Regularity of \(C^ 1\) solutions of the Hamilton-Jacobi equation.
Under some regularity conditions for a Hamiltonian, it is shown that \(C^1\) solutions of the Hamilton-Jacobi equations are necessarily \(C^{1,1}\). Similar results were obtained before by \textit{P.L. Lions} [Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, 69, Boston-London-Melbourne: Pitman Advanced Publishing Program (1982; Zbl 0497.35001)], for the particular case when the Hamiltonian depends on the fibre coordinates only. The results obtained by the author are stronger, in Theorem 3.2, it is shown that the Lipschitz constant of the derivative of the solution of the Hamilton-Jacobi equation can be locally controlled. This implies some results regarding the compactness of sets of \(C^1\) solutions of Hamilton-Jacobi equation.
scientific article; zbMATH DE number 2124650
scientific article