A double-phase Neumann problem with $p=1$

Abstract

We study a double-phase Neumann problem with non-homogeneous boundary conditions, where the lowest exponent $p$ is equal to 1. The existence of a solution is established as the limit of solutions to corresponding double-phase problems with $p>1$. We also provide a variational characterization of the limit.

18 pages

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• Double phase image restoration
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• The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data
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• The double phase Dirichlet problem when the lowest exponent is equal to 1