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Neumann type boundary conditions for Hamilton-Jacobi equations. (English) Zbl 0599.35025
The concept of viscosity solutions to first order fully nonlinear partial differential equations, first introduced by M. G. Crandall and the author [Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)] is extended here to problems with Neumann boundary conditions. This problem naturally arises in deterministic optimal control with state constraints and elsewhere. Since viscosity solutions are not required to have derivatives of any sort the problem is to make sense out of a condition like $$\partial u/\partial v=0.$$
The following definition is given. Let $$u\in C({\bar \Omega})$$. Then u is a viscosity subsolution of $$H(x,u,Du)=0$$ in $$\Omega$$, $$\partial u/\partial v=0$$ on $$\partial \Omega$$ if for every $$\phi \in C^ 1({\bar \Omega})$$, if u-$$\phi$$ has a local max at $$x_ 0$$ in $${\bar \Omega}$$, then $$H(x_ 0,u(x_ 0),D\phi (x_ 0))\leq 0$$ if $$x_ 0\in \Omega$$, $$H(x_ 0,u(x_ 0)$$, $$D\phi (x_ 0))\leq 0$$ if $$x_ 0\in \partial \Omega$$ and $$\partial \phi /\partial v(x_ 0)\geq 0$$. For u to be a viscosity supersolution replace max by min and reverse the inequalities. A solution is both a sub and supersolution. The theory of viscosity solutions is then extended to this case.
Applications to control theory and differential games and an ergodic result are also presented.
Reviewer: E.Barron

##### MSC:
 35F20 Nonlinear first-order PDEs 35D05 Existence of generalized solutions of PDE (MSC2000) 35F30 Boundary value problems for nonlinear first-order PDEs
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##### References:
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