Boundary regularity of solutions to the evolutionary p-Laplacean system
Abstract:
We consider the regularity of solutions to the Dirichlet problem for the singular p-Laplacean parabolic system, p ∈ (1, 2), in a bounded domain Ω. The analysis of this kind of system is strictly related to the study of non-stationary non-Newtonian fluids such as shear-thinning fluids. We address different aspects of the regularity theory for solutions of such systems, in particular higher space-time regularity in Sobolev spaces and Hölder continuity of the gradient of the solution. We get a first global integrability of second order derivatives, $D^2 u ∈ L^2 (ε, T ; L^p (Ω)), ε > 0$, provided that p > max{ 2N/(N+2), 3/2 }. It is interesting to note that the proof of such results does not rely on any kind of localization. Further, we obtain higher integrability results of second order derivatives, together with higher integrability results for the time derivative, under some further restrictions on the range of p. Space-time global Hölder continuity of the gradient of weak solutions follows as a consequence.
The results are part of a joint work with Paolo Maremonti (Seconda Università degli Studi di Napoli).
04.12.12
09:00
Peter Bella
( Max Planck Institute for Mathematics in the Sciences, Leipzig )
Wrinkling of a stretched annular elastic thin sheet - identification of the optimal scaling law for the energy of a ground state
Abstract:
In [Bella & Kohn: Wrinkles as the result of compressive stresses in an annular thin film, to appear in CPAM] we identified the optimal scaling law of the minimum of the elastic energy of a stretched annular thin elastic sheet. To obtain the optimal upper bound (up to a possible prefactor) we constructed a deformation with a cascade of wrinkles with changing period, but it was not clear whether the optimal construction requires a similar cascade. In fact, in the physics community it is assumed that it suffices to consider wrinkles with single wavenumber. Motivated by this, together with Felix Otto we considered a toy model where we identified the optimal prefactor in the energy scaling and expressed this prefactor as a minimum of a much simpler variational problem (where the ground state of this variational problem obviously uses a cascade of wrinkles with changing period).
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