On degenerate elliptic operators of non-divergent form in bounded domain

Abstract

In this work we prove several inequalities, which provide a lower bound for the norm of an elliptic operator in non--divergent form in a bounded domain with power degeneration along the entire boundary. Earlier similar operators were studied in the case when they were initially defined in divergent form or reduced to such a form. In contrast, the coefficients of the operators we study are generally non--differentiable and cannot be reduced to divergent form. Only in the final section of the paper, in order to study the solvability of the corresponding differential equations, the differentiability is assumed for the coefficients of operator, and the corresponding adjoint operator is studied.

We first study degenerate elliptic operators of general form and prove an inequality for them in which the sum of norm of the action of operator and the norm of the function itself with some power weight in the space $L_2$ is bounded from below by the norm of function itself in a weighted Sobolev type space. We then consider the case, in which the elliptic operators are weakly positive. For such operators, we prove an inequality in which the real part of the scalar product of the action of operator and the function itself is bounded from below. In the final section we assume that weakly positive elliptic operators have strong degeneration along the entire boundary of the domain. For such operators involving a parameter $\lambda$, we first prove an inequality in which the norm of the action of operator is bounded from below by the norm of function itself in the underlying functional space. This inequality is then proved for the adjoint operator, and as a consequence, a result on the unique solvability of the corresponding differential equation is established.

The technique developed in this paper is based on the extension of some known results for elliptic operators with constant coefficients to the case of operators with degeneracy using auxiliary integral inequalities.