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

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