We prove regularity of weakly m-polyharmonic maps (extrinsic or intrinsic) from domains in IR n of dimension n = 2m ≥ 4 to compact Riemannian manifolds, thus extending a previous result by Wang for the case m = 2. Moreover, we prove smoothness of Hölder continuous weakly polyharmonic maps for domains in IR n of dimension n ≥ 2m.

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.

By constructing examples which are explicit up to solving an ODE, we prove that singularities of first kind exist for the harmonic map and Yang-Mills heat flows. As a by-product, we also get a simplified proof of Ratto's/Ding's theorem about the existence of harmonic Hopf constructions. Singularities of first kind in the harmonic map heat flowLet M m , N n ⊂ IR n+k be compact Riemannian manifolds. We say that u : [0, T ) × M → N is a solution of the harmonic map heat flow if it satisfies the system of differential equationswhere ∆ M is the Laplace-Beltrami operator of M and A N the (vectorvalued) second fundamental form of N ; Du denotes the total derivative of u with respect to the space variables. This flow was used in the fundamental work [ES] of Eells and Sampson to prove existence of harmonic maps to manifolds of nonpositive sectional curvature. They actually proved that, given smooth initial data u(0, · ) = u 0 : M → N , the harmonic map heat flow has a smooth solution and f := lim i→∞ u(t i , · ) defines a harmonic map (homotopic to u 0 via the flow) for a suitable sequence t i → ∞. This may fail if N has positive sectional curvatures somewhere. Actually, there exists T > 0 such that u is smooth on [0, T ) × M , and if T is maximal, then lim t→T sup M |Du(t, x)| = ∞, in which case we say that u blows up at t = T .Thus a natural question is: Does blow-up happen at finite time, and can we understand the nature of the singularity? That blow-up at finite time

We consider the gradient flow of higher order elliptic functionals of the type Ð M jD m uj 2 for maps from a compact Riemannian manifold M to R n with image contained in another compact manifold, called the extrinsic polyharmonic map heat flow. We prove that smooth initial values can be continued as an eternal solution to the flow if M is of dimension < 2m. In the critical case dim M ¼ 2m, we find a unique eternal weak solution which is smooth except possibly for finitely many times. A singularity can occur only if a ''bubble'' separates, using up a certain amount of energy. 2000 Mathematics Subject Classification. 58E20, 35K30 E m ðuÞ :¼ 1 2 ð M jD m uj 2 : By D m we mean the m-th total derivative of u : M ! R n (as opposed to u : M ! N), that is we consider extrinsic energies. The direct method of the Calculus of Variations can be used to produce minimizers of E m in W m; 2 . Of course, E 1 is simply the Dirichlet integral, and the minimizers (or, more generally, stationary points) of E 1 are harmonic maps M ! N. The Euler-Lagrange equation for E 1 is the well-known harmonic map equation D M u ¼ traceðII N uÞðDu; DuÞ; where D M is the Laplace-Beltrami operator of M and II N is the second fundamental form of N. Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/27/15 12:07 PM M hD m u; D m ji; from which we infer the Euler-Lagrange relation div m D m u ? N:Here the divergence is taken with respect to the Riemannian metric of M. Introducing a local orthonormal frame fn i g i¼dim Nþ1...n of T ? N, we locally find functions l i such that div m D m u ¼ X n i¼dim Nþ1 l i n i u:Multiplying by n i u, we findmÀlÀ1 u d ðDn i uÞDuÞ: Andreas Gastel 502 Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/27/15 12:07 PM The extrinsic polyharmonic map heat flow in the critical dimension 503 Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/27/15 12:07 PM

We consider nonlinear elliptic systems of divergence type with Dini continous coefficients and prove a partial regularity result for weak solutions. Our method of proof is based on a generalization of the technique of harmonic approximation. Our result is optimal in the sense that in the case of Hölder continous coefficients we establish the optimal Hölder exponent for the derivative of the weak solution on its regular set.