Andreas Gastel

PARTIAL REGULARITY FOR ALMOST MINIMIZERS OF QUASI-CONVEX INTEGRALS ∗

We consider almost minimizers of variational integrals whose integrands are quasi-convex. Under suitable growth conditions on the integrand and on the function determining the almost minimality, we establish almost everywhere regularity for almost minimizers and obtain results on the regularity of the gradient away from the singular set. We give examples of problems from the calculus of variations whose solutions can be viewed as such almost minimizers.

Elliptic Systems, Singular Sets and Dini Continuity

Abstract We estimate the size of the singular set of solutions to non-linear elliptic systems of the form where the vector field a satisfies a Dini-type continuity condition with respect to the variables (x, u).

A Family of Expanding Ricci Solitons

The Ricci flow is a natural evolution equation for Riemannian metrics,introduced by Richard S. Hamilton in 1982. A family \({\left( {g\left( {t, \cdot } \right)} \right)_{t \in I}}\) of metrics on a Riemannian manifold M, depending on a time parameter \(t \in I \subseteq \mathbb{R} \) is a solution to the Ricci flow if it solves the equation

Nonuniqueness for the Yang–Mills heat flow

\frac{\partial }{{\partial t}}g\left( {t,\cdot} \right) = - 2Ricg\left( {t,\cdot} \right),

CONSTRUCTION OF HARMONIC MAPS BETWEEN SPHERES BY JOINING THREE EIGENMAPS

where Ric g(t, •) is the Ricci tensor associated with the evolving metric g(t, •). In general, a solution of the Ricci flow starting with smooth initial data will not possess a smooth continuation for all time. The formation of singularities has been discussed extensively in Hamilton’s article [H], and there is a particular type of solutions which is expected (and in some cases known) to appear as parabolic blowup limit of Ricci flows around a singularity, namely Ricci solitons. Ricci solitons are solutions to the Ricci flow for which there exist scalars σ(t) and diffeomorphisms Ψ t :M → M such that

Nonlinear elliptic systems with Dini continuous coefficients

g\left( {t,\cdot} \right) = \sigma \left( t \right)\Psi _t^* g\left( {T,\cdot} \right){\text{ for all t}} \in {\text{I and some fixed T}} \in {\text{I}}{\text{.}}