We study maximum principles for a class of linear, degenerate
elliptic differential operators of the second order. 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 prove a Poincare inequality, which then allows
to define the functional setting where to study weak solutions for
equations and inequalities involving this class of operators. A
good example of such an operator is the Grushin operator, 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 and a nonexistence result for
classical solutions of semilinear equations with subcritical growth
defined on the whole space. We use here the method of moving
planes, implemented just in the directions parallel to the
degeneracy set of the Grushin operator.
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