Maximum principles and applications for a class of degenerate elliptic linear operators

In this thesis 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 hypotheses on the principal part and on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on classical solutions of the Dirichlet problem for the linear equation. We also prove a Poincaré inequality, which allows us to define the functional setting where we study weak solutions for equations and inequalities involving this class of operators. Then we prove an existence and uniqueness result for weak solutions of the Dirichlet problem on bounded domains of R^n and a Weak Maximum Principle for weak solutions of differential inequalities,involving this class of operators. 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 pioneering result of Gidas-Ni-Nirenberg, and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space, which is a generalization of the celebrated result of Gidas-Spruck and Chen-Li. 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.