Regularity for the planar optimal p-compliance problem
Bohdan Bulanyi, Antoine Lemenant
TL;DR
This work analyzes the planar optimal $p$-compliance problem in dimension $2$, proving partial $C^{1,\alpha}$ regularity for all $p\in(1,\infty)$ and extending known results from the linear case $p=2$ to the nonlinear regime via a compactness-based energy-decay approach at flat points. The authors develop a robust variational framework, including capacities, a dual formulation with a stress field $\sigma$, and a priori bounds, to obtain existence, stability, and regularity results. They establish $\mathcal{H}^1$-Ahlfors regularity, absence of loops, and $C^{1,\alpha}$ regularity at $\mathcal{H}^1$-a.e. points, under a mild integrability condition on the forcing term $f$ (specifically $f\in L^q(\Omega)$ with $q>q_1$). The techniques bridge the $p=2$ theory with the high-$p$ limit and lay groundwork toward higher-dimensional extensions of free-boundary-type problems involving $p$-Laplacian energies.
Abstract
In this paper we prove a partial $C^{1,α}$ regularity result in dimension $N=2$ for the optimal $p$-compliance problem, extending for $p\not = 2$ some of the results obtained by A. Chambolle, J. Lamboley, A. Lemenant, E. Stepanov (2017). Because of the lack of good monotonicity estimates for the $p$-energy when $p\not = 2$, we employ an alternative technique based on a compactness argument leading to a $p$-energy decay at any flat point. We finally obtain that every optimal set has no loop, is Ahlfors regular, and $C^{1,α}$ at $\mathcal{H}^1$-a.e. point for every $p \in (1 ,+\infty)$.