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SUMMARY:Julian Fischer (IST Austria)
DTSTART:20220906T160000Z
DTEND:20220906T170000Z
DTSTAMP:20260423T035036Z
UID:OSGA/122
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OSGA/122/">M
 ultiphase Mean Curvature Flow: Uniqueness Properties of Weak Solution Conc
 epts and Phase-Field Approximations</a>\nby Julian Fischer (IST Austria) a
 s part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nTopology chan
 ges occur naturally in geometric evolution equations like mean curvature f
 low. As classical solution concepts break down at such geometric singulari
 ties\, the use of weak solution concepts becomes necessary in order to des
 cribe topological changes. For two-phase mean curvature flow\, the theory 
 of viscosity solutions by Chen-Giga-Goto and Evans-Spruck provides a conce
 pt of weak solutions with basically optimal existence and uniqueness prope
 rties. In contrast\, the uniqueness properties of weak solution concepts f
 or multiphase mean curvature flow had remained mostly unexplored.\n\nBy in
 troducing a novel concept of "gradient flow calibrations"\, we establish a
  weak-strong uniqueness principle for multiphase mean curvature flow: Weak
  (BV) solutions to multiphase mean curvature flow are unique as long as a 
 classical solution exists. In particular\, in planar multiphase mean curva
 ture flow\, weak (BV) solutions are unique prior to the first topological 
 change. As basic counterexamples show\, the uniqueness of evolutions may f
 ail past certain topology changes\, demonstrating the optimality of our re
 sult.\n\nIn the last part of the talk\, we discuss further applications of
  our new concept\, including the quantitative convergence of diffuse-inter
 face (Allen-Cahn) approximations for multiphase mean curvature flow.\n
LOCATION:https://researchseminars.org/talk/OSGA/122/
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