Christoph Scheven

Higher integrability in parabolic obstacle problems

Abstract. In this paper we establish the self-improving property of integrability for parabolic variational inequalities satisfying an obstacle constraint and involving possibly degenerate respectively singular operators in divergence form. In particular, our results apply to the model case of the variational inequality associated to the parabolic p-Laplacean operator. Thereby we do not impose any monotonicity assumption in time on the obstacle function.

Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients

Abstract. We establish partial regularity for solutions to systems modeling electro-rheological fluids in the stationary case. As a model case our result covers the low order regularity of systems of the type where denotes the symmetric part of the gradient , denotes the pressure, the not necessarily continuous coefficient is a bounded non-negative -function and the variable exponent function fulfills the logarithmic continuity assumption, i.e., we assume that for the modulus of continuity of the exponent function there holds To be more precise, we prove Hölder continuity of the solution outside of a negligible set. Moreover, we show that and the pressure belong to certain Morrey spaces on the regular set of , i.e., the set where is Hölder continuous. Note that under such weak assumptions partial Hölder continuity for the gradient cannot be expected. Our result is even new if the coefficient is continuous.

Elliptic obstacle problems with measure data: Potentials and low order regularity

We consider obstacle problems with measure data related to elliptic equations of p-Laplace type, and investigate the connections between low order regularity properties of the solutions and non-linear potentials of the data. In particular, we give pointwise estimates for the solutions in terms of Wol potentials and address the questions of boundedness and continuity of the solution.

The evolution of H-surfaces with a Plateau boundary condition

Abstract In this paper we consider the heat flow associated to the classical Plateau problem for surfaces of prescribed mean curvature. To be precise, for a given Jordan curve Γ ⊂ R 3 , a given prescribed mean curvature function H : R 3 → R and an initial datum u o : B → R 3 satisfying the Plateau boundary condition, i.e. that u o | ∂ B : ∂ B → Γ is a homeomorphism, we consider the geometric flow ∂ t u − Δ u = − 2 ( H ∘ u ) D 1 u × D 2 u in B × ( 0 , ∞ ) , u ( ⋅ , 0 ) = u o on B , u ( ⋅ , t ) | ∂ B : ∂ B → Γ is weakly monotone for all t > 0 . We show that an isoperimetric condition on H ensures the existence of a global weak solution. Moreover, we establish that these global solutions sub-converge as t → ∞ to a conformal solution of the classical Plateau problem for surfaces of prescribed mean curvature.

The higher integrability of weak solutions of porous medium systems

Abstract In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.

Higher differentiability of solutions of parabolic systems with discontinuous coefficients

We consider weak solutions u : ΩT → R to parabolic systems of the type ut − div a(x, t, Du) = 0 in ΩT = Ω × (0, T ), where the function a(x, t, ξ) satisfies standard p-growth and ellipticity conditions for p 2 with respect to the gradient variable ξ. We study the regularity of the solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps x → a(x, t, ξ) under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives Dxa(·, ·, ξ) are contained in the class L(0, T ; L(Ω)), where the integrability exponents α, β are coupled by p(n + 2) − 2n 2α + n β = 1. In the particular case α = β = p(n + 2)/2, our assumption reduces to Dxa ∈ L loc (ΩT ). The aim of this paper is to prove a higher differentiability result of the solutions in the spatial directions as well as the existence of a weak time derivative ut ∈ Lp/(p−1) loc (ΩT ).

Weak solutions to the heat flow for surfaces of prescribed mean curvature

In this talk we establish the existence of global weak solutions to the heat flow for surfaces of prescribed mean curvature, i.e. the existence for the Cauchy-Dirichlet problem to parabolic systems of the type { ∂tu−∆u = −2(H ◦ u)D1u×D2u in B × (0,∞), u = uo on ∂par ( B × (0,∞) ) , where H : R3 → R is a bounded continuous function satisfying an isoperimetric condition, B the unit ball in R2 and u : B× (0,∞)→ R3. As one of the possible applications we show that the problem has a solution with values inBR ⊂ R3, whenever uo(B) ⊆ BR and furthermore there holds ∫ {ξ∈BR:|H(ξ)|≥ 3 2R } |H| dξ < 9π 2 , |H(a)| ≤ 1 R for a ∈ ∂BR. The results that will be presented in the talk are joint work with Verena Bogelein und Chrsitoph Scheven from Erlangen. FRANK DUZAAR, DEPARTMENT OF MATHEMATICS, UNIVERSITY ERLANGEN–NURNBERG, CAUERSTR. 11, 91056 ERLANGEN, GERMANY E-mail address: frank.duzaar@zuv.uni-erlangen.de Date: December 29, 2011. 1