scholarly journals MINIMAL GRAPHS WITH DISCONTINUOUS BOUNDARY VALUES

2009 ◽  
Vol 86 (1) ◽  
pp. 75-95 ◽  
Author(s):  
ROBERT HUFF ◽  
JOHN MCCUAN
Keyword(s):  
Boundary Component ◽  
Global Solutions ◽  
Jump Discontinuity ◽  
Boundary Values ◽  
Minimal Graph ◽  
Closed Curves ◽  
Minimal Graphs ◽  
Minimal Surface Equation ◽  
Discontinuous Boundary ◽  
Annular Domains

AbstractWe construct global solutions of the minimal surface equation over certain smooth annular domains and over the domain exterior to certain smooth simple closed curves. Each resulting minimal graph has an isolated jump discontinuity on the inner boundary component which, at least in some cases, is shown to have nonvanishing curvature.

scholarly journals P-harmonic dimensions on ends

1993 ◽  
Vol 132 ◽  
pp. 131-139
Author(s):  
Michihiko Kawamura ◽  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Boundary Component ◽  
Harmonic Functions ◽  
Ideal Boundary ◽  
Boundary Values ◽  
Hölder Continuous ◽  
Closed Curves ◽  
Open Riemann Surface ◽  
Relative Boundary ◽  
Bounded Harmonic Functions

Consider an end Ω in the sense of Heins (cf. Heins [3]): Ω is a relatively non-compact subregion of an open Riemann surface such that the relative boundary ∂Ω consists of finitely many analytic Jordan closed curves, there exist no non-constant bounded harmonic functions with vanishing boundary values on ∂Ω and Ω has a single ideal boundary component. A density P = P(z)dxdy (z = x + iy) is a 2-form on Ω∩∂Ω with nonnegative locally Holder continuous coefficient P(z).

scholarly journals Nonparametric minimal surfaces inR3whose boundaries have a jump discontinuity

1988 ◽  
Vol 11 (4) ◽  
pp. 651-656 ◽  
Author(s):  
Kirk E. Lancaster
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Variational Solution ◽  
Jump Discontinuity ◽  
Boundary Values ◽  
Surface Equation ◽  
Radial Limits ◽  
Minimal Surface Equation ◽  
Upper Limits ◽  
Locally Convex

LetΩbe a domain inR2which is locally convex at each point of its boundary except possibly one, say(0,0),ϕbe continuous on∂Ω/{(0,0)}with a jump discontinuity at(0,0)andfbe the unique variational solution of the minimal surface equation with boundary valuesϕ. Then the radial limits offat(0,0)from all directions inΩexist. If the radial limits all lie between the lower and upper limits ofϕat(0,0), then the radial limits offare weakly monotonic; if not, they are weakly increasing and then decreasing (or the reverse). Additionally, their behavior near the extreme directions is examined and a conjecture of the author's is proven.

scholarly journals On Ramsey-Minimal Infinite Graphs

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Jordan Barrett ◽  
Valentino Vito
Keyword(s):  
Ramsey Theory ◽  
Infinite Graphs ◽  
Minimal Graph ◽  
Finite Graphs ◽  
Minimal Graphs ◽  
Finite Case ◽  
Ramsey Minimal Graph ◽  
General Conditions

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.

scholarly journals For the minimal surface equation, the set of solvable boundary values need not be convex

1996 ◽  
Vol 53 (3) ◽  
pp. 369-372
Author(s):  
Frank Morgan
Keyword(s):  
Minimal Surface ◽  
Smooth Domain ◽  
Boundary Values ◽  
Surface Equation ◽  
Minimal Surface Equation

One might think that if the minimal surface equation had a solution on a smooth domain D ⊂ Rn with boundary values φ, it would have a solution with boundary values tφ for all 0 ≤ t ≤ 1. We give a counterexample in R2.

scholarly journals Boundary properties of fractional objects: Flexibility of linear equations and rigidity of minimal graphs

