Published Paper
Inserted: 31 jan 2019
Last Updated: 31 jan 2019
Journal: Nonlinear Analysis
Volume: 178
Pages: 86--109
Year: 2019
Doi: 10.1016/j.na.2018.07.011
Abstract:
We revisit the question of existence and regularity of minimizers to the weighted least gradient problem with Dirichlet boundary condition
\[
\inf\left\{\int_{\Omega}\textrm{a}(x)
Du
:\, u\in BV(\Omega),\; u|_{\partial\Omega}=g\right\},
\]
where $g\in C(\partial\Omega)$, and $\textrm{a}\in C^2(\bar{\Omega})$ is a weight function that is bounded away from zero. Under suitable geometric conditions on the domain $\Omega\subset\mathbb{R}^n$, we construct continuous solutions of the above problem for any dimension $n\geq 2$, by extending the technique in [P. Sternberg, G. Williams, and W.P. Ziemer, Existence, uniqueness, and regularity for functions of least gradient, J. Reine Angew. Math., 430 (1992), pp. 35-60.] to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfaces in the conformal metric $\textrm{a}^{2/(n-1)}I_n$. This result complements the approach in [R.L. Jerrard, A. Moradifam, and A.I. Nachman, Existence and uniqueness of minimizers of general least gradient problems, J. Reine Angew. Math., 734 (2018), pp. 71-97] since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities. The proof relies on an application of a strict maximum principle for sets with area minimizing boundary established by Leon Simon in [L. Simon, A strict maximum principle for area minimizing hypersurfaces, J. Differential Geom., 26 (1987), pp. 327-335].
Keywords: least gradient problem, weighted perimeter, barrier condition
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