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Register a new userOn the Calder`on problem in periodic cylindrical domain with partial Dirichlet and Neumann data
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We consider the Calder`on problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution.
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We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.
We consider homogenization of the scalar wave equation in periodic media at finite wavenumbers and frequencies, with the focus on continua characterized by: (a) arbitrary Bravais lattice in $mathbb{R}^d$, $d!geqslant!2$, and (b) exclusions i.e. voids that are subject to homogenous (Neumann or Dirichlet) boundary conditions. Making use of the Bloch wave expansion, we pursue this goal via asymptotic ansatz featuring the spectral distance from a given wavenumber-eigenfrequency pair (within the first Brillouin zone) as the perturbation parameter. We then introduce the effective wave motion via projection(s) of the scalar wavefield onto the Bloch eigenfunction(s) for the unit cell of periodicity, evaluated at the origin of a spectral neighborhood. For generality, we account for the presence of the source term in the wave equation and we consider -- at a given wavenumber -- generic cases of isolated, repeated, and nearby eigenvalues. In this way we obtain a palette of effective models, featuring both wave- and Dirac-type behaviors, whose applicability is controlled by the local band structure and eigenfunction basis. In all spectral regimes, we pursue the homogenized description up to at least first order of expansion, featuring asymptotic corrections of the homogenized Bloch-wave operator and the homogenized source term. Inherently, such framework provides a convenient platform for the synthesis of a wide range of wave phenomena in metamaterials and phononic crystals. The proposed homogenization framework is illustrated by approximating asymptotically the dispersion relationships for (i) Kagome lattice featuring hexagonal Neumann exclusions, and (ii) pinned square lattice with circular Dirichlet exclusions. We complete the numerical portrayal of analytical developments by studying the response of a Kagome lattice due to a dipole-like source term acting near the edge of a band gap.
We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: $$-triangle u +operatorname{div}(umathbf{b}) =f quadtext{ and }quad -triangle v -mathbf{b} cdot abla v =g$$ in a bounded Lipschitz domain $Omega$ in $mathbb{R}^n$ $(ngeq 3)$, where $mathbf{b}:Omega rightarrow mathbb{R}^n$ is a given vector field. Under the assumption that $mathbf{b} in L^{n}(Omega)^n$, we first establish existence and uniqueness of solutions in $L_{alpha}^{p}(Omega)$ for the Dirichlet and Neumann problems. Here $L_{alpha}^{p}(Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig (1995, JFA) and Fabes-Mendez-Mitrea (1998, JFA) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(partialOmega)$.
We consider the linear second order PDOs $$ mathscr{L} = mathscr{L}_0 - partial_t : = sum_{i,j =1}^N partial_{x_i}(a_{i,j} partial_{x_j} ) - sum_{j=i}^N b_j partial_{x_j} - partial _t,$$and assume that $mathscr{L}_0$ has nonnegative characteristic form and satisfies the Olev{i}nik--Radkeviv{c} rank hypoellipticity condition. These hypotheses allow the construction of Perron-Wiener solutions of the Dirichlet problems for $mathscr{L}$ and $mathscr{L}_0$ on bounded open subsets of $mathbb R^{N+1}$ and of $mathbb R^{N}$, respectively. Our main result is the following Tikhonov-type theorem: Let $mathcal{O}:= Omega times ]0, T[$ be a bounded cylindrical domain of $mathbb R^{N+1}$, $Omega subset mathbb R^{N},$ $x_0 in partial Omega$ and $0 < t_0 < T.$ Then $z_0 = (x_0, t_0) in partial mathcal{O}$ is $mathscr{L}$-regular for $mathcal{O}$ if and only if $x_0$ is $mathscr{L}_0$-regular for $Omega$. As an application, we derive a boundary regularity criterion for degenerate Ornstein--Uhlenbeck operators.
This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $dots$, $a_{n-1}$ of the heat trace asymptotic expansion. In particular, we explicitly give the expressions for the first four coefficients. These coefficients are spectral invariants which provide precise information concerning the volume and curvatures of the boundary of the manifold and some physical quantities by the magnetic Steklov eigenvalues.
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