Related papers: Weiss monotonicity and capillary hypersurfaces

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,\alpha}$ orientable submanifolds which only…

In this paper we study surfaces in R^3 that arise as limit shapes in a class of random surface models arising from dimer models. The limit shapes are minimizers of a surface tension functional, that is, they minimize, for fixed boundary…

It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg-Landau equations $-\Delta u_\varepsilon +\varepsilon^{-2}(|u_\varepsilon|^2-1)u_\varepsilon = 0$, the energy and vorticity concentrate…

We establish local regularity results for minimizers of autonomous vectorial integrals of Calculus of Variations, assuming $\psi$-growth conditions and imposing $\varphi$-quasiconvexity only in an asymptotic sense, both in the sub-quadratic…

In this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange…

Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we…

We describe a general correspondence between weighted minimal surfaces in $\mathbb{R}^3$ and weighted maximal surfaces with some admissible singularities in $\mathbb{L}^3$, for a class of functions $\varphi$ which provides the corresponding…

We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L^2-topology. We also show that the rescaled Allen-Cahn functionals approximate this relaxed functional in the sense of…

We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function…

Motivated by questions on the delocalization of random surfaces, we prove that random surfaces satisfying a Lipschitz constraint rarely develop extremal gradients. Previous proofs of this fact relied on reflection positivity and were thus…

It has been found in numerical experiments that when one removes a sector from an elastic sheet and glues the edges of the sector back together, the resulting configuration is radially symmetric and nearly conical. We make a rigorous…

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $\alpha$, under volume constraint, where the strength of the nonlocal interaction is…

We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume inte- gral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove…

In this paper we study $\varphi$-minimal surfaces in $\mathbb{R}^3$ when the function $\varphi$ is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in $\mathbb{R}^2$. We…

For the Alt-Caffarelli problem, we study free boundary regularity of energy minimizers. In six dimensions, we show that free boundaries are analytic for generic boundary data. In general, we improve previous generic Hausdorff dimensions of…

In this paper we introduce the branched $\alpha$-flows on closed surfaces with Euler characteristic \(\chi \leq 0\). Based on the strict convexity of the branched $\alpha$-potentials, we establish the long time existence and convergence of…

In this paper, we first introduce quermassintegrals for capillary hypersurfaces in the half-space. Then we solve the related isoperimetric type problems for the convex capillary hypersurfaces and obtain the corresponding Alexandrov-Fenchel…

In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form \[ \mathfrak{F}( v, \Omega )= \int_{\Omega} \! F(x, Dv(x)) \, dx, \] where, for ${n > 2}$ and $N\ge 1$, $\Omega$ is a bounded…

Let $E \subset \Omega$ be a local almost-minimizer of the relative perimeter in the open set $\Omega \subset \mathbb{R}^{n}$. We prove a free-boundary monotonicity inequality for $E$ at a point $x\in \partial\Omega$, under a geometric…

Conventional wetting theories on rough surfaces with Wenzel, Cassie-Baxter, and Penetrate modes suggest the possibility of tuning the contact angle by adjusting the surface texture. Despite decades of intensive study, there are still many…