Calculus of Variations and Geometric Measure Theory
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L. Beck - M. Bulíček - J. Málek - E. Suli

On the existence of integrable solutions to nonlinear elliptic systems and variational problems with linear growth

created by beck on 09 Jan 2018



Inserted: 9 jan 2018
Last Updated: 9 jan 2018

Journal: Arch. Ration. Mech. Anal.
Volume: 225
Number: 2
Pages: 717-769
Year: 2016
Doi: 10.1007/s00205-017-1113-4

ArXiv: 1601.01907 PDF


We investigate the properties of certain elliptic systems leading, a~priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic Uhlenbeck structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously as in the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed.

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