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.
Let M and N be smooth Riemannian manifolds, M of the dimension n ≥ 2 with nonempty boundary, and N compact without boundary. We consider stationary harmonic maps u ∈ H 1 (M, N ) with a free boundary condition of the type u(∂M) ⊂ , given a submanifold ⊂ M. We prove partial boundary regularity, namely H n−2 (sing(u)) = 0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold .
We consider extrinsically biharmonic maps from an open domain R m into an arbitrary compact submanifold N R L . We show that the singular set of a minimizing biharmonic map has Hausdorff dimension at most m 5. Furthermore, we give conditions on the target manifold under which the dimension can be reduced further and conditions under which similar results hold for maps that are only stationary biharmonic or stable stationary biharmonic. The proof is based on the analysis of defect measures by tools of geometric measure theory. A byproduct of the proof is the theorem that weak limits of stationary biharmonic maps are weakly biharmonic.for all variations of the form u t D N .u C tW / with W 2 C 1 cpt .; R L /, where N W U ı ! N denotes the nearest-point-retraction from a neighborhood U ı R L of N onto N . Smooth biharmonic maps are characterized by 2 u ? N on , cf. (2.1).A weakly biharmonic map u 2 W 2;2 .; N / is called stationary biharmonic if it satisfies (1.1) additionally for variations of the domain, that is u t .x/ D u.x C t .x// with 2 C 1 cpt .; R m /.
In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.
We prove existence results for the obstacle problem related to the porous medium equation. For sufficiently regular obstacles, we find continuous solutions whose time derivative belongs to the dual of a parabolic Sobolev space. We also employ the notion of weak solutions and show that for more general obstacles, such a weak solution exists. The latter result is a consequence of a stability property of weak solutions with respect to the obstacle.
Mathematics Subject Classification
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.