Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy solutions converge to a harmonic map as t goes to infinity (asymptotic stability), extending previous work down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m=3, involving (among other tools) a "normal form" for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schroedinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m=2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even "eternal oscillation".Comment: 34 page

Finite time blow-up for the Yang-Mills heat flow in higher dimensions

We consider the L"2-gradient flow associated with the Yang-Mills functional, the so-called Yang-Mills heat flow. In the setting of a trivial principal SO(n)-bundle over R"n in dimension n greater than 4, we show blow-up in finite time for a class of SO(n)-equivariant initial connections. (orig.)SIGLEAvailable from TIB Hannover: RR 1596(430) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem. (orig.)Available from TIB Hannover: RO 5389(341) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Partial regularity for nonlinear elliptic systems

We consider nonlinear elliptic systems of divergence type. We provide a new method for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. This method is applied in two situations: that of quasilinear elliptic systems with inhomogeneity obeying the natural growth condition, and that of fully nonlinear homogeneous systems. In the latter case our methods extend previous partial regularity results, establishing the optimal Hoelder exponent for the derivative of a weak solution on its regular setSIGLEAvailable from TIB Hannover: RR 1596(389) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

On the Gribov copy problem for the Coulomb gauge

We consider the problem of gauge degeneracy. In particular, for connections on three-dimensional Euclidean space with the structure group SU(2), we show that a large class of spherically symmetric connections in the Coulomb gauge have distinct gauge copies in the Coulomb gauge. (orig.)Available from TIB Hannover: RR 1596(429) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Existence and regularity for higher dimensional H-systems

SIGLEAvailable from TIB Hannover: RR 1596(347) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Geometric evolution equations in critical dimensions

We make a qualitative comparison of phenomena occurring in two different geometric flows: the harmonic map heat flow in two space dimensions and the Yang-Mills heat flow in four space dimensions. Our results are a regularity result for the degree-2 equivariant harmonic map flow, and a blow-up result for an equivariant Yang-Mills-like flow. The results show that qualitatively differing behaviours observed in the two flows can be attributed to the degree of the equivariance

On variational models for quasi-static Bingham-fluids

SIGLEAvailable from TIB Hannover: RO 5389(412) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

A mixed boundary value problem for energy minimizing harmonic maps

We consider the regularity of maps between Riemannian manifolds which are energy minimizing amongst maps satisfying a mixed boundary condition. We impose the partially free boundary condition that a relatively open subset of the boundary be mapped into a closed submanifold of the target manifold, together with a fixed (Dirichlet) boundary condition on a second, disjoint, relatively open subset of the boundary. When the boundaries of the two subsets under consideration intersect perpendicularly, we demonstrate that no singularities can occur on the common boundary. In particular, this enables us to give an example of a harmonic map into Euclidean space which is energy minimizing with respect to such a mixed boundary condition, and which has a singularity in the (relative) interior of the free boundary. (orig.)Available from TIB Hannover: RO 5389(372) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Partial regularity for almost minimizers of quasiconvex integrals

We consider almost minimizers of variational integrals whose integrands are quasiconvex. 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. (orig.)SIGLEAvailable from TIB Hannover: RR 1596(428) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman