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The Dirichlet problem for the minimal surface equation on unbounded planar domains

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Harold Rosenberg
Harold Rosenberg
Harold Rosenberg
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Ricardo Sa Earp at Pontifícia Universidade Católica do Rio de Janeiro
Ricardo Sa Earp
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Abstract

On considère le problème de Dirichlet pour l'équation de surface minimale sur Ω: (1+q^2)r−2pqs+(1+p^2)t=0 en prenant des valeurs prescrites sur δΩ. On étudie le cas de domaines non bornés généraux et des données limites non linéaires We prove some existence and uniqueness results of the solutions of the minimal equation in two independent variables, when the ambient is the Euclidean space. We prove a maximum principle inside a band or proper sector. We prove a Phragmèn-Lindelöf type theorem. We prove existence over some non-convex domains. We prove a modulus of continuity result. We are not able to publish full-text.
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From References: 12
From Reviews: 1
MR1010767 (90m:35072) 35J65 (35B45 53A10)
SaEarp,Ricardo (F-PARIS7-MI);Rosenberg,Harold (F-PARIS7-MI)
TheDirichletproblemfortheminimalsurfaceequationonunboundedplanardomains.
J.Math.Pures Appl.(9) 68 (1989),no.2, 163–183.
Let Ube an infinite strip in R2,fa bounded uniformly continuous real function on the boundary
of Uand Fa solution of the associated Plateau problem, i.e. a continuous minimal extension
of fto U. The main results of this article are as follows. (0) The existence of Fis derived
from existence, comparison and compactness theorems for bounded domains of H. Jenkins and
J. Serrin [Arch. Rational Mech. Anal. 21 (1963), 321–342; MR0190811 (32 #8221)], using a
compact exhaustion of U. (1) Fsatisfies a maximum principle, namely: sup |F|= sup |f|. (2) F
is uniformly continuous on U. (3) A Phragm´
en-Lindel¨
of theorem holds for the minimal surface
operator in U. (4) Fis unique. Results (1) and (2) also hold when Uis a proper sector (i.e. with
vertexanglelessthanπ); such results would not holdforharmonicextensions(example:F(x, y) =
xy in a quadrant). Results (0) to (4) hold more generally when Uis the union of compact convex
subset with finitely many disjoint half strips attached to its boundary (a so-called convex city
map). Result (3) complements the Bernstein result of R. Langevin, G. Levitt and Rosenberg [Duke
Math. J. 55 (1987), no. 4, 985–995; MR0916132 (89h:53022)]; the authors have a preprint on its
extension to a slab in Rn,n≥3. Comparison theorems when Uis an exterior domain were carried
out by Langevin and Rosenberg [ibid. 57 (1988), no. 3, 819–828; MR0975123 (90c:53025)] and
by R. Krust [ibid. 59 (1989), no. 1, 161–173; MR1016882 (90i:49050)].
ReviewedbyPhilippeDelano¨
e
c
Copyright American Mathematical Society 1990, 2013

Supplementary resource (1)

