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.     

Hypoellipticity of certain infinitely degenerate second order operators

In this paper we establish the hypoellipticity without loss of weak derivatives of second order linear operators comprised by a linear combination, with infinite vanishing coefficients, of subelliptic operators in separate spaces. This generalizes previously known results.     

A family of degenerate elliptic operators: Maximum principle and its consequences

In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.     

Spectral asymptotics of elliptic strongly degenerate operators of arbitrary order

We compute the principal term of the spectral asymptotics for elliptic operators of an arbitrary order which is degenerate at the boundary of the domain. The degree of the degeneracy is such that the order of the decrease of the eigenvalues of the boundary-value problems is different from the classical one and the asymptotic coefficient depends on the form of the boundary conditions (strong degeneracy).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 25–30, 1976.In conclusion, the author expresses his deep gratitude to M. Z. Solomyak for his guidance in the preparation of this paper.     

The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.     

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 degenerate elliptic equations of high order and pseudodifferential operators with degeneration

Problems for high-order degenerate elliptic equations in a half-space are studied. Coercive a priori estimates and existence theorems for solutions of such problems in special weighted Sobolev-type spaces are obtained. The norms in these spaces are defined with the help of a special integral transform. Pseudodifferential operators with degeneration constructed using a special integral transform are studied. Pseudodifferential operators with degeneration are used to factorize the symbol of a high-order degenerate elliptic operator and to construct a separating operator of the Leray–Sakamoto type.     

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.