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Fundamental solutions and singular shocks in scalar conservation laws.

Emmanuel Chasseigne — 2003

Revista Matemática Complutense

We study the existence and non-existence of fundamental solutions for the scalar conservation laws u + f(u) = 0, related to convexity assumptions on f. We also study the limits of those solutions as the initial mass goes to infinity. We especially prove the existence of so-called Friendly Giants and Infinite Shock Solutions according to the convexity of f, which generalize the explicit power case f(u) = u. We introduce an extended notion of solution and entropy criterion to allow infinite shocks...

Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Guy Barles;  Emmanuel Chasseigne;  Cyril Imbert — 2011

Journal of the European Mathematical Society

This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...

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