Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the p-Laplacian
Masoud Bayrami-Aminlouee, Morteza Fotouhi, Henrik Shahgholian · Journal of Differential Equations · 2024
For a given constant λ > 0 and a bounded Lipschitz domain D ⊂ R n ( n ≥ 2 ), we establish that almost-minimizers of the functional J ( v ; D ) = ∫ D ∑ i = 1 m | ∇ v i ( x ) | p + λ χ { | v | > 0 } ( x ) d x , 1 < p < ∞ , where v = ( v 1 , ⋯ , 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.