Paper VI.

27

1.7 Bibliographical NotesMost of the material in this chapter is classical, and so as to not overly burdenthe reader with bibliographical references, I have decided to be sparing ofsuch. Instead I would like to direct the reader to the sources from which I havelearned — that is, [Eva10] and [GT01] for a PDE viewpoint and [AG01] fora potential theoretic ditto; [Ran95] for a focus on connections to holomorphicfunctions, and [Hör07] for an emphasis on connections to convexity. Finally, Iwould like to recommend Chapter 11 of [CS05] for a good presentation of thetype of problems studied in Section 1.4.

28

Part II:On the Appended Papers

2. Parabolic Partial Differential Equations

The equation

ddt

u =n

∑j=1

∂ 2

∂x2ju, (2.0.1)

is called the heat equation. As is the case with the Laplace equation, the heatequation has numerous applications in both pure and applied mathematics. Atan intuitive level, the heat equation can be thought to model evolution of thedistribution of heat in a homogeneous material, but it can be used to modelother situations where the density of a quantity tends towards equilibrium.

There is an obvious connection between the Laplace equation and the heatequation since a solution to the former equation in Ω ⊂ Rn also solves thelatter in Ω×R. Furthermore, it can be shown (see, for example, [Jos13, Sec-tion 5.2-5.3]) that solutions to the heat equation converge to solutions to theLaplace equation as time tends to infinity. To some extent this might serve asa intuitive explanation to the guiding principle that every theorem about solu-tions to the Laplace equation ought to have a counterpart for solutions to theheat equation. Of course, in the theorems about solutions to the heat equation,the time variable has to be taken into account, and often these theorems aremore complicated both to state and to prove.

In sharp contrast to the Laplace equation, which is symmetric in all coordi-nates, the heat equation has one odd variable, t, which has to be given specialattention. For example, if u(x, t) is a solution to the heat equation, the functionu(−x, t) is also a solution, but the function u(x,−t) is in general not a solution.This can intuitively be understood as if the equation distinguishes between fu-ture and past. Similarly, if r ∈ R, the function u(rx,rt) is in general not asolution, but u(rx,r2t) is. This leads to the definition of a parabolic cylinder,

Cr(x0, t0) = B(x0,r)× (t0− r2, t0 + r2),

which, in the analysis of the heat equation, plays the role that Euclidean ballsplay in the analysis of the Laplace equation. For this reason, the natural metricfor the heat equation is the following

dp((x, t),(y,s)

)=√|x− y|+(t− s)2.

A good reference for the basic properties of the heat equation is the textbookby Evans, see [Eva10]. For a more extensive study of the heat equations andits generalisations, see the book by Lieberman [Lie96].

31

The heat equation is the model equation for a large family of equations ofthe form

Hu = ut −∇ ·A(x, t,∇u) = 0, (2.0.2)

where A : Rn×R×Rn→Rn is a vector field satisfying some regularity condi-tions to be specified below. Such equations are called parabolic second orderPDEs of divergence type. The most well-studied version of H is the case ofa linear A with bounded and measurable coefficients. That is, the case whenA(x, t,∇u) = A(x, t)∇u for some matrix A(x, t) = ai, j(x, t) with measurablecoefficients such that for some β ≥ 1

1β|ξ |2 ≤

n

∑i, j=1

ai, j(x, t)ξiξ j ≤ β |ξ |2 , (2.0.3)

for all ξ ∈ Rn and (x, t) ∈ Rn+1. In this setting, the parabolic versions ofthe problems of Section 1.4 in domains of the type Ω× (0,T ), where Ω isa Lipschitz domain, were proved by Fabes et al. [FGS86], and Fabes andSafonov [FS97]; see also [FSY99] and [SY99] for related results. In Paper Iwe extend these results to non-linear parabolic equations and in Paper II weextend them to a class of linear parabolic equations failing to satisfy (2.0.3).

2.1 Paper IIn Paper I, we study the boundary behaviour of positive weak solutions to(2.0.2) in ΩT = Ω× (0,T ), that vanish continuously on the lateral boundaryST = ∂Ω× [0,T ]. We here assume that Ω is an NTA domain, see Definition19. Regarding the structural assumptions on A, the main contribution of ourpaper, is that we allow A to be non-linear. However, in order for a non-intrinsicHarnack inequality to hold true, we need to assume that the non-linearity is oflinear growth.

What we precisely assume is that A : Rn×R×Rn→ R is measurable, that

ηk 7→ A(x, t,η1, . . . ,ηn)

is continuous, for every k = 1, . . . ,n and for almost all fixed (x, t) ∈ Rn, andthat there exists a constant β ,1≤ β < ∞, such that

(i) |A(x, t,η)| ≤ β |η |,

(ii)(A(x, t,η)−A(x, t,ξ )

)·(η−ξ

)≥ 1

β|η−ξ |2.

(2.1.1)

for (x, t,η) ∈ Rn×R×Rn, ξ ∈ Rn. Under these assumptions, we can proveparabolic counterparts to Theorem 15 and Theorem 16. That is, we prove thefollowing theorems.

