# Convex integration and the ${L}^{p}$ theory of elliptic equations

Kari Astala; Daniel Faraco; László Székelyhidi Jr.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

- Volume: 7, Issue: 1, page 1-50
- ISSN: 0391-173X

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topAstala, Kari, Faraco, Daniel, and Székelyhidi Jr., László. "Convex integration and the $L^p$ theory of elliptic equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.1 (2008): 1-50. <http://eudml.org/doc/272298>.

@article{Astala2008,

abstract = {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 on the method of convex integration, as used by S. Müller and V. Šverák in [30] for the construction of counterexamples to regularity in elliptic systems, combined with the staircase type laminates introduced in [15].},

author = {Astala, Kari, Faraco, Daniel, Székelyhidi Jr., László},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

language = {eng},

number = {1},

pages = {1-50},

publisher = {Scuola Normale Superiore, Pisa},

title = {Convex integration and the $L^p$ theory of elliptic equations},

url = {http://eudml.org/doc/272298},

volume = {7},

year = {2008},

}

TY - JOUR

AU - Astala, Kari

AU - Faraco, Daniel

AU - Székelyhidi Jr., László

TI - Convex integration and the $L^p$ theory of elliptic equations

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

PY - 2008

PB - Scuola Normale Superiore, Pisa

VL - 7

IS - 1

SP - 1

EP - 50

AB - 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 on the method of convex integration, as used by S. Müller and V. Šverák in [30] for the construction of counterexamples to regularity in elliptic systems, combined with the staircase type laminates introduced in [15].

LA - eng

UR - http://eudml.org/doc/272298

ER -

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