Viscosity solutions of Hamilton-Jacobi equations at the boundary

Abstract

When considering classical solutions of boundary value problems for nonlinear first-order scalar partial differential equations, one knows that there are parts of the boundary of the region under consideration where one cannot specify data and would not expect to require data in order to prove uniqueness. Of course, classical solutions of such problems rarely exist in the large owing to the crossing of characteristics. The theory of a sort of generalized solution—called "viscosity solutions"—for which good existence and uniqueness theorems are valid has been developed over the last few years. In this note we give some results concerning parts of the boundary on which one need not know (prescribe) viscosity solutions to be able to prove comparison (and hence uniqueness) results. In this context, this amounts to identifying boundary points with the property that solutions in the interior which are continuous up to the boundary are also viscosity solutions at the boundary point. Examples indicating the sharpness of the results are given.