32

Theorem 24. Let u be a non-negative solution of Hu = 0 in ΩT vanishingcontinuously on ST . Let 0 < δ < minr0/2,

√T be a fixed constant. Suppose

(x0, t0) ∈ ST such that δ 2 < t0 < T −δ 2 and assume that r < δ/2. Then thereexists c = c(n,β ,M,diam(Ω),T,δ ), 1≤ c < ∞, such that

u(x, t)≤ cu(Ar(x0), t0

)whenever (x, t) ∈ΩT ∩Cr(x0, t0)

Since H is non-linear, there can be no measure generating the solution toHu = 0 from the boundary data, as is the case with the harmonic measure forthe Laplace equation, see Theorem 14. Still, given a solution u as in The-orem 24, one can construct a measure encoding the boundary behaviour ofu. Namely, one extends u as a subsolution in Rn× [0,T ] by putting u ≡ 0on (Rn \Ω)× [0,T ]. It then follows from Riesz’s representation theorem andthe maximum principle that there exists a unique, locally finite, positive Borelmeasure µ on Rn× [0,T ], with support on ∂Ω× [0,T ], such that

¨θdµ =−

t2ˆ

t1

ˆ

G

A(x, t,∇u) ·∇θdxdt +

t2ˆ

t1

ˆ

G

u∂

∂ tθdxdt

−ˆ

G

u(x, t2)θ(x, t2)dx+ˆ

G

u(x, t1)θ(x, t1)dx (2.1.2)

whenever 0≤ t1 < t2 ≤ T , G⊂Ω and θ ∈C∞0 (Rn× [0,T ]).

For this measure µ , we establish the following theorem concerning the dou-bling property.

Theorem 25. Let u be a non-negative solution of Hu = 0 in ΩT that is con-tinuous on the closure of ΩT and vanishes continuously on ST . Let µ be themeasure associated to u as in (2.1.2). Let 0 < δ < minr0/2,

√T be a fixed

constant. Suppose that (x0, t0) ∈ ST is such that δ 2 < t0 < T −δ 2 and assumethat r < δ/2. Then there exists c = c(n,β ,M,diam(Ω),T,δ ), 1≤ c < ∞, suchthat

µ(∆(x0, t0,2r)

)≤ cµ

(∆(x0, t0,r)

),

where ∆(x0, t0,r) = ST ∩Cr(x0, t0)

Since Theorem 24 and Theorem 25 are proved under very weak assump-tions on A, we are forced to work with the few general principles which areknown to hold true.The most important of these are the (non-intrinsic) par-abolic Harnack inequality, the comparison principle, the Caccioppoli energyestimate, and the fact that a solution to the Dirichlet problem with Höldercontinuous boundary data is itself Hölder continuous up to the boundary.

33

For the parabolic analogues of Theorem 17 and Theorem 18, we need toassume extra regularity of the vector field A. We assume that the vector fieldη 7→ A(x, t,η) is continuously differentiable in ηk, for each k = 1, . . . ,n andfor almost all (x, t) ∈ Rn+1, for η ∈ Rn \ 0. Furthermore, we assume thatthere exists a constant β , 1≤ β < ∞, such that

i)∣∣∣∣ ∂Ai

∂η j(x, t,η)

∣∣∣∣≤ β ,

ii)n

∑i, j=1

∂Ai

∂η j(x, t,η)ξiξ j ≥ β

−1|ξ |2, (2.1.3)

iii) A(x, t,λξ ) = λA(x, t,ξ ),

whenever (x, t) ∈Rn×R and ξ ,η ∈Rn \0 and for all λ ≥ 0. A straightfor-ward calculation shows that the conditions i) and ii) of (2.1.3) imply (2.1.1).Furthermore, to make sure that solutions to Hu= 0 have pointwise defined gra-dients, we also make the qualitative assumption that A(x, t,η) is Hölder con-tinuous in (x, t) with respect to the parabolic distance for all fixed η ∈Rn\0.With these assumptions, we can for each pair w1,w2 of solutions to Hu = 0construct a linear uniformly parabolic operator H with bounded and measur-able coefficients such that the difference w1−w2 is a solution to the equationHu = 0. This enables us to use modified results from [FSY99] and [SY99] onlinear parabolic equations on the difference w1−w2. Carefully choosing theright w1 and w2, we manage to prove the following theorems.

Theorem 26. Let u,v be non-negative solutions of Hu = 0 in ΩT vanishingcontinuously on ST and assume in addition that v ≤ u in ΩT . Let 0 < δ <minr0/2,

√T be a fixed constant. Suppose (x0, t0) ∈ ST such that δ 2 <

t0 < T − δ 2 and assume that r < δ/2. Then there exists a constant c =c(n,β ,M,diam(Ω),T,δ ), 1≤ c < ∞, such that

1c

u(Ar(x0), t0

)−v(Ar(x0), t0

)v(Ar(x0), t0

) ≤ u(x, t)−v(x, t)v(x, t)

≤ cu(Ar(x0), t0

)−v(Ar(x0), t0

)v(Ar(x0), t0

) ,

whenever (x, t) ∈ΩT ∩Cr(x0, t0).

