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International audience; Both optimal transport and minimal surfaces have received much attention in recent years. We show that the methodology introduced by Kantorovich on the Monge problem can, surprisingly, be adapted to questions involving least area, e.g. Plateau type problems as investigated by Federer.

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space SL ˜ 2 ( R ) with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in SL ˜ 2 ( R ) contained between two bounded entire minimal graphs separated by vertical distance less than 1 + 4 τ 2 π has multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a family of complete embedded minimal surfaces which are non-proper in SL ˜ 2 ( R ) .

We study the minimization problem for the Yang–Mills energy under fixed boundary connection in supercritical dimension n ≥ 5 . We define the natural function space A G in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A G can be also interpreted as a space of weak connections on a “real measure theoretic version” of reflexive sheaves from complex geometry. We prove the existence of weak solutions to the Yang–Mills Plateau problem in the space A G . We then prove the optimal regularity result for solutions of this Plateau problem. On the way to prove this result we establish a Coulomb gauge extraction theorem for weak curvatures with small Yang–Mills density. This generalizes to the general framework of weak L 2 curvatures previous works of Meyer–Riviere and Tao–Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.

This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C n with n ≥ 3 . The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [23] are taken from our earlier arxiv post [22] . We split [22] into two independent articles to avoid it being too long.

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger–Wang, Chau–Weinkove and the author solved this global problem in W 4 , p under some restrictions on the sign or integrability of the affine mean curvature. We remove these restrictions in this paper and obtain W 4 , p solution to the second boundary value problem when the affine mean curvature belongs to L p with p greater than the dimension. Our self-contained analysis also covers the case of Abreu's equation.

In this paper, we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order~$s \in (0,1/2)$ on a bounded domain $\Omega$. Such a Plateau problem of order $s$ can be reinterpreted as a Dirichlet problem for a nonlocal, nonlinear, degenerate operator of order $s + 1/2$. We prove that our numerical scheme converges in $W^{2r}_1(\Omega)$ for all $r, Comment: Lemma 4.1 was removed due to a mistake in its proof. This only affects the new Theorem 4.2 (convergence), whose proof has been corrected

The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization @c of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization @k of the unit circle. The Dirichlet energy of the harmonic extension of @[email protected][email protected] has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.

Consider the Plateau problem for spacelike surfaces with constant mean curvature in three-dimensional Lorentz–Minkowski space L 3 and spanning two circular axially symmetric contours in parallel planes. In this paper, we prove that rotational symmetric surfaces are the only solutions. We also give a result on uniqueness of spacelike surfaces of revolution with constant mean curvature as solutions of the exterior Dirichlet problem under a certain hypothesis at infinity.

For the minimal surfaces in Rn with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence of the initial datum.

LET y be a rectifiable Jordan curve in three-dimensional euclidean space. Answering an old question, whether y can bound a surface with minimal area, Douglas [l I] and Rad6[45] (independently) found a minimal surface spanning y which is parametrized by the disk. This minimal surface has minimal area among all Lipschitz maps from the disk into R3 which span y. The question whether this solution has branch points or not was finally settled by Osserman[42], who proved that there are no interior “true” branch points, and by Gulliver[lS], who proved that there are no interior “false” branch points. In 1948, Morrey[35] devised a new method to solve the Plateau problem for a map from the disk into a “homogeneously regular*’ Riemannian manifold. Moreover, he proved the interior regularity of the map in case the ambient manifold is regular, and that the map is real analytic if the ambient manifold is real analytic. The arguments of Osserman and Gulliver in addition show that Morrey’s solution has no interior branch point when the ambient manifold is three-dimensional. In 1951, Lewy[29] showed that if the Jordan curve y is also real analytic, in a real analytic manifold, then any minimal surface with boundary y is real analytic up to the boundary. Hence in this case Morrey’s map is real analytic map on the closed disk. (For a proof of Lewy’s Theorem in a general real analytic manifold, see HiIdebrandt[23].) In 1969, Hildebrandt[23] proved that the Douglas solution is smooth up to the boundary if the Jordan curve is smooth and regular. (Further improvements are due to Kinderlehrer [25], Nitsche [40], and Warschawski[55].) In [22], Heinz and Hildebrandt extended Hildebrandt’s result to minimal surfaces in general Riemannian manifolds. Once we have boundary regularity, it makes sense to ask whether Douglas’ or Morrey’s solution of the Plateau problem has a boundary branch point or not, To date, this problem has not been settled. The first partial result in this direction is due to Nitsche[40], who showed that there are only a finite number of boundary branch points for minimal surfaces with smooth boundary in R3. This was then generalized by Heinz and Hildebrandt to smooth manifolds. Gulliver and Lesley[l6] have also observed that the Douglas-Morrey solution for a real analytic curve in a real analytic manifold has no boundary branch point, using the previously mentioned result of Lewy. Despite all these results, an interesting topological question remained unsolved, namely, under what conditions is the Douglas solution an embedded surface? It has been generally conjectured that when the Jordan curve is extremal, i.e. lies on the

Heat pulse profiles of vitreous silica show abrupt intensity decrease and wavy structures beyond a critical heat power at a few K, which is caused by liquid helium film boiling on the heater. These anomalous features are discussed from the viewpoint of the phonon localization picture for the plateau problem.

The chemical potential at a curved or faceted surface is proportional to the surface divergence of the previousll-defined vector function ξ, lhich represents the energetics of both isotropic and anisotropic surfaces in a convenient manner. This relationship is sholn to be a generaliled reformulation of the eluations of Gibbs, Thompson, Herring and Johnson, but leads in addition to nel alternative formulas in lhich certain underlling slmmetries become evident. Applications of the vector formalism include analltic proof of the lulff theorem, distribution of torlue on a facet, formulation of the anisotropic Plateau problem, and the eluilibrium shapes of particles at surfaces and grain boundaries.