Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the \(p\)-Laplacian (Q6644977)

In this paper, the authors prove Lipschitz regularity of weakly coupled vectorial almost-minimizers for the \(p\)-Laplacian. More precisely, let \(\lambda>0\) and \(D\subset\mathbb{R}^n\) be a bounded Lipschitz domain. The authors consider almost minimizers of the functional\N\[\NJ(\vec{v};D)=\int_D\sum_{i=1}^m|\nabla v_i(x)|^p+\lambda\chi_{\{|\vec{v}|>0\}}(x)dx, \ 1<p<\infty,\N\]\Nover the set of admissible functions\N\[\N\mathcal{K}=\{\vec{v}\in W^{1,p}(D;\mathbb{R}^m) : \vec{v}=\vec{g} \text{ on } \partial D, \ v_i\ge 0\}\N\]\Nwhere \(\vec{g}=(g_1,\dots,g_m)\) is the boundary data with \(0\le g_i\in W^{1,p}(D)\), \(\vec{v}=(v_1,\ldots,v_m)\). The main result of the paper states that such almost minimizers have optimal Lipschitz regularity in compact subsets of \(D\). This is done by first proving local Hölder continuity of the almost minimizers, following ideas from \textit{C. De Filippis} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 194, Article ID 111464, 25 p. (2020; Zbl 1436.35155)]. Using blow-up arguments, the authors obtain Lipschitz regularity. Finally, the authors address the boundary Lipschitz regularity of \(\vec{v}\).