We demonstrate that solving the classical problems mentioned in the title on quadrature domains when the given boundary data is rational is as simple as the method of partial fractions. A by-product of our considerations will be a simple proof that the Dirichlet-to-Neumann map on a double quadrature domain sends rational functions on the boundary to rational functions on the boundary. The resul...

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, consider Ω=Ω0∖B‾r, where Br is ball centered at origin with radius r>0 and Ω0⊂Rn, n⩾2, an open, bounded set Lipschitz boundary, such that B‾r⊂Ω0. We impose Steklov condition on outer boundary Robin involving positive L∞ function β(x) inner boundary. Then, first σβ(Ω) its main properti...

Under suitable assumptions on the potential of the nonlinearity, we study the existence and multiplicity of solutions for a Steklov problem involving the p(x)-Laplacian. Our approach is based on variational methods.

We analyse the spectrum of the D-dimensional Poincaré invariant effective string model of Polchinski and Strominger. It is shown that the leading terms beyond the Casimir term in the long distance expansion of the spectrum have a universal character which follows from the constraint of Poincaré invariance.

The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein ...

We propose Steklov geometry processing, an extrinsic approach to spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace–Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, while many previous extrinsic methods lack theoretical justi cation. Instead, we propose a systematic approach by considering the Steklov eigenva...

We consider an elliptic equation −∆u + u = 0 with nonlinear boundary conditions ∂u ∂n = λu + g(λ, x, u), where g(λ, x, s) s → 0, as |s| → ∞. In [1, 2] the authors proved the existence of unbounded branches of solutions near a Steklov eigenvalue of odd multiplicity and, among other things, provided tools to decide whether the branch is subcritical or supercritical. In this work we give condition...

Andrei Aleksandrovich Gonchar was born on November 21, 1931, in Leningrad (now St. Petersburg), Russia. He graduated from Moscow State (Lomonosov) University (MSU) in 1954. Under the supervision of S.N. Mergelyan, he defended his Candidate’s (Ph.D.) thesis at the same university in 1957, and then his doctoral dissertation (D.Sc.) at the Steklov Mathematical Institute in Moscow in 1964. From 197...