Can one see the fundamental frequency of a drum?

We establish two-sided estimates for the fundamental frequency (the lowest eigenvalue) of the Laplacian in an open subset G of R^n with the Dirichlet boundary condition. This is done in terms of the interior capacitary radius of G which is defined as the maximal possible radius of a ball B which has a negligible intersection with the complement of G. Here negligibility of a subset F in B means that the Wiener capacity of F does not exceed gamma times the capacity of B, where gamma is an arbitrarily fixed constant between 0 and 1. We provide explicit values of constants in the two-sided estimates.Comment: 18 pages, some misprints correcte

The H\"older-Poincar\'e Duality for Lq,pL_{q,p}-cohomology

We prove the following version of Poincare duality for reduced $L_{q,p}$-cohomology: For any $1<q,p<\infty$, the $L_{q,p}$-cohomology of a Riemannian manifold is in duality with the interior $L_{p',q'}-cohomology for$1/p+1/p'=1$,$1/q+1/q'=1$.Comment: 21 page

Second-order L2L^2-regularity in nonlinear elliptic problems

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the existence and square integrability of the weak derivatives of the nonlinear expression of the gradient under the divergence operator. This provides a nonlinear counterpart of the classical $L^2$-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are established. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required. If the domain is convex, no regularity of its boundary is needed at all

Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds

We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds M of finite volume. Sharp conditions ensuring L q (M) and L ∞ (M) bounds for eigenfunctions are exhibited in terms of either the isoperimetric function or the isocapacitary function of M.

Well-posed elliptic Neumann problems involving irregular data and domains

Nonlinear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented. Résumé Nous considérons des problèmes de Neumann pour des équations elliptiques non linéaires dans domaines éventuellement non réguliers et avec des données peu régulières. Un équilibre entre l’intégrabilité de la donnée et l’(ir)régularité du domaine nous permet d’obtenir l’existence, l’unicité et la dépendance continue de solutions généralisées. L’irrégularité du domaine est décrite par des inegalités “isocapacitaires”. Nous donnons aussi des applications à certaines classes de domaines. 1 2 1 Introduction and main result

Cubature of multidimensional Schrödinger potential based on approximate approximations

We report here on some recent results obtained in collaboration with V. Maz'ya and G. Schmidt cite{LMS2017}. We derive semi-analytic cubature formulas for the solution of the Cauchy problem for the Schrödinger equation which are fast and accurate also if the space dimension is greater than or equal to 3. We follow ideas of the method of approximate approximations, which provides high-order semi-analytic cubature formulas for many important integral operators of mathematical physics. The proposed method is very efficient in high dimensions if the data allow separated representations