The Probabilistic Brosamler Formula for Some Nonlinear Neumann Boundary Value Problems Governed by Elliptic Possibly Degenerate Operators

This paper concerns with boundary value problems as { L u+a0u = f in Ω, < ∇u, γ > +c0|u|m−1u = g on ∂Ω, where L is an elliptic possibly degenerate second order operator, a0, c0 are positive function, γ is an oblique exterior vector and m 1 . By means of some arguments close to the Dynamics Programming we prove that the viscosity solution admits a representation formula that can be considered as an extension of probabilistic Brosamler formula of linear Neumann boundary value problems governed by uniformly elliptic operators. Although other generalizations are possible, by simplicity we limit this contribution to the presence of nonlinear terms exclusively on the boundary of the domain. We emphasize that any uniforme ellipticity assumption is required in the paper. Mathematics subject classification (2010): 35D40, 35J65, 60J60.