We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.

Boundary decay estimates for solutions of fourth-order elliptic equations

We obtain integral boundary decay estimates for solutions of second- and fourth-order elliptic equations on a bounded domain with smooth boundary. These improve upon previous results of this type in the second-order case and, we believe, are new in the fourth order case. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary. AMS Subject Classification: 35J40 (35P15)