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Quasiconvexity in the Riemannian setting

Aurora Corbisiero, Chiara Leone, Carlo Mantegazza

TL;DR

The paper addresses extending the classical Morrey–Acerbi–Fusco theory of quasiconvexity and lower semicontinuity from Euclidean spaces to the Riemannian setting. It introduces a Riemannian notion of quasiconvexity for functions on the bundle $\mathscr{L}(TM,\mathbb{R}^m)$, employing an exponential-map-based transport and a vanishing correction term to compare perturbations at a fixed tangent space. The authors prove an equivalence: $f$ is Riemannian quasiconvex if and only if the integral functional $F(u,\Omega)=\int_{\Omega} f(du)\,d\mu$ is sequentially lower semicontinuous with respect to the weak$^*$ topology on $W^{1,\infty}(\Omega,\mathbb{R}^m)$ for every open bounded $\Omega\subset M$, effectively generalizing Acerbi–Fusco to the Riemannian context. They also outline future directions, including extensions to $W^{1,p}$, integration with manifold-valued map results, and the development of additional convexity notions on manifolds.

Abstract

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional \[ F(u, Ω) = \int_Ω f(du) \, dμ\] with respect to the weak$^*$ topology of $W^{1,\infty}(Ω, \mathbb{R}^m)$, for every bounded open subset $Ω\subseteq M$.

Quasiconvexity in the Riemannian setting

TL;DR

The paper addresses extending the classical Morrey–Acerbi–Fusco theory of quasiconvexity and lower semicontinuity from Euclidean spaces to the Riemannian setting. It introduces a Riemannian notion of quasiconvexity for functions on the bundle , employing an exponential-map-based transport and a vanishing correction term to compare perturbations at a fixed tangent space. The authors prove an equivalence: is Riemannian quasiconvex if and only if the integral functional is sequentially lower semicontinuous with respect to the weak topology on for every open bounded , effectively generalizing Acerbi–Fusco to the Riemannian context. They also outline future directions, including extensions to , integration with manifold-valued map results, and the development of additional convexity notions on manifolds.

Abstract

We introduce a notion of quasiconvexity for continuous functions defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold and , naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional with respect to the weak topology of , for every bounded open subset .
Paper Structure (4 sections, 5 theorems, 63 equations)

This paper contains 4 sections, 5 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Quasiconvexity in the Riemannian setting
  3. Quasiconvexity and semicontinuity
  4. Some remarks and possible research directions

Key Result

Theorem 1.2

Let $f: \mathbb{R}^n\times\mathbb{R}^m\times\mathbb{R}^{{n\times m}}\to \mathbb{R}$ be a Carathéodory function satisfying for every $x\in \mathbb{R}^n$, $s\in \mathbb{R}^m$ and $\xi\in \mathbb{R}^{{n\times m}}$, where $a:{{\mathbb R}}^n\to{{\mathbb R}}$ is nonnegative and locally summable and $b:{{\mathbb R}}^n\times\mathbb{R}^{{n\times m}}\to{{\mathbb R}}$ is nonnegative and locally bounded. The

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2: Acerbi--Fusco
  • Theorem 1.3: Acerbi--Fusco
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 1 more