Variational problems of minimal surface type

Variational Problems of Minimal Surface Type III. The Dirichlet Problem with Infinite Data

HOWARD JENKINS • JAMES SERRIN

This is the third in a series of papers on variational problems of minimal surface type. In the first of these papers, hereafter referred to as I, we have inves- tigated the local behaviour of solutions with particular regard to the existence of an interior estimate for the gradient, and we have shown that a Harnack principle applies. The second paper (II) contains an existence and uniqueness theory for the Dirichlet problem (with infinite boundary data) for the minimal surface equation. It is our purpose here to generalize the results of II to the full class of variational problems of minimal surface type.

We consider in particular non-parametric variational problems of the form

(1) 5SF(p,q)dxdy=O , p=ux, q=uy,

where F(p, q) is a function of class C ~ throughout the p, q plane, satisfying the conditions

(2) F .>0 ,

(3) F - p Fp-q Fq> A,

(I+p2)Fp"+2pqFpq+(I+qE)Fqq <2E (4)

Vl + p2 +q21/FppFqq-Fp~ q

for some constants A and E. A solution of the variational problem (1) is then a twice continuously differentiable function u = u (x, y) satisfying the Euler-Lagrange equation

(5) fpp ux~+2fpq u~y+tqq uyr=0 .

In addition to conditions (2), (3), (4) we shall assume throughout that the second derivatives of F are uniformly HSlder continuous on compact subsets of the p, q plane.

In I it was shown that variational problems of minimal surface type are closely related to regular parametric variational problems of the form

(6) 5 ~ ~(r) ds dt=O

where r = (a, ~, co) and a, z, 09 are the Jacobian matrices

O(y, z) O(z, X) C3(X, y) c~(s, t) ' c~(s, t) ' ~(s, t) "

Variational Problems of Minimal Surface Type 305

Indeed, a solution of (6) which can be represented in the form z=u(x, y) is also a solution of a non-parametric variational problem whose integrand F necessarily satisfies conditions (2), (3), and (4). On the other hand, any variational problem (1) can be written in the form (6) where the integrand ~r is defined in the open half space co < 0 by the equation

(7) ~ z, to) = - o9 F ( - a/co, - z/co).

We shall show later that when F satisfies (2), (3), and (4), and is furthermore appropriately normalized, then the function ~- defined by (7) can be extended by continuity to the closed half space co<0. Moreover the extended function satisfies the condition ~ r ( 2 r ) = 2 ~ ' ( r ) for all 2 >0.

This extension allows us to associate with (1) a corresponding one-dimensional parametric variational problem

(8) c5 S ~o (7) dt=O

where ~=(2, p )=(dx /d t , dy/dt), and

(9) ~o(2,/~) = ~-(p, - 2 , 0).

If ? is an oriented piecewise smooth arc in the plane we may correspondingly define

00) I(~)=S~(~)dt;

this integral is intimately associated with our later study. For the moment, we simply notice that for the minimal surface equation F = V 1 +p2 +qZ and ~r o = I~1. Therefore in this case I(~) reduces to the Euclidean length of 7 (the same result holds when the integrand F is radially symmetric; see the remark in Section 2). In more general cases I(V) can be interpreted as the length of ? with respect to the non-symmetric Finsler metric defined by the function ~o (~).

In what follows it will be useful to use the language of chains. In particular, by a one-chain we shall mean a formal sum of open oriented segments. If F, A are one-chains, then the chains - F and A + F are defined in the obvious way, and A - F = A + ( - F ) . In addition, we define I(F) as the sum of the integrals (10) over the individual arcs comprising F.

We turn now to a specific description of the results of the paper. Let D be a bounded domain in E 2 whose boundary consists of a finite number of open convex arcs* together with their endpoints. To each of these arcs we assign either continuous data or the value plus infinity or the value minus infinity - the problem of Dirichlet is then to determine a solution of (1) in D which takes on this given data at the boundary.

The arcs on which the values plus infinity and minus infinity are assigned will be denoted respectively by A1, A2 . . . . and B 1, B 2 . . . . . and the remaining arcs will be named C1, C2 . . . . . We observe that each A s and each B~ must be

* A boundary arc is called convex if there is a neighbourhood of each of its points whose intersection with D is a convex domain. We observe specifically that a straight segment is a special case of a convex boundary arc.

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306 H. JENKINS ~; J. SERRIN"

straight in order for the problem to be solvable at all. Indeed, according to the straight line lemma (Section 5) if one of these arcs were not straight the solution would necessarily be infinite in a neighbourhood of this arc, an obvious absurdity. In what follows, then, it will be assumed that all the As and B~ are straight.

Now let ~ be a polygonal domain lying in D, whose vertices are drawn from the set of endpoints of the arcs A t, B~ and C~. If the perimeter of ~ is given a counterclockwise orientation, then the formal sum of its oriented boundary segments forms a one-chain F. Moreover F can be uniquely expressed in the forms

F = A - A ' = B - B '

where A and B are composed of those arcs of F which are in the families {Ai} and {B~} respectively. The following main result then holds.

Theorem. I f the family of arcs {C~} is not empty, the Dirichlet problem stated above is solvable if and only if

(,) I (A)<I(A') , I ( - B ) < I ( - B ' )

.for each polygon # chosen as above. I f the family {Ci} is empty the result is the same, except that if # coincides with D condition (,) should be replaced by I(A)= I ( - B ) .

