Spectral Geometry of Partial Differential OperatorsMichael Ruzhansky1,2,3, Makhmud Sadybekov4, and Durvudkhan Suragan5

1Imperial College London, 2Ghent Analysis & PDE Center, Ghent University, 3Queen Mary University ofLondon, 4Institute of Mathematics and Mathematical Modeling, 5Nazarbayev University

DescriptionThe aim of Spectral Geometry of Partial Differential Operators is to providea basic and self-contained introduction to the ideas underpinning spectral ge-ometric inequalities arising in the theory of partial differential equations.Historically, one of the first inequalities of the spectral geometry was the min-imisation problem of the first eigenvalue of the Dirichlet Laplacian. Nowa-days, this type of inequalities of spectral geometry have expanded to manyother cases with numerous applications in physics and other sciences. Themain reason why the results are useful, beyond the intrinsic interest of geo-metric extremum problems, is that they produce a priori bounds for spectralinvariants of (partial differential) operators on arbitrary domains.

Content1 Function spaces2 Foundations of linear operator theory3 Elements of the spectral theory of differential operators4 Symmetric decreasing rearrangements and applications5 Inequalities of spectral geometry5.1 Introduction5.2 Logarithmic potential operator5.3 Riesz potential operators5.4 Bessel potential operators5.5 Riesz transforms in spherical and hyperbolic geometries5.6 Heat potential operators5.7 Cauchy-Dirichlet heat operator5.8 Cauchy-Robin heat operator5.9 Cauchy-Neumann and Cauchy-Dirichlet-Neumann heat operators

Spectral geometryLet us consider Riesz potential operators

(Rα,Ωf)(x) :=

∫Ω

|x− y|α−df(y)dy, f ∈ L2(Ω), 0 < α < d, (1)

where Ω ⊂ Rd is a set with finite Lebesgue measure.Rayleigh-Faber-Krahn inequality: The ball Ω∗ is the maximiser of the firsteigenvalue of the operatorRα,Ω among all domains of a given volume, i.e.

0 < λ1(Ω) ≤ λ1(Ω∗)

for an arbitrary domain Ω ⊂ Rd with |Ω| = |Ω∗|.Hong-Krahn-Szegö inequity: The maximum of the second eigenvalueλ2(Ω) of Rα,Ω among all sets Ω ⊂ Rd with a given measure is approachedby the union of two identical balls with mutual distance going to infinity.

Richard Feynman’s drum

The picture from planksip.org

It is proved that the deepest bass note is produced by the circular drumamong all drums of the same area (as the circular drum). Moreover,one can show that among all bodies of a given volume in the three-dimensional space with constant density, the ball has the gravitationalfield of the highest energy.

Rex is minimising the strain energy by circling

Open Access Book

InformationQR code for downloading the bookmichael.ruzhansky@ugent.besadybekov@math.kzdurvudkhan.suragan@nu.edu.kzPlease visit:ruzhansky.org and analysis-pde.org