Second order regularity for solutions to anisotropic degenerate elliptic equations

TL;DR

The paper develops second-order regularity theory for solutions to degenerate anisotropic elliptic equations driven by a Finsler norm $H$ and an Orlicz-type growth $A$, addressing the quasilinear system $- abla\cdot(A(H(\nabla u)) H(\nabla u) \nabla H(\nabla u))=f$. By working in the Orlicz-Sobolev framework and exploiting structural parameters $m_{A}$ and $M_{A}$, it proves local $C^{1,\alpha}$ regularity and, away from the gradient-zero set, $W^{2,2}_{loc}$ regularity, accompanied by weighted estimates for the inverse of $A(H(\nabla u))$. It further establishes second-order regularity for a family of stress-like fields $A(H(\nabla u))^{(k-1)/m_{A}} H(\nabla u) \nabla H(\nabla u)$ in $W^{1,2}_{loc}$ under precise conditions on $m_{A}$, $M_{A}$ and $k$, including several regimes and isotropic specializations. These results generalize and extend known isotropic and Uhlenbeck-type theories, with potential implications for anisotropic materials and applications in continuum mechanics and image processing.

Abstract

We consider solutions to degenerate anisotropic elliptic equations in order to study their regularity. In particular we establish second-order estimates and enclose regularity results for the stress field. All our results are new even in the euclidean case.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• Definition 2.4
• Proposition 3.1
• proof