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First order Hamilton-Jacobi equations with integro-differential terms. II: Existence of viscosity solutions. (Equations d’Hamilton-Jacobi du premier ordre avec termes intégro- différentiels. II: Existence des solutions de viscosité.) (French) Zbl 0742.45005
Both papers are devoted to the equation $$F(x,u,Du,Bu)=0$$, where $$u=u(x)\in\mathbb{R}$$ ($$x\in\mathbb{R}^ N$$) is an unknown function, $$Du$$ its Fréchet differential, and $Bu(x)=\int\int^ 1_ 0 (Du(x+tz)- Du(x),z)dt d\mu_ x(z)$ is the Lévy integro-differential operator with a measure $$\mu_ x(z)$$ ($$x\in\mathbb{R}^ N$$, $$z\in\mathbb{R}^ N$$) satisfying $\sup_ x\int_{| z|\geq\varepsilon}| z| d\mu_ x(z)<\infty, \qquad \sup_ x\int_{| z|\leq\varepsilon}| z| d\mu_ x(z)\to 0,$ for every $$\varepsilon>0$$, $$\varepsilon\to 0+$$. As the crucial concept, the sub- (super-) solution is introduced by the requirement that the inequality $$F(x_ 0,u(x_ 0),D\varphi(x_ 0),B\varphi(x_ 0))\leq 0$$ ($$\geq 0$$) should be valid for every function $$\varphi$$ such that $$u-\varphi$$ has a global maximum (minimum) at $$x_ 0$$. The inequality $$u\leq v$$ between sub- and super-solutions together with appropriate regularization given by certain sup- and inf-convolutions yield the uniqueness and comparison theorems for solutions.
In the second part, the Perron method is used for the proof of the existence. At the end, the methods are applied to the above equation with a different operator $Bu(x)=\int(e^{u(x)}-e^{u(x+z)})d\mu_ x(z),$ where $$\mu_ x(z)$$ is a measure of compact support.
Reviewer: J.Chrastina (Brno)

##### MSC:
 45K05 Integro-partial differential equations 70H20 Hamilton-Jacobi equations in mechanics
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##### References:
 [1] M.G. Crandall, P.L. Lions. On existence and uniqueness of solutions of Hamilton-Jacohi Pquations. (1984). Eonlinear -4nal. T.M.A. [2] Ishii H., On existence and uniqueness of viscosity solutions of fully non linear second-order elliptic P.D. E’s. [3] DOI: 10.1215/S0012-7094-87-05521-9 · Zbl 0697.35030 [4] DOI: 10.1080/03605308308820301 · Zbl 0716.49023 [5] Evans L. C., Indiana Univ. math. J. 33 pp 797– (1984) [6] Crandall M. G., J. Math. Soc. [7] DOI: 10.1016/0022-0396(85)90084-1 · Zbl 0506.35020
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