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Regularizing properties of the double layer heat potential and shape analysis of a periodic problem
Full text linkThis 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
Regularizing properties of space‐periodic layer heat potentials and applications to boundary value problems in periodic domains
No full textTangential derivatives and higher-order regularizing properties of the double layer heat potential
Full text linkWe prove an explicit formula for the tangential derivatives of the double layer heat potential. By exploiting such a formula, we prove the validity of a regularizing property for the integral operator associated to the double layer heat potential in spaces of functions with high-order derivatives in parabolic Hölder spaces defined on the boundary of parabolic cylinders which are unbounded in the time variable
Time dependent boundary norms for kernels and regularizing properties of the double layer heat potential
Full text linkWe introduce a class of norms for time dependent kernels on the boundary of Lipschitz parabolic cylinders and we prove theorems of joint continuity of integral operators upon variation of both the kernel and the density function. As an application, we prove that the integral operator associated to the double layer heat potential has a regularizing property on the boundary
A few results on permittivity variations in electromagnetic cavities
Full text linkWe study the eigenvalues of time-harmonic Maxwell's equations in a cavity
upon changes in the electric permittivity of the medium. We prove
that all the eigenvalues, both simple and multiple, are locally Lipschitz
continuous with respect to . Next, we show that simple eigenvalues
and the symmetric functions of multiple eigenvalues depend real analytically
upon and we provide an explicit formula for their derivative in
. As an application of these results, we show that for a generic
permittivity all the Maxwell eigenvalues are simple.Comment: Added references and corrected some minor typo
The functional analytic approach for quasi-periodic boundary value problems for the Helmholtz equation
Full text linkWe lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boundary value problems for the Helmholtz equation. This consists in introducing a quasi-periodic fun-damental solution and the related layer potentials, showing how they are used to construct the solutions of quasi-periodic boundary value prob-lems, and how they behave when we perform a singular perturbation of the domain. To show an application, we study a nonlinear quasi-periodic Robin problem in a domain with a set of holes that shrink to points
Perturbation analysis of the effective conductivity of a periodic composite
Full text linkWe consider the effective conductivity λeff of a periodic two-phase composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. Then we study the behavior of λeff upon perturbation of the shape of the inclusions, of the periodicity structure, and of the conductivity of each material
The first Grushin eigenvalue on cartesian product domains
Full text linkIn this paper, we consider the first eigenvalue.1(O) of the Grushin operator.G :=.x1 + |x1|2s.x2 with Dirichlet boundary conditions on a bounded domain O of Rd = R d1+ d2. We prove that.1(O) admits a unique minimizer in the class of domains with prescribed finite volume, which are the cartesian product of a set in Rd1 and a set in Rd2, and that the minimizer is the product of two balls Omega(*)(1).subset of R-d1 and O-* (2)subset of R-d2. Moreover, we provide a lower bound for | Omega(*) (1) | and for lambda(1)( O-* (1) x O-* (2)). Finally, we consider the limiting problem as s tends to 0 and to +8.GR-T
Semiclassical estimates for eigenvalue means of Laplacians on spheres
Full text linkWe compute three-term semiclassical asymptotic expansions of counting
functions and Riesz-means of the eigenvalues of the Laplacian on spheres and
hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically
for Riesz-means we prove upper and lower bounds involving asymptotically sharp
shift terms, and we extend them to domains of . We also prove a
Berezin-Li-Yau inequality for domains contained in the hemisphere . Moreover, we consider polyharmonic operators for which we prove
analogous results that highlight the role of dimension for P\'olya-type
inequalities. Finally, we provide sum rules for Laplacian eigenvalues on
spheres and compact two-point homogeneous spaces
Dependence of the layer heat potentials upon support perturbations
Full text linkWe prove that the integral operators associated with the layer heat potentials depend smoothly upon a parametrization of the support of integration. The analysis is carried out in the optimal Holder setting
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