In the first case the solution is unique, in the second, unique up to an additive constant.

For the special case of the minimal surface equation this result reduces to Theorem 4 of II. As in that case it is possible to single out particular situations where the results are especially simple. For example, if D is a quadrilateral with sides A~, C1, A2, C2, in that order, then the necessary and sufficient condition for a solution of the corresponding Dirichlet problem to exist becomes

I(A1)+ I ( A 2 ) < I ( - C i ) + I ( - C 2 ) ,

that is, the sum of the generalized lengths of the sides on which the data plus infinity is prescribed should be less than the sum of the generalized lengths of the sides (orientation reversed) on which continuous data is prescribed. If the sides of D are A1, B1, A2, B2, in that order, then the condition is slightly different but equally simple, namely

I(A1)+ I (A2)=I ( -B1)+ I ( - B 2 ) .

Should both families {As} and {B~} each contain at most one segment, then condition (*) is automatically fulfilled in view of the triangle inequality (Section 5). As a special case of this, the Dirichlet problem admits a unique solution for arbitrary continuous data assigned on each arc of the boundary of D. (A similar result, but for convex domains D, was established by FINN for equations of minimal surface type.)

We remark finally that the formulation of the problem allows D to be multiply connected, and even to have re-entrant boundary segments. Naturally re-entrant


segments are to be considered as having two sides, on each of which data is assigned.

The first sections of the paper develop and assemble preliminary material, much of which has importance and significance in its own right. Once this material is available, the proof of the main result can be given by exactly the same method as in the minimal surface case. Indeed, the matter becomes essentially axiomatic, and it suffices simply to outline the procedure. This we do in Section 6. In the earlier parts of the paper we shall make free use of the results of I and II in order to shorten the exposition.

1. A Preliminary Existence Theorem

Let u=u(x, y) be a solution of (5) in a domain D. In reference I the authors showed that I Vu [ and 1172 u I can be estimated at a point P in terms of a bound on ] u l and the distance from P to the boundary of D. Since we are assuming that the coefficients of the Euler equations (5) are Hrlder continuous on compact subsets of the p, q plane, Schauder type estimates may then be used to prove the existence of a bound on the Hrlder moduli of the second derivatives of u. Thus by standard procedures we obtain the following

Compactness Principle. Let {u~} be a uniformly bounded sequence of solutions qf (5) in a domain D. Then there exists a subsequence which converges to a solution in D, the convergence being uniform on every compact subset of D.

According to a result in the Appendix the vector 17F=(Fp, F~) has bounded length. Consequently the following result also holds ([7], pp. 205 and 209):

General Maximum Principle. Let D be a bounded domain and let E be a finite set of points on the boundary aD. Suppose that ul and u2 are two solutions of (5) in D such that lira inf(ul - u 2 ) > 0 as one approaches any point of dD-E . Then ul >=u2 in D.

It is well known (see for example [8], Theorem VI) that the Dirichlet problem for (5) can be solved in strictly convex domains for sufficiently smooth boundary data. With the aid of the compactness principle and the maximum principle this result can easily be generalized to the case where the assigned data on the boundary is continuous. Finally, by using the Perron method (II, pp. 339-340) we may infer the existence of solutions in bounded domains which assume prescribed boundary values at each boundary point at which a barrier exists.

We now show that a barrier exists at each point where the boundary is convex. Let Q be such a point. It is sufficient to find, for each pair of numbers M and 6, a neighbourhood N of Q and a non-negative function w=w(x ,y) defined in N c~ D such that

i) N is contained in a disc of radius ~ about Q, ii) w> M on aNnD,

iii) w = 0 at Q,

iv) w is a super-solution and - w is a sub-solution of (5).

According to reference [5], in order for w to be a super-solution and - w to be a sub-solution it is sufficient that z=w(x, y) defines a surface whose mean

20*

308 H. JENKINS&J. SERRIN:

curvature H and Gaussian curvature K are negative and satisfy the inequality H2[IKI>E2-1. (For completeness we shall give an analytical proof of this result in the Appendix, independent of the argument in reference [5].)

For the particular function

[1 22x 2 y \ v(x ,y)=~t- - ;u logcos-~-- - logcos- :~-- ) , 2>1, 6 > 0

a calculation yields H < 0, K < 0 in the rectangle I x I < 6 n/42, [y I < 6 n/4, and

H 2 ( [ / 2 - l / V 2 , 2 > (V:-~-) 2 4 [ K ~ - ~ 2 ~ 2y =

1 - sin ----S-- sin /~

If 2 is now chosen greater than (E+V~--1-1) 2, the above criterion will be satisfied and v will be a super-solution in the rectangle for any positive number tS. A barrier at Q may now be constructed by a rotation and translation of the function w = M + v restricted to the domain lYl < 2 x < 6 n / 4 , cf. II, p. 324. We have thus proved the following

PreBmlnary Existence Theorem. Let D be a bounded domain whose boundary consists of a finite number of open convex arcs Ci together with their endpoints. Then there exists a solution of (1) in D which takes on assigned bounded continuous data on the arcs C v

This result remains true even without the assumption that the data is bounded, as we have remarked at the end of the introductory section.

2. Basic Lemmas

Here we shall establish some results which are more specific to our purposes than those of the preceding section.

