Related papers: Pseudospectra of Semiclassical Boundary Value Prob…
Let -\Delta denote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 \Delta - 1 in the semiclassical limit h \to 0+. We give a new proof that yields not…
For a class of Bellman equations in bounded domains we prove that sub- and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a…
This work intends to prove that strong instabilities may appear for high order geometric optics expansions of weakly stable quasilinear hyperbolic boundary value problems, when the forcing boundary term is perturbed by a small amplitude…
In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak…
We prove the spectral invariance of the algebra of classical pseudodifferential boundary value problems on manifolds with conical singularities in the Lp-setting. As a consequence we also obtain the spectral invariance of the classical…
We study the existence of Feller semigroups arising in the theory of multidimensional diffusion processes. We study bounded perturbations of elliptic operators with boundary conditions containing an integral over the closure of the domain…
We consider a periodic pseudodifferential operator $H=(-\Delta)^l+A$ ($l>0$) in $\R^d$ which satisfies the following conditions: (i) the symbol of $H$ is smooth in $x$, and (ii) the perturbation $A$ has order smaller than $2l-1$. Under…
In a series of papers, we will develop systematically the basic spectral theory of (self-adjoint) boundary value problems for operators of Dirac type. We begin in this paper with the characterization of (self-adjoint) boundary conditions…
We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we…
Regularization methods have been recently developed to construct stable approximate solutions to classical partial differential equations considered as final value problems. In this paper, we investigate the backward parabolic problem with…
We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also…
We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted $p-${L}aplacian operator with a coefficient that is {locally…
Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let $V$ denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator ${\mathcal…
We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This…
In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are…
Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. The latter are the H\"ormander…
A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown…
A regular elliptic boundary-value problem over a bounded domain with a smooth boundary is studied. We prove that the operator of this problem is a Fredholm one in the two-sided refined scale of the functional Hilbert spaces and generates a…
This paper deals with a class of singularly perturbed nonlinear elliptic problems $(P_\e)$ with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as $\e\to 0$, and the domain is…
Using semi-classical formalism and asymptotic proliferation law of periodic orbits, we obtain an analytical expressions for the two-level cluster function, spectral form factor, level spacing distribution and the number variance for…