Uniqueness of diffeomorphic minimizers of L^p-mean distortion

2025/07/28 by Zhu, Yizhe
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2507.20597

Abstract

We study the L^p-mean distortion functionals, \cal E_p[f] = ∫_\mathbb Y K^p_f(z) dz, for Sobolev homeomorphisms f: \mathbb Y\xrightarrow\rm onto \mathbb X where \mathbb X and \mathbb Y are bounded simply connected Lipschitz domains, and f coincides with a given boundary map f₀ \colon ∂ \mathbb Y → ∂ \mathbb X. Here, K_f(z) denotes the pointwise distortion function of f. It is conjectured that for every 1 < p < ∞, the functional E_p admits a minimizer that is a diffeomorphism. We prove that if such a diffeomorphic minimizer exists, then it is unique.

Citations

• On the uniqueness of extremal mappings of finite distortion
• Energy-minimal Principles in Geometric Function Theory
• Extremal mappings of finite distortion and the Radon-Riesz property
• The Hopf-Laplace equation: harmonicity and regularity

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