A new variational inequality in the calculus of variations and Lipschitz regularity of minimizers

2020 ◽  
Vol 268 (5) ◽  
pp. 2332-2367 ◽  
Author(s):  
Piernicola Bettiol ◽  
Carlo Mariconda
Keyword(s):  
Variational Inequality ◽  
Calculus Of Variations ◽  
Lipschitz Regularity ◽  
Regularity Of Minimizers

Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations

1986 ◽  
Vol 102 (3-4) ◽  
pp. 291-303 ◽  
Author(s):  
Michel Chipot ◽  
Lawrence C. Evans
Keyword(s):  
Optimal Design ◽  
Calculus Of Variations ◽  
Design Problem ◽  
Blow Up ◽  
Optimal Design Problem ◽  
Lagrange Equations ◽  
Lipschitz Regularity ◽  
Locally Lipschitz ◽  
Lipschitz Estimates

SynopsisWe demonstrate local Lipschitz regularity for minimisers of certain functionals which are appropriately convex and quadratic near infinity. The proof employs a blow-up argument entailing a linearisation of the Euler—Lagrange equations “at infinity”. As a application, we prove that minimisers for the relaxed optimal design problem derived by Kohn and Strang [3] are locally Lipschitz.

A dynamical approach to the variational inequality on modified elastic graphs

2020 ◽  
Vol 5 (1) ◽  
pp. 78-101
Author(s):  
Shinya Okabe ◽  
Kensuke Yoshizawa
Keyword(s):  
Variational Inequality ◽  
Calculus Of Variations ◽  
Minimization Problem ◽  
Elastic Energy ◽  
Existence Of Solutions ◽  
Direct Method ◽  
Gradient Flow ◽  
Unilateral Constraint ◽  
Dynamical Approach

AbstractWe consider the variational inequality on modified elastic graphs. Since the variational inequality is derived from the minimization problem for the modified elastic energy defined on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations. In this paper we prove the existence of solutions to the variational inequality via a dynamical approach. More precisely, we construct an L2-type gradient flow corresponding to the variational inequality and prove the existence of solutions to the variational inequality via the study on the limit of the flow.

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