Regularity of solutions for some variational problems subject to a convexity constraint

10.1002/cpa.3 ◽  
2001 ◽  
Vol 54 (5) ◽  
pp. 583-594 ◽  
Author(s):  
G. Carlier ◽  
T. Lachand-Robert
Keyword(s):  
Variational Problems ◽  
Regularity Of Solutions ◽  
Convexity Constraint

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.

scholarly journals A numerical approach to variational problems subject to convexity constraint

2001 ◽  
Vol 88 (2) ◽  
pp. 299-318 ◽  
Author(s):  
G. Carlier ◽  
T. Lachand-Robert ◽  
B. Maury
Keyword(s):  
Variational Problems ◽  
Numerical Approach ◽  
Convexity Constraint

scholarly journals On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity

2020 ◽  
pp. 1-21
Author(s):  
Nam Q. Le
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Approximation Problem ◽  
Variational Problems ◽  
Uniform Convexity ◽  
Convex Functionals ◽  
Convexity Constraint ◽  
Fourth Order Equations ◽  
Appl Math

Abstract We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.

scholarly journals On the regularity of solutions of one-dimensional variational obstacle problems

2018 ◽  
Vol 11 (2) ◽  
pp. 203-222 ◽  
Author(s):  
Jean-Philippe Mandallena
Keyword(s):  
Partial Regularity ◽  
Direct Method ◽  
Variational Problems ◽  
Obstacle Problems ◽  
Regularity Of Solutions ◽  
Hölder Continuous ◽  
One Dimensional ◽  
A Direct Method

AbstractWe study the regularity of solutions of one-dimensional variational obstacle problems in {W^{1,1}} when the Lagrangian is locally Hölder continuous and globally elliptic. In the spirit of the work of Sychev [5, 6, 7], a direct method is presented for investigating such regularity problems with obstacles. This consists of introducing a general subclass {\mathcal{L}} of {W^{1,1}}, related in a certain way to one-dimensional variational obstacle problems, such that every function of {\mathcal{L}} has Tonelli’s partial regularity, and then to prove that, depending on the regularity of the obstacles, solutions of corresponding variational problems belong to {\mathcal{L}}. As an application of this direct method, we prove that if the obstacles are {C^{1,\sigma}}, then every Sobolev solution has Tonelli’s partial regularity.

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