Wiener criterion for a class of degenerate elliptic operators

In this paper we give some geometric criteria (analogous to Wiener's, Poincaré's and Zaremba's criteria for the Laplacian) for the regularity of boundary points for the Dirichlet problem relative to a class of partial differtial operators of the form ∑_{j = 1}ⁿ X_j², fulfilling Hörmander's condition.     

A degenerate Neumann problem for quasilinear elliptic integro-differential operators

This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998     

The Keldys-Fichera boundary value problem for a class of nonlinear degenerate elliptic equations

This paper discusses the Keldys-Fichera boundary value problem for a kind of degenerate quasilinear elliptic equations in divergence form. The existence theorem, comparison principle and uniqueness theorem are proved. This project supported by NSFC.     

On a class of degenerate elliptic operators arising from Fleming-Viot processes

We are dealing with the solvability of an elliptic problem related to a class of degenerate second order operators which arise from the theory of Fleming-Viot processes in population genetics. In the one dimensional case the problem is solved in the space of continuous functions. In higher dimension we study the problem in spaces with respect to an explicit measure which, under suitable assumptions, can be taken invariant and symmetrizing for the operators. We prove the existence and uniqueness of weak solutions and we show that the closure of the operator in such spaces generates an analytic -semigroup. Received December 4, 2000; accepted December 9, 2000.     

The Poisson kernel for certain degenerate elliptic operators

Let $\text{L =}\text{1}\text{2}\text{∑}{}_{\mathrm{k\; =\; 1}}{}^{\mathrm{d}}\text{V}{}_{\mathrm{k}}{}^2\text{+ V}{}_0$ be a smooth second order differential operator on $\text{R}$ⁿ written in Hörmander form, and $\text{G}$ be a bounded open set with smooth noncharacteristic boundary. Under a global condition that ensures that the Dirichlet problem is well posed for L on $\text{G}$ and a nondegeneracy condition at the boundary (precisely: the Lie algebra generated by the vector fields V₀, V₁,…, V_d is of full rank on the boundary) then the harmonic measure for L starting at any point in $\text{G}$ has a smooth density with respect to the natural boundary measure. Estimates on the derivatives of this density (the Poisson kernel) similar to the classical estimates for the Poisson kernel for the Laplacian on a half space are given.     

Solvability of the dirichlet problem for a class of degenerate quasilinear elliptic equations

In a bounded domain of an n-dimensional space one considers the first boundary-value problem for second-order quasilinear elliptic equations having a divergent structure and admitting an implicit degeneracy of a definite type: viz., at the points where the solution vanishes the strong ellipticity of the equation is violated. The dependence of the principal part of the equation on the gradient of the solution is not assumed to be linear. One gives the definition of a generalized solution of the Dirichlet problem for such equations and one shows its existence under the condition of coerciveness (in a definite sense) and of pseudomonotonicity of the differential operator.     

On a variational degenerate elliptic problem

In this paper we study a class of variational degenerate elliptic problems of the form in on , where is a bounded or unbounded domain in Received January 1999     

Uniqueness properties of degenerate elliptic operators

Let ?? be an open subset of R ^d and ${ K=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j+\sum^d_{i=1}c_i\partial_i+c_0}$ a second-order partial differential operator with real-valued coefficients ${c_{ij}=c_{ji}\in W^{1,\infty}_{\rm loc}(\Omega),c_i,c_0\in L_{\infty,{\rm loc}}(\Omega)}$ satisfying the strict ellipticity condition ${C=(c_{ij}) >0 }$ . Further let ${H=-\sum^d_{i,j=1} \partial_i\,c_{ij}\,\partial_j}$ denote the principal part of K. Assuming an accretivity condition ${C\geq \kappa (c\otimes c^{\,T})}$ with ${\kappa >0 }$ , an invariance condition ${(1\!\!1_\Omega, K\varphi)=0}$ and a growth condition which allows ${\|C(x)\|\sim |x|^2\log |x|}$ as |x| ?? ?? we prove that K is L ₁-unique if and only if H is L ₁-unique or Markov unique.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

On an obstacle problem for degenerate elliptic operators involving the critical Sobolev exponent

We establish the existence of a solution to the variational inequality (the obstacle problem) (1.1) which involves the critical Sobolev exponent. This result is also extended to an obstacle problem with a lower order perturbation. Dedicated to Professor F. Browder on the occasion of his 80-th birthday     

Single phase problem with curvature for a class of fully nonlinear elliptic operators

We prove regularity of Lipschitz free boundaries of one phase problems for fully nonlinear elliptic operators where the mean curvature appears in the free boundary condition.      

Propagation of singularities in the Cauchy problem for a class of degenerate hyperbolic operators

We consider second order degenerate hyperbolic Cauchy problems, the degeneracy coming either from low regularity (less than Lipschitz continuity) of the coefficients with respect to time, or from weak hyperbolicity. In the weakly hyperbolic case, we assume an intermediate condition between effective hyperbolicity and the Levi condition. We construct the fundamental solution and study the propagation of singularities using an unified approach to these different kinds of degeneracy.     

On degenerate elliptic operators of infinite type

The author was partially supported by NSF Grant DMS 91-01161     

On global hypoellipticity of degenerate elliptic operators

We consider a class of degenerate elliptic operators on a torus and prove that global hypoellipticity is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For another class of operators we show that the zero order term may influence global hypoellipticity. Received August 13, 1997