Klaus Steffen

Parabolic Systems with Polynomial Growth and Regularity

The authors establish a series of optimal regularity results for solutions to general non-linear parabolic systems

Boundary regularity for minimizing currents with prescribed mean curvature

u_t- \mathrm{div} \ a(x,t,u,Du)+H=0,

Optimal regularity results via A-harmonic approximation

under the main assumption of polynomial growth at rate

Hausdorff-Dimension, reguläre Mengen und total irreguläre Mengen

p

Harmonic Maps with Free Boundaries

i.e.

Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals

|a(x,t,u,Du)|\leq L(1+|Du|^{p-1}), p \geq 2.

A partial regularity theorem for harmonic maps at a free boundary

They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderon-Zygmund estimates for non-homogeneous problems are achieved here.

Area minimizing hypersurfaces with prescribed volume and boundary

We prove complete boundary regularity for energy minimizing integer multiplicity rectifiablen currents in ℝn+1 of prescribed mean curvatureH with boundaryB=⊥ represented by an oriented smooth submanifold of dimensionn − 1 in ℝsun+1. We also give applications to the Plateau problem for surfaces with prescribed mean curvature.

Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds

We discuss a new approach to regularity theory for almost minimizers of variational integrals in geometric measure theory or in the classical calculus of variations. This method is direct, exhibiting the dependence of the regularity estimates on the structural data of the variational integrand in explicit form; it requires only weak growth and smoothness assumptions on the integrand; it allows a unified treatment of interior and boundary regularity; and it leads to new regularity results which give the best possible modulus of continuity for the derivative of the almost minimizer in a variety of situations.

The non-existence of branch points in solutions to certain classes of plateau type variational problems

Im Jahre 1918 erschien in den Mathematischen Annalen Band 79 ein Artikel von Felix Hausdorff, den er im Marz desselben Jahres in Greifswald vollendet hatte ([Ha]). Der Titel lautet „Dimension und auseres Mas“. In dieser Arbeit warden die nach Hausdorff benannten, von einem nichtnegativen reellen Dimensionsparameter abhangigen Mafe explizit eingefiihrt, und implizit findet sich dort auch das darauf gegriindete Konzept einer Dimensionsfunktion, die beliebige nichtnegative reelle Werte annehmen kann und heute den Namen „Hausdorff-Dimension“ tragt. Es handelt sich hier um Hausdorffs letzte Untersuchung mit Bezug zur Mastheorie und gleichzeitig um seine einzige, die mafltheoretischen Fragen im engeren Sinnegewidmet ist und nicht wie friihere seiner Arbeiten in erster Linie Grundlagenprobleme im Bereich der Mastheorie anspricht.