The free-boundary Brakke flow

We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must “pop” upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White [23] to the free-boundary setting. Using Ilmanen’s elliptic regularization procedure [10], we prove existence of free-boundary Brakke flows.

Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere 𝕊+3

In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.

Topological Methods for Boundary Mean Curvature Problem on Bn

Using an algebraic topological method and the tools of the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the the n-dimensional balls.

Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in 𝔹N

We consider the prescribed boundary mean curvature problem in 𝔹N with the Euclidean metric where ã(x) is positive and rotationally symmetric on We show that if K∽(x) has a local maximum point, then this problemhas infinitely many positive solutions that are not rotationally symmetric on 𝕊N−1.