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Regularity for Non-Linear PDEs

An isotropic quasi-linear parabolic equation: global existence and regularity at the boundary

speaker: Wolfgang Arendt (Universit├Ąt Ulm)

abstract: The talk is based on a multiplicative perturbation of the Laplacian by a measurable positive function m. a) A natural question is under which conditions this operator generates a semigroup on the space which consists of all continuous functions on the underlying euclidean domain which vanish at the boundary. Reformulated differently, the question is when the variational solutions under weak Dirichlet boundary conditions are actually continuous on the closure and takes the value 0 at the boundary. If the function m does not degenerate too much, then this is equivalent to Dirichlet regularity (or Wiener regularity) of the domain. For arbitrary, possibly very bad domains, the result is still true if m converges fast enough to 0 at the boundary. More generally, these two conditions can be combined leading to an optimal local criterion. b) In the second part we allow m to depend on the solution and its gradient. Here we allow the domain even to be unbounded. However, (so far) we establish only L2 solutions, with maximal regularity in space and time, though. Moreover, we show the solutions we obtain are global in time. The point is to use a version of Schauders fixed point theorem valid for locally convex topologies (Schaefer's fixed point theorem). This has the advantage that only local compactness is needed. Thus we need no regularity hypotheses on the boundary of the domain. Moreover, other boundary conditions than Dirichlet may be considered, still without regularity assumptions. The talk is based on the following papers:

W. Arendt, M. Chovanec: Dirichlet regularity and degenerate diffusion. Trans. Amer. Math. Soc., to appear. W. Arendt, R. Chill: Global existence for isotropic quasilinear diffusion equations in non-divergence form. Ann. della Scuola Normale Superiore di Pisa, to appear.

Mon 14 Sep, 16:30 - 17:30, Aula Dini
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