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

Sobolev Spaces In Mathematics I

Dedicated to the centenary of the outstanding mathematician of the 20th century, Sergey Sobolev, and, in a sense, to his celebrated work on a theorem of functional analysis, this title discusses such topics as: Sobolev-type inequalities on manifolds and metric measure spaces, and traces, inequalities with weights

Conductor inequalities and criteria for Sobolev type two-weight imbeddings

AbstractA typical inequality handled in this article connects the Lp-norm of the gradient of a function to a one-dimensional integral of the p-capacitance of the conductor between two level surfaces of the same function. Such conductor inequalities lead to necessary and sufficient conditions for multi-dimensional and one-dimensional Sobolev type inequalities involving two arbitrary measures. Compactness criteria and two-sided estimates for the essential norm of the related imbedding operator are obtained. Some counterexamples are presented to illustrate the peculiarities arising in the case of higher derivatives. Criteria for two-weight inequalities with fractional Sobolev norms of order l<2 are found

Optimal estimates for the gradient of harmonic functions in the multidimensional half-space

Blanchet Adrien. Dolmen de Ménouville. In: Bulletin Monumental, tome 66, année 1902. p. 408

The L^p-dissipativity of certain differential and integral operators

The first part of the paper is a survey of some of the results previously obtained by the authors concerning the Lp-dissipativity of scalar and matrix partial differential operators. In the second part we give new necessary and, separately, sufficient conditions for the Lp-dissipativity of the “complex oblique derivative” operator. In the case of real coefficients we provide a necessary and sufficient condition. We prove also the Lp-positivity for a certain class of integral operators