[Paper Review] Lipschitz regularity for orthotropic functionals with nonstandard growth conditions

Pierre Bousquet, Lorenzo Brasco|arXiv (Cornell University)|Oct 9, 2018

Nonlinear Partial Differential EquationsMathematics24 references45 citations

TL;DR

This paper establishes the local Lipschitz regularity of bounded local minimizers for orthotropic functionals with nonstandard growth conditions, where the growth exponents $ p_i $ satisfy $ 2 \leq p_1 \leq \cdots \leq p_N $, without any upper bound on the ratio $ p_N/p_1 $. The proof relies on a novel iterative Moser-type scheme combined with Caccioppoli-type inequalities and a recursive gain of integrability for the gradient, ultimately yielding $ \nabla U \in L^\infty_{\text{loc}}(\Omega) $ for any bounded local minimizer $ U $. This resolves a long-standing gap in the regularity theory for degenerate, anisotropic, nonstandard growth problems.

ABSTRACT

We consider a model convex functional with orthotropic structure and super-quadratic nonstandard growth conditions. We prove that bounded local minimizers are locally Lipschitz, with no restrictions on the ratio between the highest and the lowest growth rate.

Motivation & Objective

• . The paper aims to establish local Lipschitz regularity for bounded local minimizers of orthotropic functionals with nonstandard growth conditions.
• It seeks to close a significant gap in the regularity theory for degenerate, anisotropic PDEs with variable growth.
• The objective is to prove that $ \nabla U \in L^\infty_{\text{loc}}(\Omega) $ for any bounded local minimizer $ U $, even when $ p_N/p_1 $ is arbitrarily large.
• It aims to extend previous results that required restrictive bounds on the ratio of highest to lowest growth exponents.
• The work provides a new methodological framework for handling the strong degeneracy inherent in such functionals.

Proposed method

• . The authors employ a regularized problem and derive Caccioppoli-type inequalities to control the energy of the gradient.
• They develop a recursive iterative scheme inspired by Moser's method to gain higher integrability of the gradient.
• A key step involves constructing a sequence of exponents $ \{\beta_j^\ell\} $ that satisfy a recursive system to control the growth of the gradient.
• The method uses weighted Sobolev inequalities and carefully chosen test functions to derive uniform bounds independent of the exponent ratio.
• The proof relies on a novel iterative gain of integrability, where the exponent sequence is shown to stabilize at $ q_{j-2} $ after finitely many steps.
• A crucial component is the use of a modified version of the Michael–Simon Sobolev inequality in a higher-dimensional manifold setting, adapted to the orthotropic structure.

Experimental results

Research questions

• RQ1 . Can bounded local minimizers of orthotropic functionals with nonstandard growth be shown to be locally Lipschitz continuous without restrictions on the ratio $ p_N/p_1 $?
• RQ2 What techniques can overcome the strong degeneracy of such functionals to achieve higher regularity?
• RQ3 Is it possible to generalize the method of [4] to the non-uniform, anisotropic case with $ p_1 < p_N $?
• RQ4 Can a recursive gain of integrability be established for the gradient in the presence of variable, nonstandard growth?
• RQ5 Does the lack of upper bound on $ p_N/p_1 $ invalidate previous approaches, and if so, how can a new method be constructed?

Key findings

• . The main result establishes that any bounded local minimizer $ U \in W^{1,p}_{\text{loc}}(\Omega) $ of the orthotropic functional $ F_p $ satisfies $ \nabla U \in L^\infty_{\text{loc}}(\Omega) $, regardless of the size of $ p_N/p_1 $.
• The proof shows that the gradient gains higher integrability through a recursive sequence of exponents $ \{\beta_j^\ell\} $, which eventually stabilizes at $ q_{j-2} $, ensuring uniform bounds.
• The authors construct a sequence $ \{\varepsilon_j\} $ such that $ \prod_{j=j_0}^n (1 + \varepsilon_j) < +\infty $, which controls the growth of the iterative scheme and ensures convergence.
• The method avoids relying on Bernstein-type techniques or viscosity methods, offering a new path to Lipschitz regularity.
• The paper identifies a critical flaw in Lieberman's claimed proof of the same result, showing that his application of the Michael–Simon inequality fails due to an unverified condition.
• The result extends previous work by Demengel, Cupini–Marcellini–Mascolo, and others, particularly by removing the restriction $ p_N < p_1 + 1 $ and allowing $ p_1 \geq 2 $.