Theorem 27. Let u,v be non-negative solutions of Hu = 0 in ΩT vanishingcontinuously on ST . Let 0 < δ < minr0/2,

√T be a fixed constant. Suppose

(x0, t0) ∈ ST such that δ 2 < t0 < T − δ 2 and assume that r < δ/2. Then u/vis Hölder continuous on the closure of Ω× (δ 2,T ]. Furthermore, there existc = c(n,β ,M,r0,diam(Ω),T,δ ), 1≤ c < ∞, and α = α(H,M,diam(Ω),T,δ ),α ∈ (0,1), such that∣∣∣∣u(x, t)v(x, t)

− u(y,s)v(y,s)

∣∣∣∣≤ c(

dp((x, t),(y,s)

)r

)α u(Ar(x0, t0)

)v(Ar(x0, t0)

)whenever (x, t),(y,s) ∈ΩT ∩Cr(x0, t0).

34

2.2 Paper IIAs in Paper I, we study the boundary behaviour of weak solutions to (2.0.2) inthe domain ΩT = Ω× (0,T ), where Ω is an NTA domain, but this time for adifferent vector field A. We assume that the vector field A is of the form

A(x, t,η) = A(x, t)η ,

for some symmetric n×n matrix A(x, t) = ai, j(x, t). This means that the op-erator H under study is linear, which simplifies some arguments significantly.To compensate for this, we allow the ellipticity of A to degenerate. More pre-cisely, we assume that there exists a real valued function λ : Ω→ R in theMuckenhoupt A1+2/n class such that for some β ≥ 1

1β

λ (x) |ξ |2 ≤n

∑i, j=1

ai, j(x, t)ξiξ j ≤ βλ (x) |ξ |2 ,

for all (x, t) ∈ Rn+1 and ξ ∈ Rn. Remember that a function λ is in the A1+2/nclass if there exists a C such that(

B(x,r)λ (y)dV (y)

)( B(x,r)

λ (y)−n/2 dV (y)

)2/n

≤C, (2.2.1)

for each x ∈ Rn and r > 0. Note that such functions can both vanish and blowup, but only on sets of small measure. We will denote by Λ the smallest Csuch that (2.2.1) holds true.

This condition on the ellipticity might seem rather artificial, but in the eight-ies, a theory of elliptic equations with this kind of ellipticity condition was de-veloped by Fabes, Jerison, Kenig and Serapioni, see [FKJ83, FJK82, FKS82],and in recent years, it has found many applications. For example, Caffarelli,Salsa, and Silvestre [CS07, CSS08, Sil07] used them to study non-local op-erators, and Lewis and Nyström used them to study operators of p-Laplacetype, see [LN07, LN08a, LN08b, LN10a, LN10b, LN12a, LN12b]; see also[LLN08].

The interior theory for solutions to linear degenerate parabolic equationsof this type was developed by Chiarenza, Frasca and Serapioni [CF84, CF85,CS84a, CS84b, CS85, CS87]. They have shown that the classical parabolicHarnack inequality can not hold for degenerate parabolic equations of the typewe are studying. Since our approach to boundary regularity relies heavily onthis inequality, this introduces major obstacles. Our solution to this problemis to replace the standard Euclidean metric on Rn with a quasi-semi-metric dλ

adapted to the weight λ . By a theorem of Chiarenza and Serapioni [CS84b], aversion of the parabolic Harnack inequality holds in parabolic cylinders withrespect to the weighted metric. Using properties of Muckenhoupt weights, wethen manage to show that (standard) NTA domains also satisfies an NTA con-dition with respect to the weighted metric. In some sense, this means that we

35

can hide most of the problems associated with the degeneration of the equa-tion in the geometry, and proceed with the analysis almost as if the equationwas non-degenerate.

To describe the main results of the paper, we need some notation related tothe weighted geometry. Let

rx(R) =

(ˆB(x,R)

λ−n/2(ξ )dξ

)1/n

,

and define the following modified versions of parabolic cylinders and para-bolic boundary cylinders by

C∗R(x, t) = B(x,R)×(t− rx(R)2, t + rx(R)2),

∆∗R(x, t) =

(∂Ω×R

)∩C∗r (x, t).

Finally, let the diameter of Ω with respect to dλ be defined by

diamλ (Ω) = suprx(|x− y|) : x,y ∈Ω.

The main theorems of the paper are the following parabolic counterparts tothe theorems of Section 1.4.

Theorem 28. Let u be a non-negative solution of Hu = 0 in ΩT vanish-ing continuously on ST . There is a constant r0 = r0

(n,β ,Λ,M,r0,diam(Ω),

diamλ (Ω))

such that the following holds. If δ , 0 < δ < r0/2, is a fixed con-stant, (x0, t0) ∈ ST , δ 2 ≤ t0, and R satisfies rx0(R) < δ/2, then there existsc = c(n,β ,Λ,M,diam(Ω),T,δ ), 1≤ c < ∞ such that

u(x, t)≤ cu(AR(x0, t0)

),

whenever (x, t) ∈ΩT ∩C∗R(x0, t0).

