InteriorC 1+?-regularity of solutions of two-dimensional variational problems with obstacles

1973 ◽  
Vol 131 (3) ◽  
pp. 233-240 ◽  
Author(s):  
Stefan Hildebrandt
Keyword(s):  
Variational Problems ◽  
Regularity Of Solutions ◽  
Two Dimensional

scholarly journals Lipschitz Bounds and Nonautonomous Integrals

2021 ◽  
Author(s):  
Cristiana De Filippis ◽  
Giuseppe Mingione
Keyword(s):  
Variational Problems ◽  
Growth Conditions ◽  
Regularity Criteria ◽  
Regularity Of Solutions ◽  
Optimal Regularity ◽  
Lipschitz Regularity ◽  
Elliptic Case ◽  
Different Types ◽  
Vector Valued

AbstractWe provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

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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