To begin with, we observe that the quantities l , = F - p F p - q F q and 117F[ are bounded functions,

(II) 1 # t <_- Constant, I VF [ <= Constant, for all p, q.

The first of these inequalities is a simple consequence of the fact that #(hp, hq) is a decreasing function of h for h > 0, whence A < # < F(0). The second result is proved in the Appendix.

In view of (11), the integrand F(p, q) can be normalized by the addition of a linear function of p, q and multiplication by a constant so that

(12) IZF(0)=0, S u p [ V F [ = l , l < # < b ,

where b is a suitable constant (one can in fact take b = F(0)). This being done, we assert that there is a positive constant c such that

(13) l + c p < F < p + b , p=Vp- '2-~.


Indeed, from the third relation of (12) we have 1 <F(0)<b , and the right hand side follows at once. The left hand side is then an easy consequence of the convexity of the surface Z = F(p, q) and the fact that IZF(0)= 0.

We can now carry out the extension of the function ~" to the full half space co<0. Let r=(a, z, 0) be a vector in the plane 09=0, and let ~=(a, z) be the associated 2-vector. Putting k =(0, 0, 1), we then define

(14) ~ ' ( r )= lim ~ ( r - h -1 k)= lim h -1 F(h~)= lim -~. VF(h-~). h--* co h-~eo h-'* r

The latter limit exists since ~. 17F(h~) is a bounded function of h with a non- negative derivative.

It must be shown that the extended function ~" is continuous and positively homogeneous of degree one. With regard to the first of these properties, it is clearly sufficient to consider only points of the plane 09 =0. Letting ~ and ~1 be vectors in this plane, and letting ~ and ~ be the associated 2-vectors, we have by (7), for e > 0,

Using the fact that I VFI _-< 1 and letting h ~ 0 then yields

I - 8 k ) - I _-__ I r - I + I (r - . k ) - S (r) I.

Since ~a~(r-~k)~,~(r) as 8 ~ 0 , the continuity of ~" follows at once. Next, it is evident from (7) that ~ is positively homogeneous at least for to < 0, and the same result for a~ < 0 follows by continuity.

Remark. If F is radially symmetric, that is, depends on p, q only through the combination p2 +q2, then (13) holds in the stronger form

(13') l +p< F <p+b.

Moreover, using (12) and (14), it is clear that in this case ~-(r)=[~ I when r= (~r, z, 0). Hence ~o (q) = I q [ and I(~) is the ordinary Euclidean length of 7.

Now consider the function G(a, z) which is obtained from F(p, q) by the projective transformation

p q 1

The domain of G is a bounded convex set, and the second derivatives of G in this domain satisfy the inequalities

G~r G#,>0 , and

G~+G~ <28

where 8 is a constant depending only on the normalized integrand F (see I, p. 191, or part 2 of the Appendix).

310 H. JENKINS&J. SERRIN:

Our interest in G stems from the fact that it is the integrand of a variational problem dual to (1). Indeed, as in I put

d~=(F-qFq)dx+qFpdy, drl=pF~dx+(F-pFp)dy.

It is well known that corresponding to any given solution u(x, y) of (1) the dif- ferentials d~ and dr/are exact, and thereby define functions ~(x, y) and q(x, y) by integration. Moreover, since

~xr].~,- ~yrlx=F(F- p Fp-q F~)> 1

the mapping (x, y) --} (~, ~/) is locally one-to-one. Finally, the transformed solution u(~, t/) satisfies the equation

(16) G~uc~+2G~,ur a=ur z=u~.

In view of (15) this equation is uniformly elliptic. The functions ~, ~/ also define a mapping of any solution surface z=u(x, y)

into the ~, ~/plane. Let d( 2 =(1 +p2)dx+2pq dx dy+(1 +q2) dy2 be the Eucli- dean metric on such a surface. We assert that the ratio (d~ 2 + dtl2)/d( 2 is bounded from zero. Indeed, putting a = l +p2, b=pq, c = l +q2, e=(F_qFa)Z+(pFq)2, f =qFp(F-qFq)+p Fq(F-p Fp), g = ( F - p Fp)2 +(qFp) 2, we have

d~ 2 +drl 2 e d x 2 + 2 f d x d y + g d y 2 d( 2 - a d x Z + 2 b d x d y + c d y 2

Let a~, a 2 denote respectively the minimum and maximum of the above ratio, and put w = V 1 +p2 + q 2 , / z = F - p F p - q F q and O=pFq-qFp. After a calculation one finds

a g - 2 b f T e c F2+W2[z2+O 2 e g - f 2 g2 / , t 2 oq +0~ e = a c _ b 2 = W2 , oq ~2 = ~ = ~ .

Since ~1, 0~2 are positive we have then

1 0~ 1 "{-0~ 2 W 2 1 0 2 - - < tXl ~1 Ct2

By (12) and (13) the fight hand side is bounded by 3/c 2, so that finally

(17) d~2+dr/2 c2 d(2 >-~-.

We conclude this section with an analogue of Lemma 1 of II. We need first the following result from the theory of uniformly elliptic equations.

