BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Georg Weiss (University of Duisburg-Essen) DTSTART:20230920T080000Z DTEND:20230920T100000Z DTSTAMP:20260423T005830Z UID:NCTS-GMT/18 DESCRIPTION:Title: Rectifiability\, finite Hausdorff measure\, and compactness for non-mini mizing Bernoulli free boundaries\nby Georg Weiss (University of Duisbu rg-Essen) as part of NCTS international Geometric Measure Theory seminar\n \n\nAbstract\nWhile there are numerous results on minimizers or stable sol utions of the Bernoulli problem proving regularity of the free boundary an d analyzing singularities\, much less is known about $\\textit{critical po ints}$ of the corresponding energy. Saddle points of the energy (or of clo sely related energies) and solutions of the corresponding time-dependent p roblem occur naturally in applied problems such as water waves and combust ion theory.\n\nFor such critical points $u\\text{---}$which can be obtaine d as limits of classical solutions or limits of a singular perturbation pr oblem$\\text{---}$it has been open since [Weiss03] whether the singular se t can be large and what equation the measure $\\Delta u$ satisfies\, excep t for the case of two dimensions. In the present result we use recent tech niques such as a $\\textit{frequency formula}$ for the Bernoulli problem a s well as the celebrated $\\textit{Naber-Valtorta procedure}$ to answer th is more than 20 year old question in an affirmative way:\n\nFor a closed c lass we call $\\textit{variational solutions}$ of the Bernoulli problem\, we show that the topological free boundary $\\partial \\{u > 0\\}$ (includ ing $\\textit{degenerate}$ singular points $x$\, at which $u(x + r \\cdot) /r \\rightarrow 0$ as $r\\to 0$) is countably $\\mathcal{H}^{n-1}$-rectifi able and has locally finite $\\mathcal{H}^{n-1}$-measure\, and we identify the measure $\\Delta u$ completely. This gives a more precise characteriz ation of the free boundary of $u$ in arbitrary dimension than was previous ly available even in dimension two.\n\nWe also show that limits of (not ne cessarily minimizing) classical solutions as well as limits of critical po ints of a singularly perturbed energy are variational solutions\, so that the result above applies directly to all of them.\n\nThis is a joint work with Dennis Kriventsov (Rutgers).\n LOCATION:https://researchseminars.org/talk/NCTS-GMT/18/ END:VEVENT END:VCALENDAR