Related papers: Pseudospectra of Semiclassical Boundary Value Prob…
We study properties of pseudodifferential operators which arise in their use in boundary value problems. Smooth domains as well as intersections of smooth domains are considered.
We study the spectral behavior of higher order elliptic operators upon domain perturbation. We prove general spectral stability results for Dirichlet, Neumann and intermediate boundary conditions. Moreover, we consider the case of the…
For a class of non-selfadjoint semiclassical operators in dimension one, we get a complete asymptotic description of all eigenvalues near a critical value of the leading symbol of the operator on the boundary of the pseudospectrum.
The pseudospectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. In fact, for non-selfadjoint operators the resolvent could be very large outside the spectrum, making the…
We consider a nonlinear evolution problem with an asymptotic parameter and construct examples in which the linearized operator has spectrum uniformly bounded away from Re z >= 0 (that is, the problem is spectrally stable), yet the nonlinear…
For bounded domains $\Omega$ with Lipschitz boundary $\Gamma$, we investigate boundary value problems for elliptic operators with variable coefficients of fourth order subject to Wentzell (or dynamic) boundary conditions. Using form…
This paper is devoted to establishing results for semilinear elliptic boundary value problems where the solvability of problems subject to {\it No Flux} boundary conditions follows from the solvability of related {\it Dirichlet} boundary…
We study the pseudospectrum of a class of non-selfadjoint differential operators. Our work consists in a detailed study of the microlocal properties, which rule the spectral stability or instability phenomena appearing under small…
The purpose of this note is to review some recent results concerning the pseudospectra and the eigenvalues asymptotics of non-selfadjoint semiclassical pseudo-differential operators subject to small random perturbations.
In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries for operators whose coefficients exhibit small oscillations analogous to the…
We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Dirichlet boundary condition in non-Lipschitz domains $\widetilde{\Omega} \subset \mathbb C$. The suggested method is…
In this paper second order elliptic boundary value problems on bounded domains $\Omega\subset\dR^n$ with boundary conditions on $\partial\Omega$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic…
The paper develops a theory of spectral boundary value problems from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary conditions is…
Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…
We consider the dynamics of bodies with "active" microstructure described by vector-valued phase fields. For waves with time-varying amplitude, the associated evolution equation involves a matrix that can be non-normal, depending on the…
We consider first-order elliptic differential operators acting on vector bundles over smooth manifolds with smooth boundary, which is permitted to be noncompact. Under very mild assumptions, we obtain a regularity theory for sections in the…
We study the solvability of boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being a UMD complex Banach space. The originality of this work lies in the fact that we…
We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\sqrt{-\Delta + m^2} - m)u =|u|^{p-1}u \quad \textrm{in}~\Omega, with the Dirichlet boundary condition $u=0$ on…
The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $\alpha>0$ one can find a simple $1$-dimensional constant…
We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $\Omega \subset \mathbb C$…