On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications

Dedicated to Professor Shmuel Agmon     

Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators

We deal with an inverse obstacle problem for general second order scalar elliptic operators with real principal part and analytic coefficients near the obstacle. We assume that the boundary of the obstacle is a non-analytic hypersurface. We show that, when we put Dirichlet boundary conditions, one measurement is enough to reconstruct the obstacle. In the Neumann case, we have results only for n = 2, 3 in general. More precisely, we show that one measurement is enough for n = 2 and we need 3 linearly independent inputs for n = 3. However, in the case for the Helmholtz equation, we only need n ? 1 linearly independent inputs, for any n ≥ 2. Here n is the dimension of the space containing the obstacle. These are justified by investigating the analyticity properties of the zero set of a real analytic function. In addition, we give a reconstruction procedure for each case to recover the shape of obstacle. Although we state the results for the scattering problems, similar results are true for the associated boundary value problems.     

Littlewood-Paley functions associated to second order elliptic operators

We prove L^p-estimates for the Littlewood-Paley function associated with a second order divergence form operator L=–div A with bounded measurable complex coefficients in ⁿ.Mathematics Subject Classification (2000):42B20, 35J15The author is partially supported by NSF of China (Grant No. 10371134) and SRF for ROCS, SEM.     

Boundary value problems for asymptotically homogeneous elliptic second order operators

In this paper we show the Dirichlet and Neumann problems over exterior regions have unique solutions in certain weighted Sobolev spaces. Two applications are given: (1) The Dirichlet problem for semi-linear operators, and (2) a Helmholtz decomposition for vector fields on exterior regions.     

Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture

Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D⊆R^d with oblique reflection at ∂D if D≠R^d. For each x∈D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as ε→0 of the expected hitting time of the ε-ball centered at x and of the principal eigenvalue for L in the exterior domain formed by deleting the ball, with the oblique derivative boundary condition at ∂D and the Dirichlet boundary condition on the boundary of the ball. This operator is non-self-adjoint in general. The behavior is described in terms of the invariant probability density at x and Det(a(x)). In the case of normally reflected Brownian motion, the results become isoperimetric-type equalities.     

Essential self-adjointness of semibounded elliptic operators of second order

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 710–716, May, 1989.     

Spectral stability of Dirichlet second order uniformly elliptic operators

We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation.     

Markovian extensions of symmetric second order elliptic differential operators

Let be bounded with a smooth boundary Γ and let S be the symmetric operator in given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self‐adjoint extensions of S by providing an explicit one‐to‐one correspondence between such extensions and the class of Dirichlet forms in which are additively decomposable by the bilinear form of the Dirichlet‐to‐Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kre?n's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell‐type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.     

On the principal eigenvalue of linear cooperating elliptic systems with small diffusion

We use cone methods combined with distribution theory and blow ups to find the asymptotic limit of the principal eigenvalue of a cooperative elliptic linear system when the diffusion is small.     

On the essential self-adjointness of the general second order elliptic operators

In this paper, we give sufficient conditions for the essential self-adjointness of second order elliptic operators. It turns out that these conditions coincide with those for the Schrödinger operator on a manifold whose metric essentially depends on the principal coefficients of a given operator.

     

On the influence of second order uniformly elliptic operators in nonlinear problems

The purpose of the present paper is to discuss the role of second order elliptic operators of the type on the existence of a positive solution for the problem involving critical exponent where Ω is a smooth bounded domain in , , and λ is a real parameter. In particular, we show that if the function has an interior global minimum point x₀ such that is comparable to , where and is the identity matrix of order n, then the range of values of λ for which the problem above has a positive solution can change drastically from to .