In this paper, we prove an optimal Heintze-Karcher-type inequality for ani-sotropic free boundary hypersurfaces in general convex domains. The equality is achieved for anisotropic free boundary Wulff shapes in a convex cone. As applica-tions, we prove Alexandrov-type theorems in convex cones.
In this paper, we shall study the boundary case for the J-flow under certain geometric assumptions.
We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which propagates toward the opening of the cone with normal velocity depend- ing on its mean curvature. In addition, the contact angle between the hypersurface and the cone boundary depending on its position. First, we construct a family of radially symmetric self-similar solutions. Then we use these solutions to give a priori estimates for the solutions of the initial boundary value problems, and show their global exis-tence.
In this paper, we consider a single species model with nonlocal dispersal strategy and discuss how the dispersal rate and the distribution of resources affect the total population and survival chances by summarizing some previous results and demonstrate some relevant progress. The first topic is about the monotonicity of total population upon dispersal rate. For the nonlocal model, we prove a new result, which reveals essential difference between local and nonlocal models for certain distribution of resources. Secondly, we discuss optimal spatial arrangement for survival chances and total populations. The results for both local and nonlocal models demonstrate that the concentration of resources is beneficial for species.
We propose a delay-induced nonlocal free boundary problem by modeling the invasion of a two-stage structured species, where the nonlocal interaction is caused jointly by time delay, free boundary and the diffusion. After establishing a comparison principle and a priori boundedness estimates, we prove the local and global existence of classical solutions for the model.
We are interested in the dynamical behaviors of solutions to a parabolic- parabolic chemotaxis-consumption model with a volume-filling effect on a bounded interval, where the physical no-flux boundary condition for the bacteria and mixed Dirichlet-Neumann boundary condition for the oxygen are prescribed. By taking a continuity argument, we first show that the model admits a unique nonconstant steady state. Then we use Helly’s compactness theorem to show that the asymptotic profile of steady state is a transition layer as the chemotactic coefficient goes to infinity. Finally, based on the energy method along with a cancellation structure of the model, we show that the steady state is nonlinearly stable under appropriate perturbations. Moreover, we do not need any assumption on the parameters in showing the stability of steady state.
Let M be an n-manifold of positive sectional curvature ≥ 1. In this paper, we show that if the Riemannian universal covering has volume at least v > 0, then the fundamental group π1(M) has a cyclic subgroup of index bounded above by a constant depending only on n and v.
We use certain Morse functions to construct conformal metrics such that the eigenvalue vector of modified Schouten tensor belongs to a given cone. As a result, we prove that any Riemannian metric on compact 3-manifolds with boundary is confor-mal to a compact metric of negative sectional curvature.
The Ricci flow plays an essential role in modern geometric analysis. In this short note, we only survey some special topics of this broad and deep field. We first survey some convergence results of the Ricci flow and the Kahler Ricci flow. In partic-ular, we explain the basic idea in the proof of the Hamilton-Tian conjecture. Then we survey the recent progresses on the extension conjecture, which predicts that the Ricci flow can be extended when scalar curvature is bounded.
n this expository paper, we survey some results concerning the classifica-tion of solutions to the Liouville equation $\Delta u+e^{2 u}=0 \text { in } \mathbb{R}^{2}$.