Abstract
We study vector-valued almost minimizers of the energy functional ∫D|∇u|2+21+qλ+(x)|u+|q+1+λ-(x)|u-|q+1dx,0<q<1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _D\\left( |\ abla \ extbf{u}|^2+\\frac{2}{1+q}\\left( \\lambda _+(x)|\ extbf{u}^+|^{q+1}+\\lambda _-(x)|\ extbf{u}^-|^{q+1}\\right) \\right) dx,\\quad 0<q<1. \\end{aligned}$$\\end{document}For Hölder continuous coefficients lambda _pm (x)>0, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of “regular" free boundary points.
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More From: Calculus of Variations and Partial Differential Equations
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