Critical points for multiple integrals of the calculus of variations

1996 ◽  
Vol 134 (3) ◽  
pp. 249-274 ◽  
Author(s):  
David Arcoya ◽  
Lucio Boccardo
Keyword(s):  
Calculus Of Variations ◽  
Critical Points ◽  
Multiple Integrals

A note on relaxation with constraints on the determinant

2019 ◽  
Vol 25 ◽  
pp. 41 ◽  
Author(s):  
Marco Cicalese ◽  
Nicola Fusco
Keyword(s):  
Calculus Of Variations ◽  
Upper Bound ◽  
Lower Semicontinuity ◽  
Carathéodory Function ◽  
Incompressibility Constraint ◽  
Multiple Integrals ◽  
Quasiconvex Envelope

We consider multiple integrals of the Calculus of Variations of the form E(u) = ∫ W(x, u(x), Du(x)) dx where W is a Carathéodory function finite on matrices satisfying an orientation preserving or an incompressibility constraint of the type, det Du > 0 or det Du = 1, respectively. Under suitable growth and lower semicontinuity assumptions in the u variable we prove that the functional ∫ Wqc(x, u(x), Du(x)) dx is an upper bound for the relaxation of E and coincides with the relaxation if the quasiconvex envelope Wqc of W is polyconvex and satisfies p growth from below for p bigger then the ambient dimension. Our result generalises a previous one by Conti and Dolzmann [Arch. Rational Mech. Anal. 217 (2015) 413–437] relative to the case where W depends only on the gradient variable.

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