Global solutions to the heat flow for m-harmonic maps and regularity - Bögelein, Verena; Duzaar, Frank; Scheven, Christoph (Indiana University Mathematics Journal, 2012)
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma2.global

Partial Hölder continuity for discontinuous elliptic problems with VMO-coefficients - Bögelein, Verena; Duzaar, Frank; Habermann, Jens; Scheven, Christoph (Proceedings of the London Mathematical Society, 2011)
We establish partial Hölder continuity for vector-valued solutions u: Ω → N to elliptic systems of the type as well as for minimizers u: Ω → N of quasi-convex functionals where the structure function a, respectively, the integrand f is possibly discontinuous with respect to x. More precisely, we merely impose a uniform VMO-condition with respect to the x-dependence and continuity with respect to the u-dependence and prove Hölder continuity of the solutions, respectively, the minimizers outside of a negligible set. 2011 London Mathematical Society2011 © 2011 London Mathematical Society.
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma2.partia

Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients - Bögelein, Verena; Duzaar, Frank; Habermann, Jens; Scheven, Christoph (Advances in Calculus of Variations, 2012)
We establish partial regularity for solutions (u, π): ℝ n ×ℝ n ⊇ Ω × Σ → ℝ n × ℝ 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 (equation) where εu denotes the symmetric part of the gradient Du, π denotes the pressure, the not necessarily continuous coefficient μ is a bounded non-negative VMO-function and the variable exponent function p: Ω → (3n/n+2, ∞) fulfills the logarithmic continuity assumption, i.e., we assume that for the modulus of continuity ω p of the exponent function p there holds (equation) To be more precise, we prove Hölder continuity of the solution u outside of a negligible set. Moreover, we show that εu and the pressure belong to certain Morrey spaces on the regular set of u, i.e., the set where u 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 μ is continuous. © de Gruyter 2012.
UnivIS-Import:2015-03-09:Pub.2012.nat.dma.lma2.statio
Keywords: Electro-rheological fluids; Non-standard growth condition; Partial Hölder-and Morrey-type regularity; VMO-coefficients