Lemma 29. Let ∂pΩ = ST⋃

Ω×0. For each f ∈ C(∂pΩT ) there exists aunique weak solution u ∈C(ΩT ) to the problem

Hu = 0, in ΩT , u = f on ∂pΩT . (2.2.2)

As in the case the case with the Laplace equation, it now follows fromRiesz’s representation theorem and the maximum principle that there, for each(x, t) ∈ ΩT , exists a unique Borel measure ω(x, t) such that any solution u to(2.2.2) has the representation

u(x, t) =ˆ

∂pΩT

f (y,s)dω(x, t,y,s).

The measure ω is called the parabolic measure (associated to H). The follow-ing theorem shows that it has the doubling property.

36

Theorem 30. There is a r0 = r0(n,β ,Λ,M,r0,diam(Ω),diamλ (Ω),λ ,n),0 < r0 < r0, such that the following is true. Let 0 < δ < r0 be a fixed constant.Let (x0, t0) ∈ ST be such that 16δ 2 ≤ t0 ≤ T − δ 2, and suppose that rx0(R) <δ/2. Then there exists a constant c = c(n,β ,Λ,M,diam(Ω),T,δ ), 1≤ c≤ ∞,such that

ω(x, t,∆∗2R(x0, t0)

)≤ cω

(x, t,∆∗R(x0, t0)

),

whenever (x, t) ∈ΩT is such that t ≥ t0 +16(rx0(R)

)2.

Finally we also show the following theorem on the quotient of solutionsvanishing on the lateral boundary.

Theorem 31. There exists an r0 = r0(n,β ,Λ,M,r0,diam(Ω),diamλ (Ω)), 0 <r0 < r0, such that the following is true. Suppose that u and v are non-negativesolutions of Hu = 0 in ΩT , vanishing continuously on ST , and let δ , 0 <δ < r0, be a fixed constant. Then u/v is Hölder continuous on the closureof Ω× (δ 2,T ]. Furthermore, if (x0, t0) ∈ ST , δ 2 < t0 and rx0(R) < δ/2, thenthere exist constants c = c(n,β ,Λ,M,diam(Ω),T,δ ), 1 ≤ c < ∞, and α =α(n,β ,Λ,M,diam(Ω),T,δ ), α ∈ (0,1), such that∣∣∣∣u(x, t)v(x, t)

− u(y,s)v(y,s)

∣∣∣∣≤ c(

rx(|x− y|)+ |t− s|1/2

rx0(R)

)α u(AR(x0, t0)

)v(AR(x0, t0)

) ,whenever (x, t),(y,s) ∈ΩT ∩C∗R/c(x0, t0).

37

3. Several Complex Variables

One important source of interest for harmonic functions is their intimate rela-tionship with holomorphic functions in the complex plane. To be more pre-cise, in the plane R2 ∼= C, a function is harmonic if, and only if, it locallyis the real part of a holomorphic function. This means that many theoremsabout holomorphic functions can be proved using theorems about harmonicfunctions and vice versa. An example of this interplay is Riemann’s proof ofhis famous mapping theorem, which made crucial use of the solution to theDirichlet problem for harmonic functions (see [Gam01, pp. 406–407] for adiscussion of the method and [Rem98, pp 181–186] for a historical account ofthe proof).

In higher dimensions, this intimate connection is no longer present, as canbe seen from the fact that harmonic functions are not invariant under biholo-morphic mappings. Instead, one is lead to consider the following classes offunctions.

Definition 32. An upper semicontinuous function u is called plurisubhar-monic if for all a,b ∈ Cn, the function

ζ → u(a+ζ b

)is subharmonic (as a function in R2 ∼= C) in its domain of definition. If both uand −u are plurisubharmonic, u is called pluriharmonic.

If n = 1, plurisubharmonic means the same as subharmonic but when n≥ 2,the plurisubharmonic functions form a proper subclass of the subharmonicfunctions. It is a fairly straightforward exercise to show that a function on Ω ispluriharmonic if and only if it locally is the real part of a holomorphic function.A good general reference for plurisubharmonic functions is the book [Kli91]by Klimek. A general reference for the theory of multi-variable holomorphicfunctions is the book [Kra01] by Krantz.

3.1 Paper IIIIn the function theory of several complex variables, a central role is played bythe pseudoconvex domains.

38

Definition 33. A domain Ω⊂Cn is called pseudoconvex if there exists a pluri-subharmonic function ψ such that

z ∈Ω : ψ(z)<C ⊂⊂Ω, (3.1.1)

for every C ∈ R. Such a function is called a (pseudoconvex) exhaustion func-tion of Ω.

