Published Paper

Inserted: 20 oct 2015
Last Updated: 8 jun 2016

Journal: J. Differ. Equations
Volume: 261
Number: 3
Pages: 1904-1932
Year: 2016
Links: Link to the published version

Abstract:

We propose notions of $\mathrm{BV}$ supersolutions to (the Dirichlet problem for) the 1-Laplace equation, the minimal surface equation, and equations of similar type. We then establish some related compactness and consistency results.

Our main technical tool is a generalized product of $\mathrm{L}^\infty$ divergence-measure fields and gradient measures of $\mathrm{BV}$ functions. This product crucially depends on the choice of a representative of the $\mathrm{BV}$ function, and the proofs of its basic properties involve results on one-sided approximation and fine (semi)continuity in the $\mathrm{BV}$ context.

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