Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

If u : Ω ⊂ R d → X is a harmonic map valued in a metric space X and E : X → R is a convex function, in the sense that it generates an EVI 0 -gradient flow, we prove that the pullback E ∘ u : Ω → R is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature res...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2024-03, Vol.63 (2), Article 54
Main Authors: Lavenant, Hugo, Monsaingeon, Léonard, Tamanini, Luca, Vorotnikov, Dmitry
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Gradient flow
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Metric space
Riemann manifold
Systems Theory
Theoretical
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Summary:If u : Ω ⊂ R d → X is a harmonic map valued in a metric space X and E : X → R is a convex function, in the sense that it generates an EVI 0 -gradient flow, we prove that the pullback E ∘ u : Ω → R is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on X , while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the L q norm of E ∘ u on ∂ Ω controls the L p norm of E ∘ u in Ω for some well-chosen exponents p ≥ q , including the case p = q = + ∞ . In particular, our results apply when E is a geodesically convex entropy over the Wasserstein space, and thus settle some conjectures of Brenier (Optimal transportation and applications (Martina Franca, 2001), volume 1813 of lecture notes in mathematics, Springer, Berlin, pp 91–121, 2003).
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-024-02662-3