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

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

Second-Order Two-Sided Estimates in Nonlinear Elliptic Problems

Best possible second-order regularity is established for solutions to p-Laplacian type equations with p∈ (1 , ∞) and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L2-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all. © 2018, The Author(s)

Second-Order Two-Sided Estimates in Nonlinear Elliptic Problems

Best possible second-order regularity is established for solutions to p-Laplacian type equations with p∈ (1 , ∞) and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L2-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all. © 2018, The Author(s)

Second-Order Regularity for Parabolic p-Laplace Problems

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains. © 2019, Mathematica Josephina, Inc

Accretivity of the General Second Order Linear Differential Operator

For the general second order linear differential operatorL0=∑j,k=1najk∂j∂k+∑j=1nbj∂j+c with complex-valued distributional coefficients a j,k , b j , and c in an open set Ω ⊆ ℝ n (n ≥ 1), we present conditions which ensure that −L is accretive, i.e., Re ⟨−Lϕ, ϕ⟩≥0 for all φ ∈ C 0 ∞ (Ω). © 2019, Springer-Verlag GmbH Germany & The Editorial Office of AMS