2020 ◽  
Vol 2020 (769) ◽  
pp. 121-164 ◽  
Author(s):  
Serena Dipierro ◽  
Ovidiu Savin ◽  
Enrico Valdinoci
Keyword(s):  
Linear Equations ◽  
Normal Derivative ◽  
Small Error ◽  
Minimal Graph ◽  
Minimal Graphs ◽  
Formal Level ◽  
Boundary Properties ◽  
Laplace Problem ◽  
Fractional Type ◽  
Curvature Type

AbstractThe main goal of this article is to understand the trace properties of nonlocal minimal graphs in {\mathbb{R}^{3}}, i.e. nonlocal minimal surfaces with a graphical structure.We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum.This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error.We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.

scholarly journals The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values

2000 ◽  
Vol 130 (3) ◽  
pp. 591-602 ◽  
Author(s):  
H. A. Levine ◽  
Q. S. Zhang
Keyword(s):  
Boundary Value Problem ◽  
Global Solutions ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Values ◽  
Exterior Domains ◽  
Semilinear Heat Equation ◽  
Image Position ◽  
Initial Boundary Value

Let D be a domain in Rn with bounded complement and let n ≠ 2. For the initial-boundary value problem we prove that there are no non-trivial global (non-negative) solutions if 0 < n (p − 1) ≤ 2 and there exist both global non-trivial and non-global solutions if n (p − 1) > 2.

Estimation and Marginalization Using the Kikuchi Approximation Methods

2005 ◽  
Vol 17 (8) ◽  
pp. 1836-1873 ◽  
Author(s):  
Payam Pakzad ◽  
Venkat Anantharam
Keyword(s):  
Approximation Method ◽  
Message Passing ◽  
Belief Propagation ◽  
Iterative Algorithms ◽  
Product Distribution ◽  
Stationary Points ◽  
Minimal Graph ◽  
Minimal Graphs ◽  
Minimum Number ◽  
Free Case

In this letter, we examine a general method of approximation, known as the Kikuchi approximation method, for finding the marginals of a product distribution, as well as the corresponding partition function. The Kikuchi approximation method defines a certain constrained optimization problem, called the Kikuchi problem, and treats its stationary points as approximations to the desired marginals. We show how to associate a graph to any Kikuchi problem and describe a class of local message-passing algorithms along the edges of any such graph, which attempt to find the solutions to the problem. Implementation of these algorithms on graphs with fewer edges requires fewer operations in each iteration. We therefore characterize minimal graphs for a Kikuchi problem, which are those with the minimum number of edges. We show with empirical results that these simpler algorithms often offer significant savings in computational complexity, without suffering a loss in the convergence rate. We give conditions for the convexity of a given Kikuchi problem and the exactness of the approximations in terms of the loops of the minimal graph. More precisely, we show that if the minimal graph is cycle free, then the Kikuchi approximation method is exact, and the converse is also true generically. Together with the fact that in the cycle-free case, the iterative algorithms are equivalent to the well-known belief propagation algorithm, our results imply that, generically, the Kikuchi approximation method can be exact if and only if traditional junction tree methods could also solve the problem exactly.

scholarly journals Convex Combinations of Minimal Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Michael Dorff ◽  
Ryan Viertel ◽  
Magdalena Wołoszkiewicz
Keyword(s):  
Convex Combination ◽  
Sufficient Conditions ◽  
Gauss Map ◽  
Minimal Graph ◽  
Periodic Surface ◽  
Minimal Graphs

Given a collection of minimal graphs,M1,M2,…,Mn, with isothermal parametrizations in terms of the Gauss map and height differential, we give sufficient conditions onM1,M2,…,Mnso that a convex combination of them will be a minimal graph. We will then provide two examples, taking a convex combination of Scherk's doubly periodic surface with the catenoid and Enneper's surface, respectively.

Complete systems of surfaces in 3-manifolds

1997 ◽  
Vol 122 (1) ◽  
pp. 185-191 ◽  
Author(s):  
FENGCHUN LEI
Keyword(s):  
Finite Number ◽  
Boundary Component ◽  
Pairwise Disjoint ◽  
Complete System ◽  
Closed Surface ◽  
The Other ◽  
Closed Curves ◽  
Orientable Surfaces

A complete system (CS) [Jscr ]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr ] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr ] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr ] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr ] is a CS of surfaces for M, then any CS equivalent to [Jscr ] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds:

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