Katsiaryna Krupchyk

Transmission Eigenvalues for Elliptic Operators

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-self-adjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.

TRANSMISSION EIGENVALUES FOR OPERATORS WITH CONSTANT COEFFICIENTS

In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order

Inverse boundary value problems for the perturbed polyharmonic operator

\ge 2

Overdetermined Elliptic Systems

with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.

Inverse boundary problems for polyharmonic operators with unbounded potentials

Author(s): Krupchyk, K; Lassas, M; Uhlmann, G | Abstract: We show that a first order perturbation A(x) · D + q(x) of the polyharmonic operator (-Δ)m, m ≥ 2, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in ℝn, n ≥ 3. Notice that the corresponding result does not hold in general when m = 1.

Completion of overdetermined parabolic PDEs

AbstractWe consider linear overdetermined systems of partial differential equations. We show that the introduction of weights classically used for the definition of ellipticity is not necessary, as any system that is elliptic with respect to some weights becomes elliptic without weights during its completion to involution. Furthermore, it turns out that there are systems which are not elliptic for any choice of weights but whose involutive form is nevertheless elliptic. We also show that reducing the given system to lower order or to an equivalent system with only one unknown function preserves ellipticity.

On LpResolvent Estimates for Elliptic Operators on Compact Manifolds

We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in

Inverse Problems for Differential Forms on Riemannian Manifolds with Boundary

R^n

Absolute continuity of the periodic Schrödinger operator in transversal geometry

for the perturbed polyharmonic operator

Determining Electrical and Heat Transfer Parameters Using Coupled Boundary Measurements

(-\Delta)^m +q