arXiv subjects
Publications and source records attributed to Wojciech Górny.
In this paper we study least gradient functions in metric random walk spaces, which include as particular cases the least gradient functions on locally finite weighted connected graphs and nonlocal least gradient functions on $\mathbb{R}^N$. Assuming that a Poincaré inequality is satisfied, we study the Euler-Lagrange equation associated with the least gradient problem. We also prove the Poincaré inequality in a few settings.
We consider the two dimensional BV least gradient problem on an annulus with given boundary data $g \in BV(\partialΩ)$. Firstly, we prove that this problem is equivalent to the optimal transport problem with source and target measures located on the boundary of the domain. Then, under some admissibility conditions on the trace, we show that there exists a unique solution for the BV least gradient problem. Moreover, we prove some $L^p$ estimates on the corresponding minimal flow of the Beckmann problem, which implies directly $W^{1,p}$ regularity for the solution of the BV least gradient problem.
It is shown that solutions to the anisotropic least gradient problem for boundary data $f \in L^p(\partialΩ)$ lie in $L^{\frac{Np}{N-1}}(Ω)$; the exponent is shown to be optimal. Moreover, the solutions are shown to be locally bounded with explicit bounds on the rate of blow-up of the solution near the boundary in two settings: in the anisotropic case on the plane and in the isotropic case in any dimension.
We study existence of minimisers to the least gradient problem on a strictly convex domain in two settings. On a bounded domain, we allow the boundary data to be discontinuous and prove existence of minimisers in terms of the Hausdorff measure of the discontinuity set. Later, we allow the domain to be unbounded, prove existence of minimisers and study their properties in terms of the regularity of boundary data and the shape of the domain.
We study the planar least gradient problem with respect to an anisotropic norm $ϕ$ for continuous boundary data. We prove existence of minimizers for strictly convex domains $Ω$. Furthermore, we inspect the issue of uniqueness and regularity of minimizers only in terms of the modes of convexity of $ϕ$ and $Ω$. The results are independent from the regularity of $ϕ$.
We show existence of solutions to the least gradient problem on the plane for boundary data in $BV(\partialΩ)$. We also provide an example of a function $f \in L^1(\partialΩ) \backslash (C(\partialΩ) \cup BV(\partialΩ))$, for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.
Minimizers in the least gradient problem with discontinuous boundary data need not be unique. However, all of them have a similar structure of level sets. Here, we give a full characterization of the set of minimizers in terms of any one of them and discuss stability properties of an approximate problem.
We study two special cases of the planar least gradient problem. In the first one, the boundary conditions are imposed on a part of the strictly convex domain. In the second case, we impose the Dirichlet data on the boundary of a rectangle, an example of convex but not strictly convex domain. We show the existence of solutions and study their properties for particular cases of data.