[Paper Review] Higher integrability for variational integrals with non-standard growth

Mathias Schäffner|arXiv (Cornell University)|May 11, 2020

Nonlinear Partial Differential EquationsMathematics38 references19 citations

TL;DR

This paper establishes higher gradient integrability and partial regularity for minimizers of autonomous variational integrals with non-standard (p,q)-growth, proving that local minimizers belong to $W^{1,q}_{\text{loc}}$ under the improved condition $\frac{q}{p} < 1 + \frac{2}{n-1}$ for $n \geq 3$, extending prior results that required the stricter $\frac{q}{p} < 1 + \frac{2}{n}$. The key innovation lies in optimizing the cut-off function in a Caccioppoli-type inequality using Sobolev inequalities on $n-1$-dimensional spheres, enabling sharper integrability estimates and subsequent partial regularity results.

ABSTRACT

We consider autonomous integral functionals of the form $\mathcal F[u]:=\int_\Omega f(D u)\,dx$ with $u:\Omega o\mathbb R^N$ $N\geq1$, where the convex integrand $f$ satisfies controlled $(p,q)$-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of $\mathcal F$ assuming $\frac{q}p<1+\frac2{n-1}$, $n\geq3$. This improves earlier results valid under the more restrictive assumption $\frac{q}p<1+\frac2{n}$.

Motivation & Objective

• To establish higher gradient integrability for local minimizers of autonomous variational integrals with non-standard (p,q)-growth in the vectorial case $N > 1$, extending prior scalar results.
• To improve the known threshold for higher integrability from $\frac{q}{p} < 1 + \frac{2}{n}$ to $\frac{q}{p} < 1 + \frac{2}{n-1}$ for $n \geq 3$, thereby strengthening the regularity theory for such functionals.
• To prove partial regularity of minimizers under the same improved condition, showing that the gradient is H"older continuous on a full-measure open subset.
• To provide a new methodological approach based on optimized cut-off functions and Sobolev inequalities on $n-1$-dimensional spheres, overcoming limitations of previous Moser-iteration techniques in the vectorial setting.

Proposed method

• Derives a Caccioppoli-type inequality for $W^{1,q}_{\text{loc}}$ minimizers involving a cut-off function $\eta$, with the key estimate $\int \eta^2 |D(|Dv|^{p-2}/2 Dv)|^2 \lesssim \int |\nabla \eta|^2 (1 + |Dv|^q)$.
• Optimizes the choice of the cut-off function $\eta$ to exploit Sobolev inequalities on $n-1$-dimensional spheres, leading to improved integrability estimates beyond the classical $n$-dimensional scaling.
• Applies a regularization and approximation procedure to extend the $W^{1,q}_{\text{loc}}$ result from $W^{1,q}_{\text{loc}}$ minimizers to general local minimizers in $W^{1,1}_{\text{loc}}$.
• Uses the higher integrability result to deduce higher differentiability: $|\nabla u|^{p-2}/2 \nabla u \in W^{1,2}_{\text{loc}}$.
• Establishes an $\varepsilon$-regularity result (Lemma 4) by rescaling and analyzing blow-up sequences of minimizers, showing that small oscillation of the gradient implies improved decay of energy excess.
• Combines the $\varepsilon$-regularity result with a standard iteration argument to prove partial $C^{0,\alpha}$ regularity of the gradient on a full-measure set.

Experimental results

Research questions

• RQ1 Can the threshold $\frac{q}{p} < 1 + \frac{2}{n}$ for higher integrability of minimizers in the vectorial case $N > 1$ be improved to $\frac{q}{p} < 1 + \frac{2}{n-1}$?
• RQ2 Does the improved integrability threshold $\frac{q}{p} < 1 + \frac{2}{n-1}$ imply partial regularity of minimizers, with the gradient H"older continuous on a full-measure set?
• RQ3 Can the method of optimized cut-off functions and $n-1$-dimensional Sobolev inequalities be used to overcome the limitations of scalar Moser-iteration techniques in the vectorial setting?
• RQ4 Is the condition $\frac{q}{p} < 1 + \frac{2}{n-1}$ sharp for higher integrability and partial regularity in the vectorial case?

Key findings

• The main result establishes that every local minimizer $u \in W^{1,1}_{\text{loc}}(\Omega, \mathbb{R}^N)$ of the functional $F[u] = \int_\Omega f(Du)\,dx$ belongs to $W^{1,q}_{\text{loc}}(\Omega, \mathbb{R}^N)$ under the condition $\frac{q}{p} < 1 + \frac{2}{n-1}$ for $n \geq 3$, improving the prior threshold $\frac{q}{p} < 1 + \frac{2}{n}$.
• Higher differentiability is obtained: $|\nabla u|^{p-2}/2 \nabla u \in W^{1,2}_{\text{loc}}(\Omega, \mathbb{R}^{N \times n})$, which follows from the higher integrability and the Caccioppoli inequality.
• Improved higher integrability is deduced via Sobolev embedding: $\nabla u \in L^{\kappa p}_{\text{loc}}(\Omega, \mathbb{R}^{N \times n})$ with $\kappa = \frac{n}{n-2}$, and $\kappa p > q$ holds when $\frac{q}{p} < 1 + \frac{2}{n-2}$, which is satisfied under the main condition.
• Partial regularity is proven: there exists a set $\Omega_0 \subset \Omega$ with $|\Omega \setminus \Omega_0| = 0$ such that $\nabla u \in C^{0,\alpha}(\Omega_0, \mathbb{R}^{N \times n})$ for every $0 < \alpha < 1$, under the same condition $\frac{q}{p} < 1 + \frac{2}{n-1}$.
• The proof technique relies on a novel optimization of the cut-off function in the Caccioppoli inequality, enabling the use of Sobolev inequalities on $n-1$-dimensional spheres, which yields the improved threshold.
• The condition $\frac{q}{p} < 1 + \frac{2}{n-1}$ is shown to be sharp in the sense that it matches the known sharp condition for local boundedness of gradients in the scalar case $N=1$, suggesting optimality of the result.