Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators
The paper deals with maximum principles for a class of linear elliptic differential operators of the second order with noncharacteristic degeneracy. The aim of the paper is to extend some important results from the uniformly elliptic setting to the above class of linear differential operators. Namely, under suitable hypothesis on the degeneracy set of the operator, the weak and strong maximum principles and Hopf type lemma are proved for classical solutions in bounded domains. Some a priori estimates for the solutions of Dirichlet problems are derived. As an example, the Grushin operator is considered. A partial symmetry result for classical solutions of semilinear problems on bounded, symmetric, and suitably convex domains and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space are proved as well.
elliptic differential operators with noncharacteristic degeneracy
scientific article; zbMATH DE number 5712416