Existence and uniqueness of minimizers of general least gradient problems

Abstract

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems -under certain sharp conditions -for minimizers of the general least gradient problemand ϕ(x, ξ) is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the ξ variable. In particular we prove that if a ∈ C 1,1 (Ω) is bounded away from zero, then minimizers of the weighted least gradient problem inf u∈BV f Ω a|Du| are unique in BV f (Ω). We construct counterexamples to show that the regularity assumption a ∈ C 1,1 is sharp, in the sense that it can not be replaced by a ∈ C 1,α (Ω) with any α < 1.

General least gradient problems with obstacle

“…Proof. The proof is similar to the proof of Lemma 2.7 in [6] and we present it here for the sake of completeness. Given g ∈ L 1 (∂Ω; H n−1 ), define…”

“…Theorem 1.11 (Holder Regularity). Suppose that φ : Ω × R n −→ R satisfies C1-C5 and let Ω be a bounded, open subset of R n with C 2 boundary which the signed distance d(·) to ∂Ω satisfies the relation (6). Assume f ∈ C 0,α (∂Ω), and ψ ∈ C 0,α/2 for some 0 < α ≤ 1.…”

“…Lemma 1.1 (Lemma 2.2 in [6]). Let A ⊂ R n be a Borel set and E 1 , E 2 ⊂ R n be of locally finite perimeter with respect φ. Then…”

“…Let f ∈ BV (R n ) and ψ ∈ W 1,1 (Ω), and consider the obstacle least gradient problem Functions of least gradient was first studies by P. Sternberg, W. Graham, and W. Ziemer in [18], and the results were later extended to least gradient problems with obstacle in [19]. Due to important applications of least gradient problems in conductivity imaging, such problems have received an extensive attention in the past decade (see [5,6,7,8,9,10,11,12,13,14,20]). In this paper we will study existence and structure of minimizers, uniqueness, and regularity of minimizers of the general obstacle least gradient problem (3).…”

“…Since BV (R n ) ֒→ L 1 loc , F is coercive in BV (R n ) (a consequence of C 1 ) and weakly lower semicontinuous (see [6] for more details), it follows from standard arguments that {v n } ∞ n=1 has a subsequence converging strongly in…”