Example 34. Suppose that Ω ⊂ C. Then Ω is pseudoconvex. Indeed, it fol-lows from Lemma 12 that the function

ψ(z) = |z|− logd(z,∂Ω),

is subharmonic, and it is immediate that ψ satisfies (3.1.1).

In higher dimensions, the (real) geometric information gained from theidentification Cn ∼= R2n is no longer enough to recognise a pseudoconvex do-main. There are, for example, easy to construct pseudoconvex domains whichbecome non-pseudoconvex after a rigid rotation (in R2n), see [Kra01, p. 133].

In many ways, pseudoconvex domains are the natural domains on whichto study holomorphic functions, as these sets are so-called domains of holo-morphy. Thats is, a domain Ω is pseudoconvex if, and only if, there exists aholomorphic function on Ω that cannot be holomorphically extended to anyneighbourhood of any boundary point of Ω. This is the content of the famousLevi problem, formulated by Levi 1919 and finally solved by Oka, Bremer-mann and Norguet in the early 1950’s.

Related to the notion of pseudoconvexity is that of hyperconvexity. It wasintroduced by Stehlé [Ste74] to study a sort of generalised Levi problem calledthe Serre conjecture [Ser53] but it soon became evident that hyperconvex do-mains are natural for studying many different problems in complex analysis.See for example [Bło06] for applications to Bergman kernel, [Ceg04] for ap-plications to the complex Monge-Ampère equation, and [PS08] for applica-tions to Hardy and Bergman spaces.

Definition 35. A domain Ω⊂ Cn is called hyperconvex if there for each z0 ∈∂Ω exists a negative plurisubharmonic function ψ on Ω such that

limz→z0

ψ(z) = 0.

Such a function ψ is called a weak plurisubharmonic barrier at z0.

Since plurisubharmonic functions are subharmonic, it follows from thecomment following Theorem 11 that every hyperconvex domain is regular.It can also easily be shown that each hyperconvex domain is pseudoconvex,see for example [Bło96, Theorem 1.6]. In some sense, one can say that hyper-convexity is an amalgam of real and complex geometry.

39

Although pseudoconvex domains and regular domains are well understoodin geometric terms, hyperconvexity is still poorly understood. One could hy-pothesise that domains that are both pseudoconvex and regular would be hy-perconvex, but the following example of Diederich and Fornæss [DF77] showsthat this is not the case.

Example 36. Let T be the so-called Hartogs triangle defined by

T = (z,w) ∈ C2 : |z|< |w|< 1.

Since ϕ : (z,w) 7→ (z/w,w) is a biholomorphic mapping from T to the pseu-doconvex domain = (z,w) ∈C2 : |z|< 1,0 < |w|< 1, we see that T is pseu-doconvex. Since the complement of T is uniformly thick, it follows fromTheorem 23 that T is regular with respect to the Laplace equation.

However, if T were to be hyperconvex, any plurisubharmonic barrier at theorigin would constitute a subharmonic barrier at the origin of the punctureddisk w : (0,w) ∈ T = w ∈ C : 0 < |w| < 1. Since there exists no suchbarrier, T cannot be hyperconvex.

The Hartogs triangle is very irregular at the origin. Indeed, the boundarycannot be written as the graph of a function; it does not even separate space.It is therefore interesting to study the question whether a pseudoconvex do-main with some minimal boundary regularity is hyperconvex. Diederich andFornæss [DF77] showed that when Ω is bounded, pseudoconvex and has C2

boundary, Ω is hyperconvex. Kerzman and Rosay [KR81] showed that thesame is true when Ω is pseudoconvex, bounded and has C1 boundary, and De-mailly [Dem87] showed that it is enough that Ω is pseudoconvex, boundedand has Lipschitz boundary. In Paper III, we carefully re-examine and im-prove Demailly’s proof to extend his results beyond Lipschitz regularity. Themain theorem is the following.

Theorem 37. If Ω⊂ Cn is a bounded pseudoconvex domain with a boundarythat is C0,α for each α < 1, then Ω is hyperconvex.

In qualitative terms, this theorem signifies a major advance compared to De-mailly’s result, since Lipschitz functions are differentiable almost everywhere,whereas the class of functions in the theorem includes nowhere differentiablefunctions. It is also interesting to note that bounded domains with bound-aries that are C0,α for each α < 1 are generally not regular with respect to theLaplace equation. This shows that in this case, the pseudoconvexity imposesextra regularity on the boundary.

The method we use to prove the theorem provides us with specific estimateson the exhaustion function. In order to make these estimate presentable, wefirst need the following technical lemma, which is of some interest in its ownright.

40

Lemma 38. Let Ω ⊂ Rn be a bounded domain with a boundary that is C0,α

for all α < 1 and let w be a vector pointing inwards in the graph drawingdirection. Then there is a number ε0 > 0 and a positive function f (t) ≤ tsatisfying

log(t/ f (t)

)log(1/t)

0 as t 0, (3.1.2)

such that close to the boundary,

d(z,∂Ω)+ f (ε)≤ d(z+ εw,∂Ω), ∀ε : 0 < ε < ε0. (3.1.3)

For ease of notation, we define δ (z) = d(z,∂Ω). In this framework, our fulltheorem states the following.

