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

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posted on 2023-06-08, 15:46 authored by Vincenzo Nesi, Mariapia Palombaro, Marcello Ponsiglione
This paper deals with higher gradient integrability for s-harmonic functions u with discontinuous coefficients s, i.e. weak solutions of div(s?u)=0 in dimension two. When s 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 s 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, s1 and s2, and study the higher integrability of the corresponding gradient field |?u| 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 Ei=s-1(si). We find the optimal integrability exponent of the gradient field corresponding to any pair {s1,s2} 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.

History

Publication status

  • Published

File Version

  • Published version

Journal

Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire

ISSN

0294-1449

Publisher

Elsevier

Issue

3

Volume

31

Page range

615-638

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2013-09-17

First Open Access (FOA) Date

2016-03-22

First Compliant Deposit (FCD) Date

2016-11-16

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