Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
7222501 | Nonlinear Analysis: Theory, Methods & Applications | 2019 | 24 Pages |
Abstract
We revisit the question of existence and regularity of minimizers to the weighted least gradient problem with Dirichlet boundary condition infâ«Î©a(x)|Du|:uâBV(Ω),u|âΩ=g,where gâC(âΩ), and aâC2(ΩÌ) is a weight function that is bounded away from zero. Under suitable geometric conditions on the domain ΩâRn, we construct continuous solutions of the above problem for any dimension nâ¥2, by extending the Sternberg-Williams-Ziemer technique (Sternberg et al., 1992) to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfaces in the conformal metric a2â(nâ1)In. This result complements the approach in Jerrard et al. (2018) 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 Simon (1987).
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Authors
Andres Zuniga,