Theorem 39. Suppose that Ω is a bounded pseudoconvex domain with aboundary that is C0,α for each α < 1, and that f is as in Lemma 38. Thenfor each s ∈ (0,1/2) there exists a strictly plurisubharmonic function ϕ ∈C∞(Ω)∩C(Ω), which satisfies the following bound

−γ

(log(δ (z)/ f (δ (z))

)+1

log(1/δ (z)

) )≤ ϕ(z)≤− (1− s)

log(1/δ (z)

) , (3.1.4)

for δ (z)≤ δ0. Moreover

ddcϕ ≥

log(δ (z)2/ f (δ (z)2)

)log(1/δ (z)2)

ddc |z|2 + 1log(1/δ (z)

) ddc |z|2

for δ (z)< δ0.

3.2 Paper IVIn Paper IV we study the plurisubharmonic counterpart of the type of approxi-mation that appears in Theorem 21. That is, for a bounded domain Ω, we wantto uniformly approximate an arbitrary plurisubharmonic function u ∈ C(Ω)by smooth plurisubharmonic functions defined in neighbourhoods of Ω. Thisproblem was first studied by Sibony [Sib87], who showed that this is alwayspossible if Ω is pseudoconvex and has a smooth boundary. Later, Fornæssand Wiegerinck [FW89] showed that the same is true when Ω only has a C1

boundary, without any assumptions of pseudoconvexity.In the main theorem of the paper, we elaborate on the technique of For-

næss and Wiegerinck and extend their theorem to domains with even rougherboundaries.

41

Theorem 40. Let Ω be a bounded domain in Cn with a boundary that lo-cally is given as the graph of a continuous function. Then every functionu ∈ PSH(Ω)∩C(Ω) can be uniformly approximated on Ω by smooth pluri-subharmonic functions defined in neighbourhoods of Ω.

Keeping track of the estimates in the proof, we are also able to present aquantitative version of the theorem, where the size of the domain of definitionof the approximating function is related to the error of approximation. Werefer to the appended paper for the precise statement of that theorem.

3.3 Paper VIn this paper, we take a more general approach to the kind of approximationsstudied in Paper IV. It is based on the notion of plurisubharmonicity on com-pact sets as defined by Poletsky [Pol96] and further studied by Poletsky andSigurðsson [PS12].

To describe this notion we need som additional definitions. In the following,X will always denote a compact subset of Cn.

Definition 41. We say that u ∈ PSHo(X) if there is some neighbourhood Uof X such that u ∈ PSH(U)∩C(U).

Definition 42. We say that a probability measure µ is a Jensen measure atz ∈ X , and write µ ∈ Jz(X) if

u(z)≤ˆ

udµ, ∀u ∈ PSHo(X).

A Jensen measure is thus a probability measure for which all functions inPSHo(Ω) satisfy the mean value inequality. The idea is now that the Jensenmeasures can be used to define a notion of plurisubharmonicity on compact(possibly very small) sets without complex structure.

Definition 43. An upper semicontinuous function u belongs to PSH(X) if forevery z ∈ X ,

u(z)≤ˆ

udµ, ∀µ ∈ Jz(X).

It is easy to show that most elementary properties of plurisubharmonic (andsubharmonic) functions also hold for functions in PSH(X). The drawback ofthis definition of PSH(X) is however that it is hard to know what a Jensenmeasure looks like or what functions are plurisubharmonic on X . In PaperV, we explain a simplified version of an approximation theorem of Poletsky

42

[Pol96] which shows that any measure µ ∈ Jz(X) can be approximated in theweak* topology by measures µϕ of the form

ˆudµϕ =

12π

ˆ 2π

0u(ϕ(eiθ))dθ ,

where ϕ is a holomorphic mapping from some neighbourhood of the unit diskin C to a small neighbourhood of X . The main theorem of this paper is how-ever the following theorem, which links plurisubharmonicity on compact setsto the type of approximation considered in Paper IV.

Theorem 44. An upper semicontinuous function u on X belongs to PSH(X)if and only if there is a decreasing sequence u j ∈ PSHo(X) such that

limj→∞

u j(z) = u(z).

3.4 Paper VIIn Paper VI, we continue the line of thought from Paper IV and Paper V andtry to solve the problem of characterising those upper semicontinuous func-tions u on Ω that can be pointwise approximated by a decreasing sequence offunctions u j ∈ PSHo(Ω). It turns out that this question is closely related tothe following boundary value problem:

u ∈ PSHo(Ω),u = f on ∂Ω,

(3.4.1)

for a given upper semicontinuous function f on ∂Ω. As expected, this problemcan not be solved in all domains Ω, but some regularity must be assumed. Itis easy to see that unless we are willing to impose very strong restrictions onthe boundary of the domain, we can not expect to be able to solve the problem(3.4.1) for all boundary values. It turns out that there is a middle way to go,where the problem has a neat answer.

