• Corpus ID: 246016161

A non-parametric Plateau problem with partial free boundary

  title={A non-parametric Plateau problem with partial free boundary},
  author={Giovanni Bellettini and Roberta Marziani and Riccardo Scala},
We consider a Plateau problem in codimension 1 in the non-parametric setting. A Dirichlet boundary datum is given only on part of the boundary ∂Ω of a bounded convex domain Ω ⊂ R. Where the Dirichlet datum is not prescribed, we allow a free contact with the horizontal plane. We show existence of a solution, and prove regularity for the corresponding minimal surface. Finally we compare these solutions with the classical minimal surfaces of Meeks and Yau, and show that they are equivalent when… 

The relaxed area of $S^1$-valued singular maps in the strict $BV$-convergence

Given a bounded open set $\Omega \subset \mathbb R^2$, we study the relaxation   of the nonparametric area functional in the strict topology in $BV(\Omega;\mathbb R^2)$, and compute it for



Cartesian currents in the calculus of variations

1. Regular Variational Integrals.- 2. Finite Elasticity and Weak Diffeomorphisms.- 3. The Dirichlet Integral in Sobolev Spaces.- 4. The Dirichlet Energy for Maps into S2.- 5. Some Regular and Non

Minimal Surfaces I

. It is well known that isoperimetric regions in a smooth compact ( n +1)-manifold are themselves smooth, up to a closed set of codimension at most 6. In this note, we construct an 8-dimensional

On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions

In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three $${\mathcal {C}}^2$$ C 2 -embedded

Minimal surfaces and functions of bounded variation

I: Parametric Minimal Surfaces.- 1. Functions of Bounded Variation and Caccioppoli Sets.- 2. Traces of BV Functions.- 3. The Reduced Boundary.- 4. Regularity of the Reduced Boundary.- 5. Some

New lower semicontinuity results for polyconvex integrals

SummaryWe study integral functionals of the formF(u, Ω)=∫Ωf(▽u)dx, defined foru ∈ C1(Ω;Rk), Ω⊑Rn. The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) ≥c0¦ℳ(A)¦ for a suitable

Optimal estimates for the triple junction function and other surprising aspects of the area functional

We consider the relaxed area functional for vector valued maps and its exact value on the triple junction function u : B1(O) → R, a specific function which represents the first example of map whose

Functions of Bounded Variation and Free Discontinuity Problems

Measure Theory Basic Geometric Measure Theory Functions of bounded variation Special functions of bounded variation Semicontinuity in BV The Mumford-Shah functional Minimisers of free continuity

The $L^1$-relaxed area of the graph of the vortex map

We compute the value of the $L^1$-relaxed area of the graph of the map $u : B_l(0)\subset \mathbb R^2 \mapsto \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, for all values of $l>0$.