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].

@article{ASNSP_2008_5_7_1_1_0, author = {Astala, Kari and Faraco, Daniel and Sz\'ekelyhidi Jr., L\'aszl\'o}, title = {Convex integration and the $L^p$ theory of elliptic equations}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {1--50}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 7}, number = {1}, year = {2008}, zbl = {1164.30014}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2008_5_7_1_1_0/} }

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 SP - 1 EP - 50 VL - 7 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2008_5_7_1_1_0/ LA - en ID - ASNSP_2008_5_7_1_1_0 ER -

%0 Journal Article %A Astala, Kari %A Faraco, Daniel %A Székelyhidi Jr., László %T Convex integration and the $L^p$ theory of elliptic equations %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2008 %P 1-50 %V 7 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2008_5_7_1_1_0/ %G en %F ASNSP_2008_5_7_1_1_0

Astala, Kari; Faraco, Daniel; Székelyhidi Jr., László. Convex integration and the $L^p$ theory of elliptic equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 1, pp. 1-50. http://archive.numdam.org/item/ASNSP_2008_5_7_1_1_0/

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