Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the p-Laplacian

For a given constant λ>0 and a bounded Lipschitz domain D⊂Rⁿ (n≥2), we establish that almost-minimizers of the functional J(v;D)=∫D∑i=1m|∇v_i(x)|^p+λχ_{|v|>0}(x)dx,1<p<∞, where v=(v₁,⋯,v_m), and m∈N, exhibit optimal Lipschitz continuity in compact sets of D. Furthermore, assuming p≥2 and employing a distinctly different methodology, we tackle the issue of boundary Lipschitz regularity for v. This approach simultaneously yields an alternative proof for the optimal local Lipschitz regularity for the interior case.