Almost continuous solutions of geometric Hamilton--Jacobi equations
The Hamilton-Jacobi time dependent equation \(u_t+H(x,Du)=0\) in \(\mathbb{R}^N \times (0,+\infty)\), where \(H\) has constant sign and is homogeneous in the second argument, but does not verify any convexity or uniform continuity assumptions with respect to the state variable, is studied. The author is interested in discontinuous viscosity solutions verifying certain initial conditions with discontinuous data. More precisely, the locally bounded solutions \(u\) verifying \[ \limsup_{(x,t) \to(x_0,0)}u^* (x,t)\leq u_0^* (x_0),\;\liminf_{(x,t) \to(x_0,0)} u_*(x,t)\geq u_{0_*}(x_0) \] for any \(x_0\in \mathbb{R}^N\), where \(u_0\) is a locally bounded initial datum and \(u^*(u_*)\) denotes the upper (lower) semicontinuous envelope of \(u\), are seeked. The aim of this paper is to give representation formulae as well as uniqueness and stability results.
scientific article; zbMATH DE number 1912450
scientific article