Robin problem with measure data and singular nonlinearities on the boundary

TL;DR

This paper studies a Robin boundary value problem with measure data and singular boundary nonlinearities, establishing existence, regularity, and uniqueness of solutions, and providing a stochastic representation as a generalized nonlinear Feynman-Kac formula.

Contribution

It introduces a framework for solving Robin problems with measure data and boundary singularities, including a stochastic representation and regularity results.

Findings

Existence of positive renormalized solutions.

Stochastic representation as a nonlinear Feynman-Kac formula.

Additional regularity and uniqueness results.

Abstract

We consider the Robin problem for a uniformly elliptic divergence operator with measure data on the right-hand side of the equation and an absorption term on the boundary involving blowing up terms. We prove the existence of a positive renormalized solution and provide it stochastic representation, which can be viwed as a generalized nonlinear Feynman-Kac formula. From this representation we derive some additional regularity results for the solution and some uniqueness results.