Let v (x, y) be a positive solution of the equation

(18) a(x, y) vxx + 2b(x, y) vxy + c(x, y) vyy=O

in the domain X 2 @ y2 <r 2. Suppose moreover that (18) is uniformly elliptic, that is,

a + c ] / ~ < 2e


for some constant e. Then there exists a constant k, depending only upon e, such that

I Vv(O,O)l<k v(0, 0) r

Proof. If I vl_-<m then according to a lemma of ROSENBCOOM ([6], p. 371) we have IVy(0, O)l<Km/r, where K depends only upon e. In our case v may be unbounded; nevertheless, since it is positive we may infer from the Harnack inequality ([9], Theorem 1) that Ivl ____K' v(0, 0) for x2+y2<-_(r/2) 2, the constant K' depending only upon e. Applying ROSENBCOOM'S lemma in this disc completes the proof.

Now let u=u(x ,y ) be a solution of (5) in a convex domain D, and let S denote the surface z=u(x , y). For P~D, let d be the distance from P to the boundary of D, let P ' be any boundary point of t9 whose distance from P is d, and let ~ be the unit vector from P to P ' . Also let P* denote the point on S directly over P and let r be the geodesic distance on S from P* to the boundary of S. The following estimate then holds.

Lemma 1. For any 8 > 0 there exists a positive constant 6 such that if d/r < 6 then one of the following pairs of inequalities,

I ~ . V F - ~ ( v ) l < e and ~.Vu>O o r

I ~ . V F + ~ r ( - v ) l < e and ~.Vu<O, holds at P. *

Proof. Let the functions 4, r/be as defined above, and assume that the constants of integration have been chosen so that ~(P)=~/(P)=0. Since the mapping (x, y ) ~ ( ~ , r/) is locally one-to-one, it is clear that the functions 4, ~/ define a locally one-to-one mapping of S such that the point P* maps onto the origin. Let A be the largest circle about the origin in the 4, ~/plane in which the inverse mapping is one-to-one. We assert that the radius of A is >cr/2. To prove this, note that if the radius of A is less than cr/2 then (17) implies that the image on S of each radius of A is a curve from P* to an interior point of S. The mapping is therefore one-to-one in a neighbourhood of every boundary point of A, contra- dicting the definition of A.

To prove the lemma, we may assume without loss of generality that ~ points in the positive x-direction. Now consider the function x(~, r/) in A. We find easily that

-qFp F - p F p =~G~-G, x~- -~ - z G ~ . (19) x~= F # F #

Consequently, by direct calculation and the use of (16),

(20) G ~ x r 1 6 2 1 6 2 a=ur z=u~.

Thus x(r ~/) is a solution of a uniformly elliptic equation in A. From our construction and from the convexity of D,

x-xp<_d, (x ,y )~D,

�9 Here v is the three-vector (v 1, v 2, 0) associated with -v = (vl, vz).

312 H. JENKINS & J. SERRIN:

so that the function d+Xl,-X(~, tl) is a non-negative solution of (20) in A. Applying the result preceding Lemma 1, we find that

2k d (21) [ Vx(O, 0) 1 < - - . - - ,

c g

where k depends only on r (see (15)), that is, only on the form of the integrand F.

Now using (19) and putting 2=2kd/cr, the preceding inequality may be rewritten

2 2 p F~ -2pFFp+F2(1-).2#2)<O at P .

If we regard the left hand expression as a quadratic polynomial in Fp, then its discriminant must be non-negative. This immediately yields the inequality

(22) ]ql/p<2p at P .

Next, making use of (19), (21), (22), (12), and (13) it follows that

1 qgr Plf~l Iql <(1+1/c)2. T- = x e - T--~-~ --<lxr F Pt~

But since p + b > F this implies c

P=> ( c + 1 ) 2 - b at P .

To complete the proof, we put/3 =(p, q), p =(p, q, 0) and observe that

Division by p then yields the identity

7 p/-?"

Now by (22),/3[p approximates either ~ or - ~ when 2 is small; at the same time p beeomes arbitrarily large. Since ~ is continuous, ] VF] ____ 1, and [ #[ __< b, this proves the lemma.

3. The Conjugate Function If u is a solution of (5) in a domain D, the function ~b which arises by integrating

the exact differential d~=Fpdy-F~dx

is called the conjugate function of u. This function plays a basic role in the solution of the infinite data Difichlet problem. Clearly 117r =l VFI < 1, so that if D is a domain having a piecewise smooth boundary, then ~O is Lipschitz continuous in (boundary slits and reentrant boundary segments are taken to be two sided).

Now let ? be an oriented piecewise smooth arc lying in D, and let s denote arc length along ?. Let ~ be the fight hand unit normal on y. We assert that

dd/=~. VF<:(v) v=(vl,v2 0). ds ' '


For using (14) and the fact that I VF[ is bounded, we have

~ - ( 0 = lim h - l F ( h ~)= lim h - IF (p+ h i ) = lim ~. VF(~+ h ~) h-*oo h"* oo h"* o~

where ff = (p, q). The assertion then follows since ~. VF(~ + h ~) is a bounded increasing function of h. Similarly, letting h ~ - oo we obtain ~. VF> - ~ ( - v ) . If-/denotes the unit tangent on ~, then by (9) we have ~ ( v ) =~o (t). Consequently

and

(23)

<:o(7)

-l(-y)<~ d~b<Ify).

Suppose next that y lies in the closure of D and that the boundary of D consists of a finite number of convex arcs (as in the introduction). Then we can approximate y by piecewise smooth arcs in D, from which it follows at once that

(24) - I ( - y ) < 5 d~/<I(y).