On the minimal equation in Euclidean space
Data
January 1989
Harold Rosenberg · Ricardo Sa Earp
... One of the earliest significant contributions in this field was made by H. Rosenberg and R. Sa Earp, [20]. They proved the existence of a solution to the Dirichlet problem for the minimal surface equation in unbounded domains that are contained in a strip or a sector of the plane. ...
... Denote Λ(r) = Ω ∩ (x, y) ∈ R 2 : x 2 + y 2 = r and M (r) = sup The main purpose of this work is to extend the results in [20] and Theorem 1 to the more general setting of a Killing submersion. An oriented and connected 3dimensional manifold E admitting a non-singular Killing vector field admits a Killing submersion π : E → M onto a Riemannian surface M , connected and oriented, whose fibers are the integral curves of a non-zero Killing vector field ξ ∈ X(E). ...
... The results originally established in [20] have been further expanded upon in subsequent works. In [21], R. Sa Earp and E. Toubiana extended these results to hyperbolic space, while in [22], they considered the case of H 2 × R. The authors, along with B. Nelli, further extended these findings to the Heisenberg group in [18]. ...
Killing graphs over non-compact domains in 3-manifolds with a Killing vector field
Preprint
Full-text available
  • Jul 2023
Let $\mathbb{E}$ be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field $\xi\in\mathfrak{X}(\mathbb{E})$ whose associated one-parameter group of the isometries of $\mathbb{E}$ acts freely and properly on $\mathbb{E}$. Then, there is a Killing submersion from $\mathbb{E}$ onto a connected and orientable surface $M$ whose fibers are the integral curves of $\xi$. We solve the Dirichlet problem for the minimal surface equation over certain unbounded domains of $M$, taking piecewise continuous boundary values. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We obtain Collin-Krust type estimates in arbitrary Killing submersions with not necessary unitary Killing vector field. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable.
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... When R 2 is contained in a wedge of opening angle < , then [14] showed that any minimal graph over with zero Dirichlet data must be 0. In a similar vein, when R 2 is a wedge with opening angle ¤ ; 2 , then [11] showed any minimal graph with zero Neumann data away from the cone point must be constant (see also [13]). In [12] the second-named author and his collaborators obtained Liouville-type theorems for minimal graphs over half-spaces (for any n) with linear Dirichlet boundary value or constant Neumann boundary value. ...
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... On the other hand, after further requiring the boundary value to be bounded, Hwang [12] and Mīkljukov [17] confirmed Nitsche's conjecture independently. Moreover, when D ⊂ R 2 is the union of a compact convex subset with finitely many disjoint half strips attached to its boundary, Sa Earp and Rosenberg [20] proved the uniqueness of the solution of the minimal surface equation with bounded uniformly continuous boundary value. ...
Stability of Bernstein type theorem for the minimal surface equation
Preprint
  • Jan 2022
Let $ \Omega \subsetneq \mathbf{R}^n\,(n\geq 2)$ be an unbounded convex domain. We study the minimal surface equation in $\Omega$ with boundary value given by the sum of a linear function and a bounded uniformly continuous function in $ \mathbf{R}^n$. If $ \Omega $ is not a half space, we prove that the solution is unique. If $ \Omega $ is a half space, we prove that graphs of all solutions form a foliation of $\Omega\times\mathbf{R}$. This can be viewed as a stability type theorem for Edelen-Wang's Bernstein type theorem in \cite{EW2021}. We also establish a comparison principle for the minimal surface equation in $\Omega$.
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... When Ω ⊂ R 2 is contained in a wedge of opening angle < π, then [20] showed that any minimal graph over Ω with zero Dirichlet data must be 0. In a similar vein, when Ω ⊂ R 2 is a wedge with opening angle = π, 2π, then [6] showed any minimal graph with zero Neumann data away from the cone point must be constant (see also [11]). In [10] the second named author and his collaborators obtained Liouville type theorems for minimal graphs over half-spaces (for any n) with linear Dirichlet boundary value or constant Neumann boundary value. ...
A Bernstein Type Theorem for Minimal Graphs over Convex Cones
Preprint
  • Aug 2020
Given any $n \geq 2$, we show that if $\Omega \subsetneq \mathbb{R}^n$ is an open, convex cone (e.g. a half-space), and $u : \Omega \to \mathbb{R}$ is a solution to the minimal surface equation which agrees with a linear function on $\partial\Omega$, then $u$ must itself be linear.
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... In R 3 , if the boundary data on a wedge of angle less that π is zero (respectively bounded) then the minimal solution is zero (resp. bounded) [22]. ...
Minimal Graphs in Nil_3 : existence and non-existence results (thanks to the referee for his competent and deep analysis)
Article
Full-text available
  • Feb 2017
  • CALC VAR PARTIAL DIF
We study the minimal surface equation in the Heisenberg space, Nil_3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved for some prescribed boundary data (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbounded convex domains, taking bounded, piecewise continuous boundary value. We are able to construct a Scherk type minimal surface and we use it as a barrier to construct non trivial minimal graphs over a wedge of angle θ ∈ [ π /2 , π[, taking non negative continuous boundary data, having at least quadratic growth. In the case of an half-plane, we are also able to give solutions (with either linear or quadratic growth), provided some geometric hypothesis on the boundary data are satisfied. Finally, some open problems arising from our work, are posed. See http://link.springer.com/article/10.1007/s00526-017-1123-y
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The Dirichlet problem for the minimal surface equation on unbounded helicoidal domains of $\mathbb{R}^{m}
Preprint
Full-text available
  • Jun 2023
We consider a helicoidal group $G$ in $\mathbb{R}^{n+1}$ and unbounded $G$-invariant $C^{2,\alpha}$-domains $\Omega\subset\mathbb{R}^{n+1}$ whose helicoidal projections are exterior domains in $\mathbb{R}^{n}$, $n\geq2$. We show that for all $s\in\mathbb{R}$, there exists a $G$-invariant solution $u_{s}\in C^{2,\alpha}\left( \overline{\Omega}\right) $ of the Dirichlet problem for the minimal surface equation with zero boundary data which satisfies $\sup_{\partial\Omega}\left\vert \operatorname{grad}u_{s}\right\vert =\left\vert s\right\vert $. Additionally, we provide further information on the behavior of these solutions at infinity.
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The Dirichlet Problem in Unbounded Domains
Chapter
  • Jan 2013
This chapter deals with the Dirichlet problem for the constant mean curvature equation in the case that the domain \(\varOmega\subset \mathbb{R}^{2}\) is not bounded and \(H\not=0\). We shall characterize the solvability of the Dirichlet problem when Ω is convex. In this setting, we employ cylinders as barriers. If Ω is not convex, we use nodoids to obtain results of existence when Ω satisfies a uniform circle exterior condition. By the way, we shall obtain height estimates for cmc graphs with the same flavor to the ones obtained for bounded domains. The method of existence that we apply is a modified version of the classical Perron method of super and subsolutions. The subsolution will be a solution of the minimal surface equation, while a supersolution is replaced by a family of local upper barriers, which will be pieces of cylinders or nodoids depending on the domain.
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A plateau problem at infinity for properly immersed minimal surfaces with finite total curvature
Article
  • Jun 2001
  • INDIANA U MATH J
Given an ordered list of nonzero horizontal vectors υ1,... , υr, r ≥ 3, such that υ1 + ⋯+ υr = 0, we obtain a necessary and sufficient condition for the existence of an Alexandrov-embedded genus zero minimal r-noid M with prescribed fluxes υ1,... , υr at its r consecutive horizontal catenoidal ends: the surface M exists if and only if the closed polygonal curve whose edges are the vectors υi bounds an immersed disc in the plane. Moreover we prove that M is nondegenerate, in the sense that it does not admit nontrivial bounded Jacobi functions, and we study the uniqueness problem.
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