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Monotone approximation of differentiable convex functions with applications to general minimization problems

Petteri Harjulehto, Peter Hästö, Andrea Torricelli

TL;DR

This work addresses minimization of non-autonomous energy densities under minimal growth and coercivity assumptions and proves that minimizers satisfy the corresponding Euler–Lagrange relation or inequality in a distributional sense. The authors construct a robust regularization via a family $F_k$ of convex, $k$-Lipschitz densities using a ball-based Fenchel-conjugate extension, and show $F_k\in C^1$ with $F_k\to F$ as $k\to\infty$, preserving the normalization $F(0)=0=F'(0)$. They then apply Ekeland’s variational principle to obtain almost minimizers for the regularized problems, pass to the limit, and derive integrability properties $F^*(x, F'(x,\nabla u))\in L^1(\Omega)$ and $F'(x,\nabla u)\cdot\nabla u\in L^1(\Omega)$, together with a variational inequality $\int_\Omega F'(x,\nabla u)\cdot \nabla\eta\,dx \ge 0$ that yields the Euler–Lagrange identity/inequality for the limit minimizer. The framework extends autonomous results to broad non-autonomous growth models, including variable-exponent, double-phase, and generalized Orlicz energies, while avoiding extra regularization steps and providing a streamlined approximation approach. Overall, the paper delivers a principled, general approach to non-standard growth minimization and obstacle problems with rigorous limit passages.

Abstract

We study minimizers of non-autonomous energies with minimal growth and coercivity assumptions on the energy. We show that the minimizer is nevertheless the solution of the relevant Euler--Lagrange equation or inequality. The main tool is an extension result for convex $C^1$-energies.

Monotone approximation of differentiable convex functions with applications to general minimization problems

TL;DR

This work addresses minimization of non-autonomous energy densities under minimal growth and coercivity assumptions and proves that minimizers satisfy the corresponding Euler–Lagrange relation or inequality in a distributional sense. The authors construct a robust regularization via a family of convex, -Lipschitz densities using a ball-based Fenchel-conjugate extension, and show with as , preserving the normalization . They then apply Ekeland’s variational principle to obtain almost minimizers for the regularized problems, pass to the limit, and derive integrability properties and , together with a variational inequality that yields the Euler–Lagrange identity/inequality for the limit minimizer. The framework extends autonomous results to broad non-autonomous growth models, including variable-exponent, double-phase, and generalized Orlicz energies, while avoiding extra regularization steps and providing a streamlined approximation approach. Overall, the paper delivers a principled, general approach to non-standard growth minimization and obstacle problems with rigorous limit passages.

Abstract

We study minimizers of non-autonomous energies with minimal growth and coercivity assumptions on the energy. We show that the minimizer is nevertheless the solution of the relevant Euler--Lagrange equation or inequality. The main tool is an extension result for convex -energies.
Paper Structure (4 sections, 6 theorems, 51 equations)

This paper contains 4 sections, 6 theorems, 51 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Approximating convex functions
  3. Proof of the main result
  4. An alternative proof of differentiability

Key Result

Theorem 1.1

Let $\Omega\subset{\mathbb{R}^n}$, $n\geqslant 2$, be a bounded domain, $u_0\in W^{1,1}(\Omega)$ and $K\subset W^{1,1}_{u_0}$ be closed and convex. Assume that $F: \Omega\times {\mathbb{R}^n}\to [0,\infty)$ is Carathéodory, strictly convex, locally bounded and superlinear at infinity. Further we ass For any $\eta\in W^{1,\infty}(\Omega)$ with $\eta+K\subset K$, we have

Theorems & Definitions (13)

  • Theorem 1.1
  • Example 1.3
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5: Scorza Dragoni Theorem, p. 235, EkeT99
  • Remark 2.6
  • Theorem 3.1: Ekeland's variational principle, Theorem 1.4.1, page 29, Zal02
  • Remark 3.4
  • ...and 3 more