scholarly journals A duality result between the minimal surface equation and the maximal surface equation

2001 ◽  
Vol 73 (2) ◽  
pp. 161-164 ◽  
Author(s):  
LUIS J. ALÍAS ◽  
BENNETT PALMER
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Minimal Surface ◽  
Minkowski Space ◽  
Minimal Surfaces ◽  
Maximal Surface ◽  
Maximal Surfaces ◽  
Surface Equation ◽  
Duality Result ◽  
Minimal Surface Equation

In this note we show how classical Bernstein's theorem on minimal surfaces in the Euclidean space can be seen as a consequence of Calabi-Bernstein's theorem on maximal surfaces in the Lorentz-Minkowski space (and viceversa). This follows from a simple but nice duality between solutions to their corresponding differential equations.

The Mapping of the Main Functions and Different Variations of YH-DIE

2020 ◽  
Author(s):  
Gilbert Mason ◽  
Yuanjue Chou ◽  
Springer Pind
Keyword(s):  
Algebraic Geometry ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Minimal Surface ◽  
Access Point ◽  
Partial Differential ◽  
Surface Equation ◽  
Minimal Surface Equation ◽  
Fusion Point ◽  
Special Case

YH-DIE must have continuity . Given the basic algebraic clusters of homogeneous configurations,we can get the basic three equations: \begin{array}{l} {\mathop{\int}\nolimits_{0}\nolimits^{{x}_{i}}{\frac{{G}\left({{x}_{i}\mathrm{,}s}\right)}{{\left({{x}_{i}\mathrm{{-}}{s}}\right)}^{\mathit{\alpha}}}\mathrm{\varphi}\left({s}\right){ds}}\mathrm{{=}}{f}\left({{x}_{i}}\right)}\ ;\ {\frac{\mathrm{\partial}}{\mathrm{\partial}{x}_{i}}\left({\frac{{\mathrm{\partial}}_{{x}_{i}}G}{\sqrt{{1}\mathrm{{+}}{\left|{\mathrm{\nabla}{G}}\right|}^{2}}}}\right)\mathrm{{=}}{0}}\ ;\ {{i}\mathrm{{=}}\mathop{\sum}\limits_{{x}_{i}\mathrm{{=}}{1}}\limits^{\mathrm{\infty}}{\arccos\hspace{0.33em}\mathrm{\varphi}\left({{x}_{i}}\right)}\mathrm{{=}}{f}\left({\fbox{${Yuh}$}}\right)} \end{array} YH-DIE has become a fusion point and access point in the fields of algebraic geometry and partial differential equations, and its mapping on multidimensional algebraic clusters or manifolds is very special. The minimal surface equation is a special case.

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 A Spin-Perturbation for Minimal Surfaces

2021 ◽  
Author(s):  
Ruijun Wu
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Energy Momentum Tensor ◽  
Harmonic Maps ◽  
Momentum Tensor ◽  
Lagrange Equations ◽  
Surface Equation ◽  
Minimal Surface Equation ◽  
Volume Functional

AbstractWe investigate the coupling of the minimal surface equation with a spinor of harmonic type. This arises as the Euler–Lagrange equations of the sum of the volume functional and the Dirac action, defined on an appropriated Dirac bundle. The solutions show a relation to Dirac-harmonic maps under some constraints on the energy-momentum tensor, extending the relation between Riemannian minimal surface and harmonic maps.

FEDOROV UNIVERSAL EQUATIONS IN THE STRING AND MEMBRANE THEORIES

1990 ◽  
Vol 04 (01) ◽  
pp. 93-111
Author(s):  
N. S. SHAVOKHINA
Keyword(s):  
Euclidean Space ◽  
Minimal Surface ◽  
Minkowski Space ◽  
Matrix Equation ◽  
Minimal Surfaces ◽  
Minimal Hypersurface ◽  
Spherically Symmetric ◽  
Partial Derivatives ◽  
First Order ◽  
Rich Information

It is shown that the equation of minimal hypersurface in the Euclidean (or pseudo-Euclidean) space can be written as the universal Fedorov matrix equation with first-order partial derivatives. Time-like minimal surfaces in the pseudo-Euclidean Minkowski space describe the free motion of relativistic strings and membranes, whereas space-like minimal surfaces describe the potential in the nonlinear Born electrostatic. All of them are imaginary images of minimal surface of the Euclidean space. Spherically symmetric surfaces are found to be all the three types, the hypercatenoid of any dimensionality and its imaginary images. The Fedorov equations provide rich information on the minimal surfaces.

scholarly journals On the analyticity of generalized minimal surfaces

1971 ◽  
Vol 5 (3) ◽  
pp. 315-320 ◽  
Author(s):  
Neil S. Trudinger
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Global Regularity ◽  
Regularity Result ◽  
Classical Solutions ◽  
Surface Equation ◽  
Minimal Surface Equation

Strongly differentiable solutions of the minimal surface equation are shown to be classical solutions and consequently locally analytic. A global regularity result is also proved.

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