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Quasiconformal solutions to certain first order systems and the proof of a conjecture of G. W. Milton. (English) Zbl 0869.35019
Summary: We study fine properties of the gradient of the solution to the conductivity equation \[ \text{div}(\sigma(x)\nabla u(x))=0 \] in bounded domains. Our analysis is restricted to dimension two and it is concerned with merely measurable, elliptic coefficients. We establish sharp results on the higher order integrability of the modulus of the gradient and of the inverse of the modulus of the gradient of the solution with the aid of recent advances in the theory of quasiconformal mappings due to Astala, Eremenko and Hamilton. We also consider the first-order system associated to the second-order elliptic equation, hence defining the map \(w=(u,\widetilde u)\) and we isolate a class of Dirichlet boundary data on the function \(u\) which guarantees the quasiconformality of the mapping \(w\). This leads in particular to a geometrical characterization of the electrostatic energy. We make use of results about the critical points of solutions of elliptic equations due to Alessandrini and Alessandrini and Magnanini.

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35J15 Second-order elliptic equations
30C62 Quasiconformal mappings in the complex plane
35F05 Linear first-order PDEs
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