TY - JOUR

T1 - Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

AU - Lavenant, H

AU - Monsaingeon, L

AU - Tamanini, Luca

AU - Vorotnikov, D

PY - 2024

Y1 - 2024

N2 - If $u:\Omega \subset \mathbb{R}^d \to X$ is a harmonic map valued in a metric space $X$ and $E : X \to \mathbb{R}$ is a convex function, in the sense that it generates an EVI-gradient flow, we prove that the pullback $E \circle u : \Omega \to \mathbb{R}$ 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 $E \circ u$ on $\partial\Omega$ controls the $L^p$ norm of $E \circ u$ in $\Omega$ for some well-chosen exponents $p \geq q$, including the case $p=q=+\infty$. 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).

AB - If $u:\Omega \subset \mathbb{R}^d \to X$ is a harmonic map valued in a metric space $X$ and $E : X \to \mathbb{R}$ is a convex function, in the sense that it generates an EVI-gradient flow, we prove that the pullback $E \circle u : \Omega \to \mathbb{R}$ 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 $E \circ u$ on $\partial\Omega$ controls the $L^p$ norm of $E \circ u$ in $\Omega$ for some well-chosen exponents $p \geq q$, including the case $p=q=+\infty$. 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).

KW - EVI gradient flow

KW - Harmonic maps

KW - Ishihara property

KW - Metric geometry

KW - EVI gradient flow

KW - Harmonic maps

KW - Ishihara property

KW - Metric geometry

UR - http://hdl.handle.net/10807/267514

UR - https://link.springer.com/article/10.1007/s00526-024-02662-3

U2 - 10.1007/s00526-024-02662-3

DO - 10.1007/s00526-024-02662-3

M3 - Article

SN - 0944-2669

VL - 63

SP - N/A-N/A

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

ER -