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

Abstract

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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].

How to cite

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Astala, 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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