Moreover, if ~ is a piecewise smooth curve which bounds a subregion of D, then

(25) j" de =0. F

The next lemma shows that inequality (23) can persist even for boundary arcs of D.

Lemma 2. Let u be a solution of (5) in a domain D. Suppose in addition that u is continuous on D u y where ~ is an oriented convex boundary arc of D. Then

dq,<t(r).

Proof. By the mean value theorem

(26) (Pt - P ) " { V F ( p , ) - VF(p)) >0

when ~4=~. The result now follows exactly as in the proof of Lemma 3 of II.

Lemma 3. Let D be a domain bounded in part by a straight segment T, oriented so that the right hand normal on T is the outer normal to D. Let u be a solution of (5) in D which takes the boundary value plus infinity on T. Then

S d~k=I(T). T

If u is a solution in D which takes the boundary value minus infinity on T, then

S d~b= - I ( - W ) . T

The proof follows by using Lemma 1 to estimate S d~/, where T, is a segment T 6

in D, parallel to T and at a distance 3 from it. The reader is referred to the proof of Lemma 4 of II for the details (note that when u = oo on T it is the first inequality

314 H . JENKINS • J. SERRIN:

of Lemma 1 which applies, while if u = - oo on T then the second inequality must be used).

The same method of proof yields the following generalization of Lemma 3.

Lemma 4. Let D be a domain bounded in part by a straight segment T, oriented as in Lemma 3. Let {u,} be a sequence of solutions of (5) in D, each u, being continuous in Du T. If the sequence {u,} diverges uniformly to infinity on T while remaining uniformly bounded on compact subsets of D, or if the sequence diverges uniformly to minus infinity on compact subsets of D while remaining bounded on T, then

lim ~ d ~ , = I ( T ) . n-~oo T

On the other hand, if the sequence diverges uniformly to minus infinity on T while remaining uniformly bounded on compact subsets of D, or if the sequence diverges uniformly to plus infinity on compact subsets of D while remaining bounded on T, then

lim S d~b,= - I ( - T ) . n--* co T

4. The Straight Line Lemma and the Triangle Inequality

The existence proof of the following section depends heavily on the limit behaviour of a monotone increasing family of solutions. The main result required in the discussion of this limit behaviour is the fundamental

Straight Line Lemma. Let D be a bounded domain lying entirely to one side of a straight line L. Let the boundary of D consist of a part C disjoint from L and a remaining part C' contained in L. Suppose that u is a solution of (5) in D such that m<u< M on C. Then for any compact subset S of D u C there is a number N, depending only upon the diameter of D and the distance from S to L, such that

m - N < u < M + N in S.

This result is an easy consequence of the maximum principle together with the existence of a super-function ~b + and a sub-function ~b- which are defined in the interior of a triangle, take respectively non-negative and non-positive values on two sides of the triangle, and become respectively positively and negatively infinite at interior points of the third side. The required functions ~b + and ~b- are here supplied by the function v(x, y) constructed in Section 1. The proof is car- ried out in detail in I, Lemma 6, and also in [5], Theorem 1.

The remaining technical tool for the study of monotone increasing families of solutions is a triangle inequality for the integral I(~). If P, Q are points in the (x, y) plane, we denote by PQ the open oriented segment from P to Q. The following conclusion then holds.

Triangle Inequality. Let P, Q, R be non-collinear points in the (x, y) plane.

Then I(PR) < I(PQ) + I(QR).

The proof hinges on the auxiliary assertion that the set in the (;~, #) plane defined by ~o (2, #) = 1 is a strictly convex closed curve surrounding the origin.


Before proving this assertion, we show how it implies the required inequality. Put

a =~o(P-Q), b=~o(Q--R).

By construction (see (9) and (14)) it is evident that a and b are greater than zero. Hence if 3 x and 32 are non-collinear vectors satisfying ~o = 1 we have by convexity

a - b ~0 (a- - -~ rl + a--~-~- r21 <1 .

Setting F1 =P--Q/a, 32 = Q'-R/b and using the positive homogeneity of ~0 now yields easily

which is equivalent to the required conclusion. To prove the auxiliary assertion, consider the surface ~-(r )=1 in the half

space o9<0. As shown in I, p. 191, this surface is convex about the origin and can be represented in the form o9 =G(a , z) where the domain of G is a bounded convex set 9 with G =0 on its boundary. Therefore, using (9), the strict convexity of the set ~o = 1 is equivalent to the strict convexity of 9 . *

It is therefore enough to prove that 9 is strictly convex. Let us suppose for contradiction that this is not the case; that is, we suppose that the boundary of 9 contains a straight segment l. Without loss of generality it can be assumed that l has the form

a=ao, ~<z</~, (co>O).

We assert that FG = (G,, G~) tends to a constant value as points of I are approached from 9 .

Consider first G,. Let (no, ~o) be an arbitrary fixed point o f / , and let ~l be a number between Zo and ft. Using the convexity of G we find that

G ( a o + h , ~ l ) - G ( a o + h , zo)>(z l - zo)G~(ao+h, zo), h o < h < O .

Now by direct calculation 17G = IZF/l~, so that [IZG] < 1. Therefore, since G=O on/ , we have

G~(ao + h, Zo) < 2 ] h [/(zl - Zo).

