0000000000218375

AUTHOR

Max Engelstein

0000-0003-3234-1144

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Minimizers for the Thin One‐Phase Free Boundary Problem

2021

We consider the “thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0001 plus the area of the positivity set of that function in urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0002. We establish full regularity of the free boundary for dimensions urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0003, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for…

Pure mathematicsApplied MathematicsGeneral MathematicsDimension (graph theory)Content (measure theory)Free boundary problemBoundary (topology)Almost everywhereDirichlet's energyFunction (mathematics)Measure (mathematics)MathematicsCommunications on Pure and Applied Mathematics
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