Quasiderivative method for derivative estimates of solutions to degenerate elliptic equations

Details Quasiderivative method for derivative estimates of solutions to degenerate elliptic equations Quasiderivative method for derivative estimates of solutions to degenerate elliptic equations

2011-12-24

arxiv.org/abs/1112.5689v1

Mathematics

Probability

35 pages

Scientific paper

We introduce the concept of quasiderivative and give an example constructed by random time change, Girsanov's Theorem and Levy's Theorem. As an application, we investigate the smoothness and estimate the derivatives up to second order for probabilistic solutions to Dirichlet problems for linear degenerate elliptic partial differential equations of second order, under the assumption of non-degeneracy with respect to the normal to the boundary and an interior condition to control the moments of quasiderivatives, which is weaker than non-degeneracy. The probabilistic solution is the unique solution to the associated Dirichlet problem.

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