Least Gradient Problems with Neumann Boundary Condition

الأصل · EN

We study existence of minimizers of the least gradient problem ∈ BVg ∫Ωφ(x, Dv), where BVg={v ∈ BV(Ω): ∫∂ Ωgv=1}, φ(x,p): Ω× ⁿ → is a convex, continuous, and homogeneous function of degree 1 with respect to the p variable, and g satisfies the comparability condition ∫∂ Ω g dS=0. We prove that for every 0 ≡ g ∈ L∞(∂ Ω) there are infinitely many minimizers in BV(Ω). Moreover there exists a divergence free vector field T∈ (L∞(Ω))ⁿ that determines the structure of level sets of all minimizers, i.e. T determines Du/|Du|, |Du|- a.e. in Ω, for every minimizer u. We also prove some existence results for general 1-Laplacian type equations with Neumann boundary condition. A numerical algorithm is presented that simultaneously finds T and a minimizer of the above least gradient problem. Applications of the results in conductivity imaging are discussed.