A parabolic free boundary problem with Bernoulli type condition on the free boundary

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dc.creatorAndersson, J.
dc.creatorWeiss, G. S.
dc.date2006-08-30
dc.date.accessioned2026-07-07T07:22:19Z
dc.date.available2026-07-07T07:22:19Z
dc.descriptionConsider the parabolic free boundary problem $$ Δu - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a realistic class of solutions, containing for example {\em all} limits of the singular perturbation problem $$Δu_ε- \partial_t u_ε= β_ε(u_ε) \textrm{as} ε\to 0,$$ we prove that one-sided flatness of the free boundary implies regularity. In particular, we show that the topological free boundary $\partial\{u>0\}$ can be decomposed into an {\em open} regular set (relative to $\partial\{u>0\}$) which is locally a surface with Hölder-continuous space normal, and a closed singular set. Our result extends the main theorem in the paper by H.W. Alt-L.A. Caffarelli (1981) to more general solutions as well as the time-dependent case. Our proof uses methods developed in H.W. Alt-L.A. Caffarelli (1981), however we replace the core of that paper, which relies on non-positive mean curvature at singular points, by an argument based on scaling discrepancies, which promises to be applicable to more general free boundary or free discontinuity problems.
dc.identifierhttps://arxiv.org/abs/math/0608746
dc.identifierhttp://arxiv.org/abs/math/0608746
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/115618
dc.subjectAnalysis of PDEs
dc.subject35R35; 35K55
dc.titleA parabolic free boundary problem with Bernoulli type condition on the free boundary
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