This paper deals with higher gradient integrability for σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of in dimension two. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. When only the ellipticity is fixed and σ is otherwise unconstrained, the optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries.We focus also on two-phase conductivities, i.e., conductivities assuming only two matrix values, and , and study the higher integrability of the corresponding gradient field for this special but very significant class. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement described by the sets . We find the optimal integrability exponent of the gradient field corresponding to any pair of elliptic matrices, i.e., the worst among all possible microgeometries.We also treat the unconstrained case when an arbitrary but finite number of phases are present.
Keywords: Beltrami system, Quasiconformal mappings, Elliptic equations, Composites, Gradient integrability
@article{AIHPC_2014__31_3_615_0, author = {Nesi, Vincenzo and Palombaro, Mariapia and Ponsiglione, Marcello}, title = {Gradient integrability and rigidity results for two-phase conductivities in two dimensions}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {615--638}, publisher = {Elsevier}, volume = {31}, number = {3}, year = {2014}, doi = {10.1016/j.anihpc.2013.05.002}, mrnumber = {3208457}, zbl = {1298.30018}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.05.002/} }
TY - JOUR AU - Nesi, Vincenzo AU - Palombaro, Mariapia AU - Ponsiglione, Marcello TI - Gradient integrability and rigidity results for two-phase conductivities in two dimensions JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 615 EP - 638 VL - 31 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.05.002/ DO - 10.1016/j.anihpc.2013.05.002 LA - en ID - AIHPC_2014__31_3_615_0 ER -
%0 Journal Article %A Nesi, Vincenzo %A Palombaro, Mariapia %A Ponsiglione, Marcello %T Gradient integrability and rigidity results for two-phase conductivities in two dimensions %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 615-638 %V 31 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.05.002/ %R 10.1016/j.anihpc.2013.05.002 %G en %F AIHPC_2014__31_3_615_0
Nesi, Vincenzo; Palombaro, Mariapia; Ponsiglione, Marcello. Gradient integrability and rigidity results for two-phase conductivities in two dimensions. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, pp. 615-638. doi : 10.1016/j.anihpc.2013.05.002. http://www.numdam.org/articles/10.1016/j.anihpc.2013.05.002/
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