Regularizing properties of the double layer heat potential and shape analysis of a periodic problem

This Dissertation is devoted to the study of some integral operators arising in parabolic potential theory which are relevant in order to analyze boundary value problems for the heat equation subject to a singular perturbation of the domain by exploiting a known functional analytic approach for elliptic problems,and to the analysis of some elliptic perturbation problems with a potential theoretic approach. The Dissertation is divided into two independent parts. In the first part (Chapters 1-3) we produce new results in parabolic potential theory and, in particular, we study the mapping properties of some integral operators associated with layer heat potentials, while in the second part (Chapter 4) we investigate the behavior of an elliptic boundary value problem under domain perturbation with a potential theoretic approach. The Dissertation is organized as follows. In Chapter 1 we introduce a normed class of time dependent weakly singular kernels and we prove results of joint continuity of some parabolic integral operators upon variation both of the kernel in the above class and of the density function. Moreover we apply these results to some integral operators related to layer heat potentials. In Chapter 2 we prove an explicit formula for the tangential derivatives of the double layer heat potentials and we prove a regularizing property of the integral operator associated with the double layer heat potential. In Chapter 3 we consider space-periodic layer heat potentials and we solve some periodic boundary value problems for the heat equation. Finally, Chapter 4 is devoted to the study of the behavior of the longitudinal permeability of a periodic array of cylinders upon the perturbation of the periodicity structure and of the cross sections of the cylinders. At the end of the Dissertation we have enclosed some Appendices with some results that we have exploited

Acoustic scattering from corners, edges and circular cones

Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be non-convex. We prove that such an obstacle scatters any incoming wave non-trivially (i.e., the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of a penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone

Impact of Yves Meyer's work on Kato's conjecture

We discuss how the works of Yves Meyer, together with Raphy Coifman, on Calder{\'o}n's program and singular integrals with minimal smoothness in the seventies, paved the way not only to a solution to Kato's conjecture for square roots of elliptic operators, but also to major developments in elliptic and parabolic boundary value problems with rough coefficients on rough domains.Comment: to be publishe

Resolvent and heat kernel properties for second order elliptic differential operators with general boundary conditions on L^p

Under general (including mixed) boundary conditions, nonsmooth coefficients and weak assumptions on the spatial domain, resolvent estimates for second order elliptic operators in divergence form are proved. The semigroups generated by them are analytic, map into Hölder spaces, are positivity improving, and their heat kernels are Hölder continuous in both arguments. We regard perturbations of the elliptic operator by nonnegative potentials, by first order differential operators and multiplicative perturbations. Finally the results provide that the solutions of the corresponding linear and semilinear parabolic equations are Hölder continuous in space and time

On the numerical range of second order elliptic operators with mixed boundary conditions in Lp^p

We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on Lp in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin- instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions