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On continuity properties of monotone operators beyond the natural domain of definition

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Abstract

We study boundary value problems associated to a nonlinear elliptic system of partial differential equations. The leading second order elliptic operator provides L 2-coerciveness and has at most linear growth with respect to the gradient. Incorporating properties of Lipschitz approximations of Sobolev functions we are able to show that the problems in consideration are well-posed, in the sense of Hadamard, in W 1,p for all \({p \in (p_0,2]}\) for certain \({p_0 \in [3/2,2)}\) . For simplicity we restrict ourselves to the Dirichlet problem.

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  1. Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Miroslav Bulíček

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  1. Miroslav Bulíček
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Correspondence to Miroslav Bulíček.

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Bulíček, M. On continuity properties of monotone operators beyond the natural domain of definition. manuscripta math. 138, 287–298 (2012). https://doi.org/10.1007/s00229-011-0506-1

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  • DOI: https://doi.org/10.1007/s00229-011-0506-1

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