On functions with given boundary data and convex constraints on the gradient

2022/09/03 by Camilla Brizzi, Brizzi, Camilla
Computer Science · Mathematics · #49K24 49N15 49N60 35R70 58B20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2209.01462

openalex publication_date 2022/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ℝ^d be an open set. Given a boundary datum g on ∂Ω and a function K: Ω \toK, the family of all compact convex subsets of ℝ^d, we prove the existence of functions u:Ω→ℝ such that u=g on ∂Ω and ∇ u(x)∈ K(x) a.e. and we investigate the regularity of such solutions on the set U ⊂ Ω of points at which they all coincide.

Related

• The planar Least Gradient problem in convex domains, the case of continuous datum
• The planar Least Gradient problem in convex domains
• Global Lipschitz continuity for minima of degenerate problems
• An inequality characterizing convex domains
• Existence of minimisers in the least gradient problem for general boundary data