On two-dimensional variational problems in parametric form

1961 ◽  
Vol 8 (1) ◽  
pp. 181-206 ◽  
Author(s):  
H. B. Jenkins
Keyword(s):  
Variational Problems ◽  
Parametric Form ◽  
Two Dimensional

On two-dimensional parametric variational problems

1999 ◽  
Vol 9 (3) ◽  
pp. 249-267 ◽  
Author(s):  
Stefan Hildebrandt ◽  
Heiko von der Mosel
Keyword(s):  
Variational Problems ◽  
Two Dimensional

VARIATIONAL PROBLEMS ON UNBOUNDED TWO-DIMENSIONAL DOMAINS

1992 ◽  
Vol 02 (02) ◽  
pp. 183-201
Author(s):  
ARIE LEIZAROWITZ
Keyword(s):  
Control Theory ◽  
Large Class ◽  
Optimality Criterion ◽  
Variational Problems ◽  
Two Dimensional ◽  
Minimal Energy ◽  
Chemical Systems ◽  
Overtaking Optimality ◽  
Special Case ◽  
Class Of Functions

We consider the functional IΩ(u) = ∫Ω [ψ (u(x,y)) + ½K (∇ u)]dxdy defined for real valued functions u on ℝ2 and study its minimization over a certain class of functions u(·, ·). We look for a minimizer u⋆ which is universal in the sense that IΩ(u⋆)≤IΩ(u) for every bounded domain (in a certain class) and for every u(·, ·) which satisfies u|∂Ω=u⋆|∂Ω. This optimality notion is an extension to a multivariable situation of the overtaking optimality criterion used in control theory, and the minimal-energy-configuration concept employed in the study of certain chemical systems. The existence of such universal minimizers is established for a large class of variational problems. In the special case were K(∇ u) = ½ |∇ u|2 these minimizers are characterized as the functions u⋆(x, y)=ϕ(ax+by+c) for some explicitly computable ϕ:ℝ1→ℝ1 and constants a, b and c.

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