Non-selfadjoint Elliptic Operators Research Articles

Dirichlet’s and Thomson’s Principles for Non-selfadjoint Elliptic Operators with Application to Non-reversible Metastable Diffusion Processes

We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.

Results of Ambrosetti–Prodi type for non-selfadjoint elliptic operators

The well-known Ambrosetti–Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non-self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity result for elliptic equations in non-divergence form. We employ techniques based on the maximum principle.

Semiclassical asymptotic behavior of the spectrum of a nonselfadjoint elliptic operator on a two-dimensional surface of revolution

The asymptotics as ɛ → 0 of the spectrum of the shifted Laplace operator H = ɛΔ + ∂v on a two-dimensional compact surface of revolution homeomorphic to the sphere is described, where v is a vector field.

The principal eigencurves for a nonselfadjoint elliptic operator

In this paper we study the existence of the principal eigencurves for a nonselfadjoint elliptic operator. We obtain their variational formulation. We establish also the continuity and the differentiability of the principal eigencurves.

Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems

We consider elliptic operators A on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of A through an observation, with an exponential cost. Following the strategy of Lebeau and Robbiano (1995) [25], we deduce the construction of a control for the non-selfadjoint parabolic problem ∂ t u + A u = B g . In particular, the L 2 norm of the control that achieves the extinction of the lower modes of A is estimated. Examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of A.

Strong Stability and Non-smooth Data Error Estimates for Discretizations of Linear Parabolic Problems

We study smoothing properties of discretizations of a linear parabolic initial boundary value problem with a possibly non-selfadjoint elliptic operator. The solution at time t > 0 of this problem, as well as its time derivatives, are in L r for initial values in L s even when r > s. We show that similar strong stability results hold for discrete solutions obtained by discretizing in space by linear finite elements and in time by a class of A(Θ)-stable implicit rational multistep methods (including single step methods as a special case) with good smoothing properties, as well as for certain combinations of single step methods. Most of our results are derived from the corresponding L 2-bounds, shown by semigroup techniques, together with a discrete Gagliardo-Nirenberg inequality, and generalize previously known estimates with respect to admissible problems and time discretization methods. Our techniques make it possible to obtain, e.g., supremum norm error estimates for initial data which are only required to be in L 1.

A numerical method for backward parabolic problems with non-selfadjoint elliptic operators

A method of solution of backward parabolic problems with non-selfadjoint elliptic operators is presented. The method employs a quasisolution approach and is based on the separation of the problem into a sequence of well-posed forward problems on the entire mesh and an ill-posed system of algebraic equations on a coarser submesh. For the corresponding forward problem the continuous dependence of the solution on the initial profile is proved. From this result a stability estimate on the final time T is obtained. The estimate shows a decrease in stability of the forward (hence, the backward) problem, as the final time T is increased. Using the stability result the existence of a quasisolution of the backward problem is proved. For the solution of the intermediate non-selfadjoint forward problems a modified alternating-direction finite difference scheme is presented. The ill-conditioned system of algebraic equations is solved by using truncated singular value decomposition. The effectiveness of the method is demonstrated on a numerical test problem with exact and noisy data.

Asymptotic Behaviour of the Ground State of Singularly Perturbed Elliptic Equations

The ground state of a singularly perturbed nonselfadjoint elliptic operator $$$$ defined on a smooth compact Riemannian manifold with metric aij(x)=(aij(x))−1, is studied. We investigate the limiting behaviour of the first eigenvalue of this operator as μ goes to zero, and find the logarithmic asymptotics of the first eigenfunction everywhere on the manifold. The results are formulated in terms of auxiliary variational problems on the manifold. This approach also allows to study the general singularly perturbed second order elliptic operator on a bounded domain in Rn.

Spectral distribution of preconditioned elliptic operators

This paper is concerned with the singular value distribution and eigenvalue distribution of preconditioned non-selfadjoint elliptic operators corresponding to finite element discretizations of elliptic boundary value problems. The distribution of the singular values has an important bearing on the convergence behavior of the conjugate gradient method applied to the normal equations. The eigenvalue distribution has an important bearing on various alternative conjugate gradient like iterative methods. The main results give a precise interval on which the singular values cluster. It is also shown that the (complex) eigenvalues cluster about an interval of the real axis under suitable assumptions. The analysis is based on the theory of collectively compact operators.

A Generalized Dirichlet Principle for Second Order Nonselfadjoint Elliptic Operators

The classical Dirichlet principle states that if the spectrum of $L = \frac{1}{2}\Delta - V$ on a smooth bounded domain $D \subset R^n $ with the Dirichlet boundary condition on $\partial D$ is negative, then the unique solution $\phi _0 $ of $L\phi _0 = 0$ in D with $\phi _0 = f$ on $\partial D$, for a smooth function f, minimizes a certain energy integral. This may be easily extended to the operator $L = \frac{1}{2}\nabla \cdot a\nabla + a\nabla Q \cdot \nabla - V$. Now L is selfadjoint with respect to the density $e^{2Q} $. In this paper, we generalize this result to nonselfadjoint operators on bounded domains. We consider the solution $\phi _0 $ to $L\phi _0 = (\frac{1}{2}\nabla \cdot a\nabla + b \cdot \nabla - V)\phi _0 = 0$ in D and $\phi _0 = f \geqq 0$ on $\partial D$ under the assumption $\operatorname{Re} (\sigma (L)) < 0$ for L with the Dirichlet boundary condition on $\partial D$.

Approximation du spectre d'un operateur lineaire; application aux operateurs differentiels elliptiques non autoadjoints

To be computed, the eigenvalues of a closed linear operatorT in a Banach space are usually approximated by the eigenvalues ofT h, a linear operator approximatingT in a finite dimensional space (for example, finite difference method, Galerkin method),h is a parameter which tends to 0. This approximation is studied in [2]; stability ofT h implies the continuity of the spectrum ofT h, whenh tends to 0. We present here a new kind of sufficient condition. For that purpose, we disconnect the continuity of the spectrum ofT h into lower and upper semicontinuities. And we give two different criteria for these semi-continuities. Applications to the approximation of nonselfadjoint elliptic operators by finite difference schemes, are given.

Monotonic difference schemes for elliptic and parabolic equations in the case of a non-selfadjoint elliptic operator

THE present note proposes monotonic schemes, uniformly convergent at a rate 0( h 2), for a non-selfadjoint second order elliptic equation in an arbitrary region, and also monotonic locally one-dimensional schemes, uniformly convergent at a rate 0( h 2 + τ) for a parabolic equation with non-selfadjoint elliptic operator.

Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes

The present work extends our short communication L. Andersson, M. Mars and W. Simon, Local existence of dynamical and trapping horizons, Phys. Rev. Lett. 95 (2005), 111102. For smooth marginally outer trapped surfaces (MOTS) in a smooth spacetime, we define stability with respect to variations along arbitrary vectors v normal to the MOTS. After giving some introductory material about linear nonself-adjoint elliptic operators, we introduce the stability operator Lv and we characterize stable MOTS in terms of sign conditions on the principal eigenvalue of Lv. The main result shows that given a strictly stable MOTS S0 ⊂ Σ0 in a spacetime with a reference foliation Σt, there is an open marginally outer trapped tube (MOTT), adapted to the reference foliation, which contains S0. We give conditions under which the MOTT can be completed. Finally, we show that under standard energy conditions on the spacetime, the MOTT must be either locally achronal, spacelike or null.