Similarly, letting T z be a number between ~ and %, we get

G~(ao + h, Zo) >= - 2 [ h ]](Zo- ~2).

Consequently G~ tends uniformly to zero as one approaches compact subsets of l.

Before considering the remaining component G, of 17G, we observe that (15) implies

4 8 2 G 2 < 2 ( 2 d , 2 1) 2 2 _ G , , G ~ - G ~ , - G , , .

�9 For particular integrands F it may be quite simple to verify directly that the domain is strictly convex; in these cases the remaining argument can be avoided. A particularly important situation where this occurs is when F arises from a regular parametric variational problem, for then the surface ~ ' ( t )= 1 has everywhere positive Gaussian curvature.

The minimal surface equation provides a specific example, the domain ~ in this case being circular.


By Cauchy's inequality the first term on the right hand side cannot be larger than ( 2 r 2 - 1) 2 2 2 G~, + G~ ~, whence

I G, ~ I < I/-fz-ZT-1G~ ~ .

Consequently, for ~ < z < fl and for fixed Zo,

I G o ( a o + h , x ) - G o ( a o + h , % ) l < = l / ~ l G ~ ( ~ o + h , z ) - G ~ ( a o + h , % ) l .

Now G,(ao+h, z) is a bounded monotone function of h. Hence we can allow h to tend to zero in the above inequality, with the result that

lim Gr o + h, z )= lira Gr + h, Zo)

uniformly in z. Thus G~ tends to a constant upon approach to l.

Next we observe that the function pair (G,, G~) generates a quasi-conformal mapping of the domain ~ . To prove this, note that by (15)

2 2 2 2 2 G ~ + 2 G ~ + G ~ < ( G ~ + G ~ ) 2 < 4 8 ( G ~ G ~ - G ~ ) ,

as required (the mapping (x, y) ~ (u, v) is quasi-conformal provided that the condition ux+uy+vx2 2 2 +vy<2Klux2 vy -uy vxl is satisfied). Furthermore (G~, G,) is one-to-one since G is convex.

A contradiction now results from the fact that a one-to-one quasi-conformal mapping of a Jordan domain ~ cannot attain constant values on a non-degenerate boundary arc. (This fact can be proved by reducing it to the same assertion about conformal mappings. To be precise, let �9 be a quasi-conformal homeomorphism of a Jordan domain ~ , and let g be a conformal map of ~ ( ~ ) onto the unit disk. Then go ~ is a quasi-conformal homeomorphism of ~ onto the unit disk. By a well-known result [1, page 47] the mapping go ~ can be extended to a homeomorphism of the closure of ~ . Consequently, if �9 is constant on a non-degenerate boundary arc / , then g-~ must be constant on the circular arc go r But this is clearly impossible, and the required assertion is proved.)

We have thus proved that ~ is strictly convex, and consequently that the equation ~0 = 1 defines a strictly convex closed curve. Since the domain ~ contains the origin, it is also clear that the curve ~o = 1 surrounds the origin. This completes the proof of the triangle inequality.

Lemma 5. I f condition (*) of the main theorem is satisfied, then no two segments A t and no two segments B~ can meet at a convex corner of D.

Proof. Suppose in fact that two segments A i did meet at a convex corner of D. Let ~ be the boundary arc of D consisting of A1, A2 and their endpoints, and let be the curve of shortest length joining the outer endpoints of A1 and A2 and homotopie to ~ in ~.

Clearly ~ is a polygonal arc different from ~. Moreover, the vertices of are drawn from the endpoints of the families {As} , {Bi}, and {C~}; and the corners of y (if there are any) are convex in the same sense as the comer of ~. Now let be the polygonal domain bounded by ~ and 7. Assuming that A1 and A2 are positively oriented, the associated boundary chain of ~ has the form

F = A I + A 2 - L 1 . . . . . L.


where L1, . . . , Ln are oriented segments of y. Evidently

(27) I (L , ) +.. . +I(Ln) <I(A1) + I(A2)

(to see this, divide 9 ~ into n triangles by extending each of the segments L~ in the positive direction until it reaches ~, and then apply the triangle inequality).

Since (27) stands in contradiction to (*), and since a similar argument applies if two segments B~ meet at a convex corner of D, the lemma is proved.

5. The Maximum Principle for Infinite Data

In this section we shall give a generalization of the maximum principle to the case where the solutions involved take on infinite boundary values. This result immediately yields the uniqueness of solutions of the Dirichlet problem with infinite data.

Maximum Principle. Let D be a bounded domain whose boundary O D includes two families of open segments {A,} and {Bi}.* Let u t and u2 be two solutions of (5) in D, both of which take the boundary value plus infinity on each A~ and minus infinity on each B~.

Let E be a finite point set on OD, and let C denote the set of points of O D - E which are not in the closure of the families {A,} and {B~}. I f C is non-empty, assume that lim inf(u2 -ul)>_-0 as one approaches any point of C. If C is null, assume that u2 > ul at some point of D.

Then in either case, u2 > ul throughout D.

Proof. It can be assumed for simplicity that the set E is empty, since only minor and obvious changes are required in the proof otherwise.

We consider first the case when C is non-empty. Let e, M be fixed positive numbers with e small and M large, and define

M - 8 when U2--Ul>=M ~ ( x , y ) = u 2 - u l - e when F.<U2-Ul<M

0 when u 2 - u l __<e.

