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Relaxation of the incompressible porous media equation

László Székelyhidi Jr — 2012

Annales scientifiques de l'École Normale Supérieure

It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular T 4 configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding T 4 configurations. We then use this to construct weak solutions to the unstable interface problem (the...

Convex integration and the L p theory of elliptic equations

Kari Astala;  Daniel Faraco;  László Székelyhidi Jr. — 2008

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

This paper deals with the L p theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general L p theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions are based...

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