Definition 45. We say that Ω is P-hyperconvex, if there for each z0 ∈ ∂Ω existsa function u ∈ PSH(Ω)∩C(Ω) which is negative in Ω and vanishes at z0.

In Paper VI, we show that P-hyperconvex domains share many propertieswith hyperconvex domains, but it is easy to see that there are hyperconvexdomains that are not P-hyperconvex (take for example the unit disk in C mi-nus a closed line segment contained in the disk). In this setting we show thefollowing theorems.

43

Theorem 46. Suppose that Ω is a bounded P-hyperconvex domain and that fis an upper semicontinuous function on ∂Ω. Then the problem (3.4.1) has asolution if and only if f ∈ PSH(∂Ω).

Theorem 47. Suppose that Ω is a bounded P-hyperconvex domain and that uis an upper semicontinuous function on Ω. Then u ∈ PSH(Ω) if and only ifu|∂Ω ∈ PSH(∂Ω).

In the case that each plurisubharmonic function u ∈ C(Ω) also belongs toPSHo(Ω) (which by Theorem 40 is the case when Ω has a C0 boundary),these theorems completely characterise the boundary values of continuousplurisubharmonic functions. The problem of characterising boundary val-ues of plurisubharmonic functions is intimately connected to the study of theDirichlet problem for the complex Monge-Ampère equation, see for example[Åha07, Bło96, Ceg98]. For more results concerning boundary values of con-tinuous plurisubharmonic functions, see for example [ÅC07, Bło00, Sad82,Wik01].

44

Summary in Swedish

Den här avhandlingen handlar om randuppträdandet hos funktioner som haratt göra med partiella differentialekvationer och med komplex analys i fleravariabler. Den består av en sammanfattning och sex vetenskapliga artiklar. Devetenskapliga artiklarna kan delas in i två olika grupper baserat på de ämnende avhandlar. Den ena gruppen studerar randuppträdandet hos icke-negativalösningar till partiella differentialekvationer av parabolisk typ, och den andragruppen studerar randuppträdandet hos plurisubharmoniska funktioner.

I artiklarna I och II, som båda är skrivna i samarbete med Kaj Nyström ochOlow Sande, studeras icke-negativa svaga lösningar till ekvationer av typen

∂

∂ tu−∇ ·A(x, t,∇u) = 0, (3.4.2)

i cylindriska områden ΩT = Ω× (0,T ), där Ω är ett NTA-område (av non-tangentially accessible, på svenska ”icke-tangentiellt åtkomligt”).

I artikeln II antas operatorn A vara icke-linjär, men med linjär tillväxt. Tillhuvudresultaten hör att om u är en icke-negativ lösning till (3.4.2) med konti-nuerligt randvärde noll på den laterala randen ST = ∂Ω× [0,T ], så uppfyllerden en bakåtriktad Harnackolikhet vid laterala randen och det Rieszmått somhör till u har fördubblingsegenskapen. För den andra hälften av artikeln görsytterligare antaganden på operatorn A för att säkerställa att lineariseringen avoperatorn är likformigt elliptisk. Under dessa starkare antaganden visas att omu är som tidigare och v är ytterligare en funktion som uppfyller samma krav,så är kvoten u/v Hölderkontinuerlig ut på den laterala randen ST .

I artikeln II, studeras ekvationer av typen (3.4.2), där operatorn A är påformen A(x, t)∇u(x), för någon symmetrisk (n× n)-matris A = ai, j(x, t).Matrisen A antas inte vara likformigt elliptisk, utan tillåts degenerera. Iställetantas att det finns en funktion λ i Muckenhoupts A1+2/n-klass så at A uppfyllerföljande villkor

1β

λ (x) |ξ |2 ≤n

∑i, j=1

ai, j(x, t)ξiξ j ≤ βλ (x) |ξ |2 ,

för alla (x, t) ∈ Rn+1 och ξ ∈ Rn och för någon konstant β > 0. Det förstahuvudresultatet i artikeln är att om u är en icke-negativ lösning till en sådanekvation och har kontinuerligt randvärde noll på den laterala randen ST , såuppfyller u en bakåtriktad Harnackolikhet vid den laterala randen. Det andrahuvudresultatet är att om u är som tidigare och v är en annan funktion som

45

uppfyller samma antaganden, så är kvoten u/v Hölderkontinuerlig ut på denlaterala randen ST . Slutligen visas att det paraboliska måttet med avseende påekvationen (3.4.2) har fördubblingsegenskapen.

Artiklarna III – VI studerar randuppträdandet för plurisubharmoniska funk-tioner i det komplexa euklidiska rummet Cn.

I artikeln III, som är skriven tillsammans med Benny Avelin och Lisa Hed,studeras frågan om när ett pseudokonvext område Ω är hyperkonvext, det villsäga, har en begränsad plurisubharmonisk uttömningsfunktion. Huvudresul-tatet är att om randen till Ω lokalt ges som grafen till en funktion som är Höl-derkontinuerlig med exponent α för alla α < 1, så är Ω hyperkonvex. Precisaglobala uppskattningar på den bergänsade uttömningsfunktionen ges.

