Boundary regularity for minimizing currents with prescribed mean curvature
type: Item
instance of: scholarly article
title: Boundary regularity for minimizing currents with prescribed mean curvature
zbMATH Open document ID: 0806.49028
DOI: 10.1007/BF01206958
author: Frank Duzaar
author: Klaus Steffen
published in: Calculus of Variations and Partial Differential Equations
publication date: 1994-01-13T00:00:00Z
review text: The authors prove complete boundary regularity for energy minimizing integer multiplicity rectifiable \(n\) currents \(T\) in \(\mathbb{R}^{n+1}\) of prescribed mean curvature \(H\) with boundary \(B= \partial T\) represented by an oriented smooth submanifold of dimension \(n-1\) in \(\mathbb{R}^{n+1}\). Also, combining their boundary regularity theorem with the interior regularity theory, they formulate and prove a theorem regarding the Plateau problem for surfaces with prescribed mean curvature. The notion of energy minimality here refers to the energy functional \(E_ H\) associated with a prescribed mean curvature function \(H: \mathbb{R}^{n+1}\to \mathbb{R}\), i.e., \(E_ H(T)= M(T)- V_ H(T)\), where \(M(T)\) is the mass (= total \(n\) area) of \(T\) and \(V_ H(T)\) is the \(H\) weighted volume enclosed by \(T\) and a fixed integer multiplicity rectifiable reference \(n\) current \(T_ 0\) with \(\partial T_ 0= B\). The references contain 35 papers, but the indispensable work for the preceding results is \textit{R. Hardt} and \textit{L. Simon} [Ann. Math., II. Ser. 110, 439-486 (1979; Zbl 0457.49029)].
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 484525
zbMATH Keywords: rectifiable currents
zbMATH Keywords: energy functional
zbMATH Keywords: complete boundary regularity
zbMATH Keywords: prescribed mean curvature
zbMATH Keywords: boundary regularity
zbMATH Keywords: Plateau problem
zbMATH Keywords: energy minimality
reviewed by: Constantin Udrişte
MaRDI profile type: Publication
cites work: On boundary regularity for Plateau's problem
cites work: On the first variation of a varifold
cites work: On the first variation of a varifold: Boundary behavior
cites work: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure
cites work: Verzweigungspunkte von H-Flächen. I
cites work: Q4151298
cites work: On the Size of a Stable Minimal Surface in R 3
cites work: Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
cites work: On the existence of surfaces with prescribed mean curvature and boundary in higher dimensions
cites work: Area minimizing hypersurfaces with prescribed volume and boundary
cites work: On the existence of integral currents with prescribed mean curvature vector
cites work: Q4038322
cites work: Q4049704
cites work: \(\lambda\) minimizing currents
cites work: Q5565773
cites work: The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension
cites work: Regularity of minimizing surfaces of prescribed mean curvature
cites work: On boundary branch points of minimizing surfaces
cites work: Elliptic Partial Differential Equations of Second Order
cites work: On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand
cites work: Boundary regularity and embedded solutions for the oriented Plateau problem
cites work: Esistenza e regolarita delle ipersuperfici di curvatura media assegnata in \(R^n\)
cites work: Q5518064
cites work: Q3679817
cites work: Q3336268
cites work: On the existence of surfaces with prescribed mean curvature and boundary
cites work: The structure of singularities in solutions to ellipsoidal variational problems with constraints in \(R^3\)
cites work: Q3310565
Recommended article: Existence and regularity of hypersurfaces with prescribed mean curvature and a free boundary.
Recommended article: \(\lambda\) minimizing currents
Recommended article: On the existence of surfaces with prescribed mean curvature and boundary in higher dimensions
Recommended article: Boundary regularity for area minimizing currents with prescribed volume
Recommended article: Regularity results for solutions to the \(c\)-Plateau problem with free boundary having small singular set, and generally in codimension zero
Recommended article: On integral currents with constant mean curvature
Recommended article: Q3683142
Recommended article: Solutions to the mean curvature equation by fixed point methods
Recommended article: Q3733658
Recommended article: Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds
description: scientific article; zbMATH DE number 484525