Existence results for first-order Hamilton-Jacobi equations. (English) Zbl 0552.70012

Lions, Pierre-Louis

Ric. Mat. 32, 3-23 (1983).

This paper establishes some significant existence theorems for Hamilton- Jacobi equations. Let \(H(x,t,p)\) be a function of which it is assumed that it is at least continuous on \({\bar \Omega}\times {\mathbb{R}}\times {\mathbb{R}}^ N\), where \(\Omega\) is a bounded open set in \({\mathbb{R}}^ N\). This gives rise to the Dirichlet problem \((*)\quad H(x,u,Du)=0,\) \(u=\phi\) on \(\partial \Omega\), where u denotes a real-valued function from \({\bar \Omega}\) into \({\mathbb{R}}\) with gradient Du, and \(\phi\) is a given boundary condition. It is shown that, under appropriate conditions on H, if there exist Lipschitz, viscosity sub- and supersolutions of (*), then there exists a unique Lipschitz, viscosity solution of (*). [The terminology used in this statement is clearly explained in the text.]
The Cauchy problem is considered for some special Hamilton-Jacobi equations of the type \(\partial u/\partial t+\sum_{i,j}a_{ij}(x)(\partial u/\partial x_ i)(\partial u/\partial x_ j)=0\) in \({\mathbb{R}}^ n\times (0,+\infty)\) with \(u(x,0)=u_ 0(x),\) for all \(x\in {\mathbb{R}}^ n\), subject to the conditions \(a_{ij}\in W^{2,\infty}({\mathbb{R}}^ N)\), \(a_{ij}=a_{ji}\), \((a_{ij}(x))\geq 0\), for all \(x\in {\mathbb{R}}^ N\), and \(u_ 0\in W^{1,\infty}({\mathbb{R}}^ N)\). The results thus obtained have important implications with regard to optimal control theory and geometrical optics, and their relationship to previously established existence theorems is elucidated.

Reviewer: H.Rund

MSC:

70H05	Hamilton’s equations
35L60	First-order nonlinear hyperbolic equations
35F30	Boundary value problems for nonlinear first-order PDEs

Keywords:

viscosity solution; Cauchy problem; optimal control; geometrical optics