Partial regularity for $\mathbb{A}$-quasiconvex functionals with Orlicz growth

Technical Summary: Partial Regularity for A-Quasiconvex Functionals with Orlicz Growth

Problem Statement
The paper addresses the partial regularity of minimizers for variational functionals of the form
$$\mathcal{F}[u] := \int_{\Omega} F(Au) \, dx,$$
where $\Omega \subset \mathbb{R}^n$ is an open bounded set, $A$ is a first-order homogeneous linear differential operator with constant coefficients, and $F$ is a variational integrand. The study focuses on two specific growth regimes:

1. $\Delta_2 \cap \nabla_2$-Orlicz growth: Where the growth function $\phi$ satisfies both the doubling condition and its conjugate does as well.
2. $L \log L$-growth: A limiting case where $\phi(t) = t \log(1+t)$, which satisfies the $\Delta_2$ condition but fails the $\nabla_2$ condition.

The operator $A$ acts on maps $u: \Omega \to \mathbb{R}^k$ (or a general finite-dimensional inner product space $V$) and takes values in a space $W$. The integrand $F$ is assumed to be $\phi$-strongly $A$-quasiconvex. The central question is whether local minimizers of such functionals are $C^{1,\alpha}$-regular on a relatively open set of full Lebesgue measure (partial regularity), and under what conditions on the operator $A$ this holds.

Methodology
The primary methodological strategy, inspired by Conti & Gmeineder [20], is a reduction argument. Rather than executing a full partial regularity proof via blow-up techniques or $A$-harmonic approximation directly on the operator $A$, the author reduces the problem to known partial regularity results for functionals involving the full gradient $\nabla$.

The reduction relies on two critical components:

1. Korn-type Inequalities: The author utilizes inequalities that allow the control of the full gradient $\nabla u$ by the operator $Au$ in Orlicz spaces.
   • For the $\Delta_2 \cap \nabla_2$ case, standard Korn-type inequalities (Theorem 2.4) imply that if $Au \in L^\phi$, then $\nabla u \in L^\phi$.
   • For the $L \log L$ case, the author establishes a generalized Korn-type inequality (Theorem 5.1) based on Cianchi's work [19]. This inequality reveals a "loss of one logarithm": if $Au \in L^\phi$ with $\phi(t) = t \log(1+t)$, then $\nabla u$ is controlled in a slightly weaker space, specifically $L^1$ with a logarithmic weight, necessitating a shift in the quasiconvexity framework.
2. Transformation to Full Gradient Functionals: By defining a new integrand $G(z) = F(\pi_A(z))$, where $\pi_A$ is a linear map satisfying $Av = \pi_A(\nabla v)$, the problem is mapped to a functional depending on $\nabla v$. The author verifies that the $\phi$-strong $A$-quasiconvexity of $F$ translates into the appropriate strong quasiconvexity conditions for $G$ in the respective growth regimes.

Key Contributions and Results

• Theorem 1.1 ($\Delta_2 \cap \nabla_2$-growth): The paper establishes that for functionals with $\phi$-growth where $\phi \in \Delta_2 \cap \nabla_2$, a local minimizer $u$ is $C^{1,\alpha}_{loc}$-partially regular if and only if the operator $A$ is elliptic (i.e., its Fourier symbol $A[\xi]$ is injective for all $\xi \neq 0$).

  • The proof demonstrates that under the ellipticity assumption, the Korn-type inequality allows the reduction to the full gradient case, where partial regularity is already established for Orlicz growth (citing Diening et al. [28] and Breit & Verde [11]).
  • Conversely, if $A$ is not elliptic, the existence of non-regular functions in the kernel of $A$ (Lemma 2.2) precludes partial regularity.

• Theorem 1.2 ($L \log L$-growth): The paper extends the reduction strategy to the limiting case $\phi(t) = t \log(1+t)$. It proves that partial regularity holds if and only if $A$ is elliptic.

  • Due to the loss of regularity in the Korn inequality (Theorem 5.1), the reduction does not map directly to $C^{1,\alpha}$ results for $L^\phi$ gradients. Instead, the author adapts the framework to the space of functions of Bounded Variation (BV).
  • By utilizing recent results on $p=1$ growth and $L \log L$ growth by Gmeineder & Kristensen [44], the author shows that minimizers of the original functional correspond to local BV-minimizers of the reduced functional, thereby inheriting partial regularity.

• Coercivity and Existence (Section 3): The paper provides a rigorous treatment of the coercivity of the functionals on Dirichlet classes in Orlicz-Sobolev spaces $W^{1,\phi}_A(\Omega)$. It establishes the existence of minimizers under the $\phi$-strong $A$-quasiconvexity assumption, clarifying the equivalence between the integral definition of strong quasiconvexity and the condition involving the auxiliary $V$-function (Theorem 3.6).

Significance and Claims
The paper claims to provide a unified approach to partial regularity for a broad class of differential operators and growth conditions.

• Optimality of Reduction: The author asserts that the reduction strategy, previously applied to $p$-growth functionals by Conti & Gmeineder [20], is extended optimally to the $L \log L$ case. The paper notes that for the endpoint $p=1$ (or sub-logarithmic growth), the reduction strategy fails because $Au \in L^\phi$ does not imply $\nabla u \in L^1_{loc}$, and thus the results for $L \log L$ represent the limit of this specific methodology.
• Generalization: The results generalize existing partial regularity theories from the standard gradient case ($A=\nabla$) and the symmetric gradient case ($A=\varepsilon$) to arbitrary constant-rank elliptic operators.
• Scope: The paper explicitly defers the investigation of exponential growth integrands (the other end of the scale) to future work, noting that the reduction strategy is the primary focus.

In summary, the paper demonstrates that the ellipticity of the differential operator $A$ is the necessary and sufficient condition for the partial regularity of minimizers in the context of Orlicz growth, covering both standard $\Delta_2 \cap \nabla_2$ regimes and the critical $L \log L$ limiting case, by successfully reducing these problems to established results for full gradient functionals.