Related papers: Some remarks on singular capillary cones with free…
Let (M,g) be a smooth compact, n dimensional Riemannian manifold,with smooth n-1 dimensional boundary. We prove that the stable critical points of the mean curvature of the boundary generates solutions for a singularly perturbed elliptic…
In this paper, we consider conformal metrics on a unit 4-disc with an asymptotically hyperbolic end and possible isolated conic singularities. We define a mass term of the AH end. If the $\sigma_{2}$ curvature has lower bound…
We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $\Sigma^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact…
We consider weighted Hardy inequalities involving the distance function to the boundary of a domain in the $N$-dimensional Euclidean space with nonempty boundary. We give a lower bound for the corresponding best Hardy constant for a domain…
We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a…
In this paper, continuing our previous work, we investigate the third gap problem in the Simon conjecture for closed minimal surfaces in the unit sphere. By developing refined third-order Simons-type integral identities and establishing new…
We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption of…
In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional $E$. We show…
We prove that the boundary of the trapped region in an asymptotically Euclidean Riemannian manifold of dimension at least 3 is a stable smooth minimal hypersurface except for a singular set of codimension at least 8.
In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the cylinder, we obtain some results of…
We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes $\Omega\times \mathbb{R}$ in a gravity free environment, in the case of physical interest, that is, for bounded, open, and…
We consider the problem $\mathbf{P_{curve}}$ of minimizing $\int \limits_0^L \sqrt{\xi^2 + \kappa^2(s)} \, {\rm d}s$ for a curve $\mathbf{x}$ on $\mathbb R$ with fixed boundary points and directions. Here the total length $L\geq 0$ is free,…
A two-dimensional steady problem of a potential free-surface flow of an ideal incompressible fluid caused by a singular sink is considered. The sink is placed at the horizontal bottom of the fluid layer. With the help of the Levi-Civita…
We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded $C^2$ domains for a large class of…
This paper presents the linear Global stability analysis of the incompressible axisymmetric boundary layer on a circular cone. The base flow is considered parallel to the axis of cone at the inlet. The angle of attack is zero and hence the…
Spiral waves are a ubiquitous feature of the nonequilibrium dynamics of a great variety of excitable systems. In the limit of a large separation in timescale between fast excitation and slow recovery, one can reduce the spiral problem to…
We study the problem of conformally deforming a manifold with boundary to have vanishing {\sigma}4-curvature in the interior and constant H4- curvature on the boundary. We prove that there are geometrically distinct solutions using…
Let M,g a compact Riemannian n-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. Also,…
This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a rigid plane and with an upper boundary given by a free surface. The fluid is subject to a constant external force with a horizontal component, which…
This article is devoted to investigate the singular profile of the free boundary of two-dimensional incompressible inviscid fluid with external force near the stagnation point. More precisely, given an external force with some polynomial…