PARTIAL DIFFERENTIAL OPERATORS WITH NON-NEGATIVE CHARACTERISTIC FORM, MAXIMUM PRINCIPLES, AND UNIQUENESS FOR BOUNDARY VALUE AND OBSTACLE PROBLEMS
Paul M. N. Feehan · arXiv (Cornell University) · 2013
We prove weak and strong maximum principles, including a Hopf lemma, for classical solutions to equations defined by linear, second-order, partial differential operators with non- negative characteristic form (degenerate-elliptic operators), in the presence of a second-order boundary condition of Ventcel type along the degeneracy locus of the principle symbol of the operator on the domain boundary. We apply these maximum principles to obtain uniqueness and a priori maximum principle estimates for classical solutions to boundary value and obstacle problems defined by these degenerate-elliptic operators, again in the presence of a second-order boundary condition, for Dirichlet or Neumann boundary conditions along the complement of the degeneracy locus. We also prove weak maximum principles and uniqueness for solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.