Interior estimates of harmonic heat flow

Motivated by Giaquinta and Hildebrandt’s regularity result for harmonic mappings [M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 1982(336) (1982) 124–164, Theorems 3 and 4], we show a [Formula: see text]-regularity result of the harmonic flow between two Riemannian manifolds when the image is in a regular geodesic ball. The proof is based on De Giorgi–Moser’s iteration and Schauder estimate.

On stable CMC free-boundary surfaces in a strictly convex domain of a bi-invariant Lie group

Let [Formula: see text] be a three-dimensional Lie group with a bi-invariant metric. Consider [Formula: see text] a strictly convex domain in [Formula: see text]. We prove that if [Formula: see text] is a stable CMC free-boundary surface in [Formula: see text] then [Formula: see text] has genus either 0 or 1, and at most three boundary components. This result was proved by Nunes [I. Nunes, On stable constant mean curvature surfaces with free-boundary, Math. Z. 287(1–2) (2017) 73–479] for the case where [Formula: see text] and by R. Souam for the case where [Formula: see text] and [Formula: see text] is a geodesic ball with radius [Formula: see text], excluding the possibility of [Formula: see text] having three boundary components. Besides [Formula: see text] and [Formula: see text], our result also apply to the spaces [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. When [Formula: see text] and [Formula: see text] is a geodesic ball with radius [Formula: see text], we obtain that if [Formula: see text] is stable then [Formula: see text] is a totally umbilical disc. In order to prove those results, we use an extended stability inequality and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces.

Sobolev spaces of maps and the Dirichlet problem for harmonic maps

We give a self-contained treatment of the existence of a regular solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. No curvature assumptions on the target are required. In this route we introduce a new deformation result which permits to glue a suitable Euclidean end to the geodesic ball without violating the convexity property of the distance function from the fixed origin. We also take the occasion to analyze the relationships between different notions of Sobolev maps when the target manifold is covered by a single normal coordinate chart. In particular, we provide full details on the equivalence between the notions of traced Sobolev classes of bounded maps defined intrinsically and in terms of Euclidean isometric embeddings.

Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in {S^{n}}. In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and {\operatorname{SO}(n)}-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for {\operatorname{SO}(n)}-invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].

Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

We prove that the area of a free boundary minimal surface $\Sigma ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $\Sigma $ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows works of Brendle and Fraser–Schoen in the euclidean setting.

Stability of capillary hypersurfaces in a manifold with density

We introduce capillary hypersurfaces and its stability in a manifold with density. We prove that stable [Formula: see text]-minimal hypersurfaces with free boundary in a geodesic ball in space form with suitable radial density must be totally geodesic. We also prove two criteria for instability of the capillary hypersurfaces in a Euclidean ball with suitable density. At last, we obtain a topological restriction on strongly stable capillary surfaces in a 3-manifold with density under certain conditions. These results generalize those in a manifold with constant density.

Regularity of stable solutions to semilinear elliptic equations on Riemannian models

We consider the reaction-diffusion problem -Δgu = f(u) in ℬR with zero Dirichlet boundary condition, posed in a geodesic ball ℬR with radius R of a Riemannian model (M,g). This class of Riemannian manifolds includes the classical space forms, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish L∞, Lp, and W1,p estimates which are optimal and do not depend on the nonlinearity f. As an application, under standard assumptions on the nonlinearity λf(u), we prove that the corresponding extremal solution u* is bounded whenever n ≤ 9. To establish the optimality of our regularity results we find the extremal solution for some exponential and power nonlinearities using an improved weighted Hardy inequality.