Gradient integrability and rigidity results for two-phase conductivities in two dimensions

  • Vincenzo Nesi

    Dipartimento di Matematica “G. Castelnuovo”, Sapienza, Universitá di Roma, Piazzale A. Moro 2, 00185 Roma, Italy
  • Mariapia Palombaro

    Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, via Vetoio, Coppito, L'Aquila, Italy
  • Marcello Ponsiglione

    Dipartimento di Matematica “G. Castelnuovo”, Sapienza, Universitá di Roma, Piazzale A. Moro 2, 00185 Roma, Italy

Abstract

This paper deals with higher gradient integrability for -harmonic functions with discontinuous coefficients , i.e. weak solutions of in dimension two. When 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 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.

Cite this article

Vincenzo Nesi, Mariapia Palombaro, Marcello Ponsiglione, Gradient integrability and rigidity results for two-phase conductivities in two dimensions. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 3, pp. 615–638

DOI 10.1016/J.ANIHPC.2013.05.002