Clearly ~ is a strongly differentiable continuous function in D, vanishing in the neighbourhood of any point of C.

Also let D,a denote the domain consisting of those points of D at distance > e from the endpoints of the segments A i and B~ and >6 from the boundary of D, where ~<~. Since ~ is small, the boundary of D~a consists of straight segments L 1 , L 2 . . . . near the segments A~ and B~, circular arcs El, E2 . . . . near their endpoints, and a remaining set N near C. By taking 6 suitably near zero, we may assume in addition that

(i) ~b- 0 in a neighbourhood of N, and

(ii) Id~l /ds-dd/2/ds [ =<2e on the segments L t, L2 . . . . .

That (ii) can in fact be attained follows from Lemma 1 and the relation

d_O0= 7. VF ds

on these segments.

* It is tacitly assumed that the families {A~} and {B~} are finite.


With 6 = 3 (e) thus chosen, let

(28) J = S 17~ b" {VF(ITu2)- [TF([Tttl)} dxdy. D~ 6

By partial integration, using (5) and condition (i) above, one finds

i L~ El

From (ii) and the conditions I~l <M, I d~/ds[ < 1, we obtain further

(29) J<28M {~ ([A~I + 2 n ) + ~ (IBsl +2n)}. i i

Now Vq5 =0 almost everywhere outside the set where e < u 2 - u l <M, while in that set

V~a= Vu2- VUl.

Accordingly the integrand on the right side of (28) is non-negative (see (26)). Consider a sequence of values of ~ decreasing to zero such that the corresponding values 6(e) also decrease to zero. The set D~6 is therefore increasing, and the monotone convergence theorem may be applied to J. Thus from (29) we obtain

{17U2- g i r l} . {VF(u2)--ITF(ul)} dx dy=O

where the integration takes place over the set 0 < u 2 - u l <M. It follows that the integrand is zero in this set. But since M can be taken arbitrarily large, this in turn implies

Vus-Vul whenever u2>ul.

Hence u2 -- ux + constant in any component of D where u2 > u~. But then, assuming the existence of such a component, u s - Ux + ~: throughout D, where x >0. This contradicts the given boundary condition on C, whence we must have Us <Ul throughout D.

The same proof applies when C is empty, except that we must of course delete reference to C and N.

6. Proof of the Main Theorem

In the preceding sections we have assembled the necessary preliminary material to carry out the proof of the theorem stated in the introduction. The proof itself is exactly the same as the proof of Theorem 4 in II, and consequently it will be sufficient here simply to point out the corresponding steps.

First, the uniqueness assertion follows directly from the infinite data maximum principle of Section 5. The preliminary existence theorem which is required next (see II, p. 339) is given in Section 1. The remaining proof of Theorem 4 in II requires the following results: the maximum principle, the compactness principle, the straight line lemma, the monotone convergence theorem, results analogous to those in Section 3 here, Lemmas 6, 7, 8 of II, and the fact that no two segments A s and no two segments B~ can meet at a convex corner of D.


The only ones of these results which have not been proved above are the montone convergence theorem and the analogues of Lemmas 6, 7, 8.

The monotone convergence theorem for our ease is a consequence of the straight line lemma, the Harnack principle (I, Theorem 3), and the compactness principle. The reader is referred to I, p. 210, for a proof. The refinements of the monotone convergence theorem stated on p. 330 of I (including Lemma 6) depend only on the straight line lemma, the triangle inequality and the results of Section 3.

Lemmas 7 and 8, finally, are consequences of the maximum principle, the preliminary existence theorem, the straight line lemma, and a barrier type argument. A suitable function for the latter has already been given in Section 1. This completes the proof of our main result.

Remarks. It may appear paradoxical that we are able to prove existence of solutions of the Dirichlet problem in non-convex domains, when this problem is in general not well posed for continuous data. The explanation, however, lies in the observation that the solution is not required to fulfill any conditions at the endpoints of the arcs A s, B~, and C~.

In this connection, it is worth noting that the solutions which we obtain have many of the qualitative features of solutions of the Dirichlet problem for convex domains. In particular, the following results hold:

(i) If ul and u2 are two solutions and Uz >u l on the arcs C~, then u2 > u l throughout D.

(ii) Solutions depend continuously upon the data assigned on the arcs C~.

(iii) If the assigned boundary data has a removable discontinuity at at convex boundary point, then the solution also has a removable discontinuity there.

Appendix 1. We are to show that I VFI < Constant, using conditions (2) and (3). Consider

the surface Z =F(p, q) in the three dimensional space (p, q, Z). By (2) this surface is convex, and by (3) the tangent plane to the surface at (p, q, F(p, q)) intersects the Z-axis at the point with ordinate #=F-pFp-qFq>A. Hence it is geometri- cally evident that for p =Vp-Y-~2 > 1 the surface lies below the cone

Z=ap+A, a = M a x (F(p,q)-A}. p = l

Thus there exists a constant a' such that

(30) F < a p + a' for all p, q.

For the rest of the argument it is convenient to resort to vector notation. By the convexity of F we have for any two vectors ~ and -~1,

F(pl)-F(p)>_-(pl-p). FF(~).

