Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

In this paper we deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for the linear equation. A good example of such an operator is the Grushin operator on R^{d+k}, to which we devote particular attention. As an application of these tools in the degenerate elliptic setting, we prove a partial symmetry result for classical solutions of semilinear problems on bounded, symmetric and suitably convex domains, which is a generalization of the results of Gidas-Ni-Nirenberg (1979, 1981) and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space, which is a generalization of the result of Gidas-Spruck (1981) and Chen- Li (1991). The method we use to obtain these results is the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator.