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A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type. (English) Zbl 0644.35017

The motivation for this paper is the problem a(x)Du\(\cdot Du-b(x)\cdot Du=0\) in \(\Omega\), \(u=g\) on the boundary of \(\Omega\), where it is assumed that a(x)\(\xi\cdot \xi \geq \theta | \xi |^ 2\), for some \(\theta >0\), and b(x)\(\cdot D\psi (x)\geq 1\), for some \(\psi \in C^ 1({\bar \Omega})\). The author then considers the more general problem \(H(x,u,Du)=0\) in \(\Omega\) and proves, under hypotheses on H which are satisfied by the model problem that a viscosity solution is unique.
Reviewer: E.Barron

MSC:

35F30 Boundary value problems for nonlinear first-order PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35L60 First-order nonlinear hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:

[1] M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), no. 2, 487 – 502. · Zbl 0543.35011
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