Nonconvex degenerate Hamilton-Jacobi equations
In this paper, the equation \[ H(x,\nabla u)= 0\tag{1} \] in a bounded open domain \(\Omega\) of \(\mathbb{R}^N\) is studied. It is well known that if there is a strict \(C^1\) subsolution \(\psi\) of (1) and the zero-sublevel sets \({\mathcal Z}(x)\) of \(H(x,\cdot)\) are strictly starshaped with respect to \(\nabla\psi(x)\), then the solution of (1) coupled with a Dirichlet boundary condition is unique provided that it exists. The authors require only the existence of a \(C^1\) subsolution and the main achievement is a characterzation of the maximal solution, or more generally subsolution, of (1) satisfying a Dirichlet boundary condition by means of viscosity test functions. They weaken the structure conditions on the Hamiltonian: in particular the \(C^1\) subsolution \(\psi\) is not necessarily the null function and \({\mathcal Z}(x)\) does not equal \(\{\nabla\psi(x)\}\) on \({\mathcal K}= \{x\in\Omega: H(x, \nabla\psi(x))= 0\}\). Moreover, \(H\) does not verify any convexity assumption; the lack of this property is replaced by the requirement that a certain inequality holds on the boundary of \({\mathcal Z}(x)\) near \({\mathcal K}\). In the last section, a stability property for singular supersolutions is established.
scientific article; zbMATH DE number 2075716