I artikeln IV, som även den är skriven tillsammans med Benny Avelin ochLisa Hed, studeras likformig approximation av plurisubharmoniska funktioneru∈C(Ω) med glatta plurisubharmoniska funktioner definierade i omgivningarav Ω. I artikelns huvudsats visas att om Ω är begränsat och om ∂Ω lokaltges som grafen till en kontinuerlig funktion, så är sådan approximation alltidmöjlig. Uppskattningar på storleken på approximanternas definitionsområdeni termer av approximationsfelet ges också.

Med hjälp av Jensenmått konstruerade utifrån analytiska diskar har Po-letsky, se [Pol96], introducerat ett sätt att tala om plurisubharmonicitet förfunktioner definierade på kompakta mängder. I artikel V, som är skriven isamarbete med Rafał Czyz och Lisa Hed, visas hur Poletskys klass av Jen-senmått sammanfaller med en klass av Jensenmått som konstrueras med enindirekt metod. I artikelns huvudsats visas att de funktioner som är plurisub-harmoniska på en kompakt mängd X enligt Poletskys definition är precis defunktioner som går att punktvis approximera med en avtagande följd glattaplurisubharmoniska funktioner definierade i omgivningar av X .

I artikeln VI, som är skriven i samarbete med Lisa Hed, fortsätter studiet avden sorts approximation som studerats i artiklarna IV och V. Detta görs genomatt koppla ihop frågan om approximerbarhet av en plurisubharmonisk funk-tion med lösbarheten för ett Dirichletproblem för plurisubharmoniska funk-tioner definierade på kompakta mängder. För att studera detta Dirichletpro-blem introduceras en ny delklass av de hyperkonvexa områdena. Ett begränsathyperkonvext område Ω sägs vara P-hyperkonvext om det har en begränsaduttömningsfunktion som är plurisubharmonisk på den kompakta mängden Ω.Artikelns första huvudsats är att i ett P-hyperkonvext område går Dirichlet-problemet med avseende på plurisubharmoniska funktioner på Ω att lösa om,och endast om, randdatat är plurisubharmoniskt som en funktion på randen.Den andra huvudsaten är att en uppåt halvkontinuerlig funktion u på Ω som ärplurisubharmonisk i Ω går att punktvis approximera med en avtagande följdglatta plurisubharmoniska funktioner definierade på omgivningar till Ω om,och endast om, restriktionen av u till ∂Ω är plurisubharmonisk.

46

Acknowledgements

When finishing my thesis, I have realised that not only am I bringing five yearsof graduate studies to an end, but also a twenty-year educational endeavourthat started at the primary school in Björna, Örnsköldsvik, at a time that seemsso very recent and yet so long ago. Thinking about this, the idea is not far-fetched that this also marks the definite end of my adolescence. You cannot bea doctor of philosophy and keep believing you are a child. All these thoughtsmake me humble and grateful for all the people that have helped and supportedme along the way. I would therefore like to use this opportunity to extend mydeepest thanks and appreciation to all of you who have been there, supportingme. As it would be impossible to mention you all by name, I hope that youalready know who you are. However, I would like to extra mention some ofthose who have had the most direct influence on the work presented in thisthesis.

In particular, I would like to thank my thesis supervisor, Kaj Nyström. Notonly has he shared his time and his profound mathematical knowledge withme, he has also taught me the value of self-discipline. I would also like tothank my assistant supervisor, Maciej Klimek, for his inspirational supportand guidance, and Anders Fällström who was my supervisor during my timeat Umeå University; without his inspiration and encouragement, I would prob-ably never have become a mathematician.

I would also like to thank all my colleagues at the Department of Mathe-matics at Uppsala University for a both intellectually and socially stimulatingworking environment. Special thanks go out to Benny, Björn, Marcus, andOlow who have put up with the meticulously ordered chaos of my office workspace, and to Cissi for doing her best to widen my mathematical horizons.

I am very grateful to the complex analysis research group in Umeå for intro-ducing me to mathematical research, and showing me that mathematics alsocan be a social activity. Thank you Lisa for taking your time teaching memathematical research and for travelling to conferences with me. Thank youPer for sharing your excitement for research and for your constant interest formy mathematical development. Thank you Anders, Berit, Linus, and Urban.

I would would also like to extend my thanks to all my former colleagues atthe department of Mathematics and Mathematical Statistics at Umeå Univer-sity. You made working overtime indistinguishable from hanging out with myfriends.

I am also very thankful to my fellow undergraduate students at Umeå Uni-versity. You both made studying so much fun, and made me focus on moreimportant things than school. Thank you all!

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I would not have become the one I am today without the help and guid-ance of my family. Thank you Mamma and Pappa for always believing in me,supporting me, and encouraging me. There is no way for me to properly ex-press my deep gratitude. Thank you Anneli, Daniel, Hans, Annika and Annettfor letting me pretend to be some kind of role model. Thank you Linda foreverything.

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