Almost classical solutions of Hamilton-Jacobi equations
The authors study the existence of almost classical solutions of fairly general Hamilton-Jacobi equations on open subsets of \(\mathbb R^d\) and on \(d\)-dimensional smooth manifolds for \(d \geq 2\). Let \(d \geq 2\) and let \(\Omega\) be an open subset of \(\mathbb R^d\). Let \(F : \mathbb R \times \Omega \times \mathbb R^{d} \rightarrow \mathbb R\) and \(u_{0} : \partial \Omega \rightarrow \mathbb R\) be continuous functions. We say that \(u : \overline \Omega \rightarrow \mathbb R\) is an almost classical solution of \(F(u(x),x,\nabla u(x))=0\) with Dirichlet condition \(u|_{ \partial \Omega} = u_{0}\) if {\parindent=4mm \begin{itemize}\item[{\(\bullet\)}] \(u(x) = u_{0}(x)\), \(x \in \partial \Omega\); \item[{\(\bullet\)}] for every \(x \in \Omega\), \(u\) is differentiable at \(x\) and \(F(u(x),x,\nabla u(x)) \leq 0\); \item[{\(\bullet\)}] \(u\) satisfies \(F(u(x),x,\nabla u(x))=0\) for almost every \(x \in \Omega\) in the sense of Lebesgue measure on \(\mathbb R^d\). \end{itemize}} The authors obtain the following. Theorem. Let \(d \geq 2\), let \(\Omega\) be an open subset of \(\mathbb R^d\) and let \(F : \mathbb R \times \Omega \times \mathbb R^{d} \rightarrow \mathbb R\) be a continuous function. Suppose that {\parindent=7mm \begin{itemize}\item[(A)] there exists a continuous function \(u_{0} : \overline \Omega \rightarrow \mathbb R\) such that \(u_{0}\) is \(C^{1}\)-smooth on \(\Omega\) and \(F(u_{0}(x),x,\nabla u_{0}(x)) \leq 0\) \((x \in \Omega)\); \item[(B)] for each compact set \(K \subseteq \Omega\) there exist an \(M_{K}>0\) and an \(\alpha_{K} > 0\) such that for every \(x \in K\), \(u \in [0,\alpha_{K}]\) and \(p \in \mathbb R ^{d}\) satisfying \(\|p\| \geq M_{K}\) we have \(F(u_{0}(x)+u,x,p) > 0\). \end{itemize}} Then there exists an almost classical solution \(u\) of \(F(u(x),x,\nabla u(x))=0\) with Dirichlet condition \(u|_{ \partial \Omega} = u_{0}\). Moreover, if \(u_{0}\) is \(C^{1}\)-smooth on \(\mathbb R^d\) then \(u\) can be extended to a differentiable function on \(\mathbb R^d\). Analogous results are obtained for similarly general Hamilton-Jacobi equations on \(d\)-dimensional smooth manifolds for \(d \geq 2\). As a very special corollary of their results, for every Riemannian manifold \(M\) of dimension \(\geq 2\) the authors prove the existence of a differentiable function \(u\) on \(M\) which satisfies the eikonal equation \(\| \nabla u(x) \|_{x}=1\) almost everywhere on \(M\).
scientific article; zbMATH DE number 5509271
scientific article