scholarly journals Minimal Surfaces: A Derivation of the Minimal Surface Equation for an Arbitrary ${\mathbb{C}}^2$ Coordinate Chart

2001 ◽  
Vol 13 (3) ◽  
pp. 145-153
Author(s):  
Sarah Field Griffin
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Coordinate Chart ◽  
Surface Equation ◽  
Minimal Surface Equation

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.

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.

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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