Setting ~ =~ +h VF(~) and using (30) then yields

h lVF(~)12<a(l~l +hlVF(~)l)+a'--F(p).


If we now divide both sides by h and let h tend to infinity there results 117FI <a, and the proof is complete.

2. Here we shall give a short demonstration of (15). First of all, it is apparent that we can assume q =0 without loss of generality. Now

Fp , F_ G G,= -~ , ( la=F--pFp-qF~),

" It 1 a

whence by direct calculation and evaluation at q = 0

(31) G#,,+G~, _ (F2+p2Ff)Fpp+21tpF~Fpq+!a2Fqq

In view of (2), (12), and (13), the right hand side is less than

4b (1-'[-p2)Fpp'-FF~q

"Iherefore, using (4), we find that (15) holds with g=4bE/c. This argument also allows us to give a direct proof that (4) holds when F

arises from a regular parametric variational problem, independent of the rather complicated technique used in I. We first observe that if F arises from a regular variational problem then the function G(a, ~) which represents the (normalized) surface ~'(r) = 1 in non-parametric form can be extended to a domain 9 ' properly including 9 , and G, , G , , - G 2 , > 0 in 9 ' . Thus there must exist some constant g for which (15) holds.

On the other hand, by (2)

121zpFqepql< 2+c 2 p2F2 2 = 2 Fpp+-2- '~ kt2Fqq"

Hence using (12) and (13) the right hand side of (31) is greater than

c 2 V ~ _ Min ( 1 , 2 - ~ c ) . (1-]-p2)FpP-I-Fqq

V 1 - - 4 - P V F , , F , : F L "

Consequently (4) holds with E = 5 b 2 8/c 2 provided F denotes the normalized integrand. But this completes the proof, since the left hand side of (4) is invariant under normalization.

3. Let Lu=Auxx+2Buxy+Cuyy

be a quasilinear elliptic operator satisfying the minimal surface type condition

(l + p2)A+ 2 p q B + ( l +qZ)c < 2 E , (E>I ) .


Also let u =u(x , y) be a function such that the surface z =u(x , y) has negative mean and Gaussian curvatures satisfying

H 2 IKI =>E2-1"

Then Lu<O, that is, u is a super-solution.

Proof. Let P be a point at which Lu is to be evaluated. By obvious invariance properties, we may assume without loss of generality that q = u y = 0 at P. Intro- ducing the classical r, s, t notation for the second derivatives of u, the assertion then reduces to showing

Ar+2Bs+Ct<O provided that r t<s 2 and

(I+p2)A+C <2E, V1-- V

If we now write A'=2A, C'=2-~C,

(l+p2)r+t < _2[/-~rZf. ]//1 + p-~[ /~-- r t

r'=2r, t'=2-1t where 2=Vl+p:, the assertion reduces further (after dropping the primes) to proving that

Ar+2Bs+Ct<O

when A + C < 2 E V A C - B 2 and r+t< -2 V~-T-1Vs~-rt . In showing this, invariance properties allow an even further reduction to the

case where s = 0 , r > 0 , t<0 . Hence we must finally establish the inequality

(32) A r + C t < 0

when A + C < 2 E [ / ~ and r+t<-2V~--f- l[ /Zlt l . The last conditions imply

V Therefore

and (32) follows at once.

The same proof establishes that L u > 0 when H > 0 , K < 0 , and I-I2/IKI >E 2-1. Thus for the original function u we have L(-u)>O and - u is a sub-solution.

Acknowledgement: This work was partially supported by the United States Air Force Office of Scientific Research under Grants AF-AFOSR-883-67 and AF-AFOSR-1301-67.

R e f e r e n c e s

1. AHLmRS, L., Lectures on Quasiconformal Mappings. Van Nostrand, New York 1966. 2. FXNN, R., New estimates for equations of minimal surface type. Arch. Rational Mech. Anal.

14, 337--375 (1963). 3. JENKXNS, H., & J. SERPaN, Variational problems of minimal surface type, I. Arch. Rational

Mech. Anal. 12, 185--212 (1963).

21 Arch. Rat ional Mech. Anal. , Vol. 29

322 H. JENKINS 8z J. SERRIN: Variational Problems of Minimal Surface Type

4. JENKINS, H., & J. SERRIN, Variational problems of minimal surface type, II. Boundary value problems for the minimal surface equation. Arch. Rational Mech. Anal. 21, 321- 342 (1966).

5. JENKINS, H., Super-solutions for quasi-linear elliptic equations. Arch. Rational Mech. Anal. 16, 402--410 (1964).

6. MEYERS, N., On a class of non-uniformly elliptic quasi-linear equations in the plane. Arch. Rational Mech. Anal. 12, 367--391 (1963).

7. NITSCHE, J. C. C., ~ber ein verallgcmeinertes Dirichletsches Problem fiir die Minimalfliichen- gleichung und hebbare Unstetigkeiten ihrer L6sungen. Math. Ann. 158, 203--214 (1965).

8. NIRENBERG, L., On non-linear elliptic partial differential equations and H61der continuity. Comm. Pure Appl. Math. 6, 103-- 156 (1953).

9. SERRIN, J., On the Harnack inequality for linear elliptic equations. J. d'Analyse Math6matique 4, 292-- 308 (1956).

Department of Mathematics University of Minnesota

Minneapolis

(Received December 14, 1967)

Variational problems of minimal surface type

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