Related papers: On a Problem Posed by Brezis and Mironescu

By Federer and Fleming there exist at least one mass-minimizing normal current in every real-valued homology class of a Riemannian manifold. However the regularity of the mass-minimizing currents and their distributions may generally be…

In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Eulcidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition…

For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$, we prove an inequality regarding the geometric $L^2$-norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and…

For any non-elementary, torsion-free hyperbolic group, we provide a correspondence between the left-invariant Gromov-hyperbolic metrics on the group that are quasi-isometric to a word metric, and continuous reparameterizations of the…

Sobolev spaces are a natural framework for the analysis of problems in partial differential equations and calculus of variations. Some physical and geometric contexts, such as liquid crystals models and harmonic maps, lead to consider…

We study the immersed boundary problem in 2-D. It models a 1-D elastic closed string immersed and moving in a fluid that fills the entire plane, where the fluid motion is governed by the 2-D incompressible Navier-Stokes equation with a…

This paper examines the Balanced Submodular Flow Problem, that is the problem of finding a feasible submodular flow minimizing the difference between the flow values along the edges. A min-max formula is given to the problem and an…

Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by…

In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are…

E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the…

We consider vectorial problems in the calculus of variations with an additional pointwise constraint. Our admissible mappings ${\bf n}:\mathbb{R}^k\rightarrow \mathbb{R}^d$ satisfy ${\bf n}(x)\in M$, where $M$ is a manifold embedded in…

We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve…

We consider the approximation of minimal geodesics between two closed sets in $\mathbb{R}^D$ endowed with a smooth Riemannian metric. The continuous problem is formulated as the minimization of the energy functional over piecewise smooth…

Let $\Sigma$ be a closed orientable hyperbolic surface. We introduce the notion of a \textit{geodesic current with corners} on $\Sigma$, which behaves like a geodesic current away from certain singularities (the "corners"). We topologize…

We generalize McCann's theorem of optimal transport to a submanifold setting and prove Michael-Simon-Sobolev inequalities for submanifolds in manifolds with lower bounds on intermediate Ricci curvatures. The results include a variant of the…

In this paper we study the quasilinear equation $- \ep^2 \Delta u-\Delta_p u=f(u)$ in a smooth bounded domain $\Omega$ with Dirichlet boundary condition. For $\ep \geq 0$, we review existence of a least energy nodal solution and then…

We prove that tangent cones at singular boundary points of a two-dimensional current almost area minimizing are unique. Following the ideas exposed by White in [8], the result is achieved by combining a suitable epiperimetric inequality and…

This paper is devoted to exploring a new minimax approach by introducing a characteristic mapping family which is invariant under the smooth descending flow for initial value. The minimax approach is self-contained, and its features are…

We consider the simplest optimal control problem with one nonregular mixed inequality constraint, i.e. when its gradient in the control can vanish on the zero surface. Using the Dubovitskii--Milyutin theorem on the approximate separation of…

New solutions to the classical equations of motion of a bosonic matrix-membrane are given. Their continuum limit defines 3-manifolds (in Minkowski space) whose mean curvature vanishes. Part of the construction are minimal surfaces in S^7,…