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Gradient integrability and rigidity results for two-phase conductivities in two dimensions
journal contribution
posted on 2023-06-08, 15:46 authored by Vincenzo Nesi, Mariapia Palombaro, Marcello PonsiglioneThis 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.
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Publication status
- Published
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- Published version
Journal
Annales de l'Institut Henri Poincaré C, Analyse Non LinéaireISSN
0294-1449Publisher
ElsevierExternal DOI
Issue
3Volume
31Page range
615-638Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2013-09-17First Open Access (FOA) Date
2016-03-22First Compliant Deposit (FCD) Date
2016-11-16Usage metrics
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