Hopf-type estimates for viscosity solutions to concave-convex Hamilton-Jacobi equations
The Cauchy problem for the simplest Hamilton-Jacobi equation NEWLINE\[NEWLINE{\partial u\over\partial t}+ f\Biggl({\partial u\over\partial x}\Biggr)= 0\quad\text{in }D= \{t>0, x\in\mathbb{R}^n\},\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0, x)= \Phi(x)\quad\text{on }\{t= 0, x\in\mathbb{R}^n\}\tag{2}NEWLINE\]NEWLINE is considered. If \(f\) is a (concave-convex) function given by a special representation \(f(p',p'')= H_1(p')+ H_2(p'')\) on \(\mathbb{R}^{n_1}\times \mathbb{R}^{n_2}\), with \(n_1+ n_2= n\), \(H_1\) concave and \(H_2\) convex, and \(\Phi\) is uniformly continuous, Bardi and Faggian have recently found explicit pointwise upper and lower bounds of Hopf type for the (unique) viscosity solution of (1)--(2). Using the ``index of non-convexity'' and a classical minimax theorem, \textit{M. Bardi} and \textit{S. Faggian} [SIAM J. Math. Anal. 29, No. 5, 1067-1086 (1998; Zbl 0914.35026)] have thus obtained a representation formula for the viscosity solution, at least for short times (when the upper and lower bounds are equal). In this note, the author extends Bardi-Faggian's estimates (loc. cit.) to the case of general concave-convex Hamiltonians. Some remarks on their estimates simplified in a particular case where \(\Phi\) is Lipschitz continuous are given.
scientific article; zbMATH DE number 1660526