On lower semicontinuity and relaxation

2001 ◽  
Vol 131 (3) ◽  
pp. 519-565 ◽  
Author(s):  
Irene Fonseca ◽  
Giovanni Leoni
Keyword(s):  
Energy Density ◽  
Calculus Of Variations ◽  
Lower Semicontinuity ◽  
Multiple Integrals ◽  
Image Position

Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

On lower semicontinuity and relaxation

2001 ◽  
Vol 131 (3) ◽  
pp. 519-565
Author(s):  
Irene Fonseca ◽  
Giovanni Leoni
Keyword(s):  
Energy Density ◽  
Calculus Of Variations ◽  
Lower Semicontinuity ◽  
Multiple Integrals ◽  
Image Position

Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

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.

Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations

2001 ◽  
Vol 131 (1) ◽  
pp. 155-184 ◽  
Author(s):  
Ali Taheri
Keyword(s):  
Field Theory ◽  
Calculus Of Variations ◽  
Sufficient Conditions ◽  
Local Minimizer ◽  
Local Minimizers ◽  
Indirect Approach ◽  
The Subject ◽  
Multiple Integrals ◽  
Image Position ◽  
The Right

Let Ω ⊂ Rn be a bounded domain and let f : Ω × RN × RN×n → R. Consider the functional over the class of Sobolev functions W1,q(Ω;RN) (1 ≤ q ≤ ∞) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function u0 and f to ensure that u0 provides an Lr local minimizer for I where 1 ≤ r ≤ ∞. The case r = ∞ is somewhat known and there is a considerable literature on the subject treating the case min(n, N) = 1, mostly based on the field theory of the calculus of variations. The main contribution here is to present a set of sufficient conditions for the case 1 ≤ r < ∞. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ‘directional convergence’.

Lower semicontinuity of multiple integrals and the Biting Lemma

1990 ◽  
Vol 114 (3-4) ◽  
pp. 367-379 ◽  
Author(s):  
J. M. Ball ◽  
K.-W. Zhang
Keyword(s):  
Integral Representation ◽  
Calculus Of Variations ◽  
Lower Semicontinuity ◽  
Young Measure ◽  
Weak Limit ◽  
Periodic Case ◽  
Quasiconvex Functions ◽  
General Sequence ◽  
Weak Lower Semicontinuity ◽  
Multiple Integrals

SynopsisWeak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.

Relaxation of multiple integrals in the spaceBV(Ω, RP)

1992 ◽  
Vol 121 (3-4) ◽  
pp. 321-348 ◽  
Author(s):  
Irene Fonseca ◽  
Piotr Rybka
Keyword(s):  
Energy Density ◽  
Surface Energy ◽  
Total Energy ◽  
Surface Energy Density ◽  
Multiple Integrals ◽  
Image Position

SynopsisA characterisation of the surface energy density for the relaxation in V(Ω; Rp) of the functionalis obtained. A lemma of De Giorgi is used to modify a sequence near the boundary without increasing its total energy.

scholarly journals A Differential Geometry Associated with Dissipative Systems

1965 ◽  
Vol 8 (4) ◽  
pp. 433-451 ◽  
Author(s):  
M. A. McKiernan
Keyword(s):  
Differential Geometry ◽  
Calculus Of Variations ◽  
Dissipative Systems ◽  
Image Position ◽  
Problem Of Lagrange

Consider the following problem of Lagrange in the calculus of variations: relative to differentiable curves xi(t) satisfying xi(t0) = xi0 and xi(t1) = xi1 find a curve minimizing1

The general double integral problem in the calculus of variations

1933 ◽  
Vol 29 (2) ◽  
pp. 207-211
Author(s):  
R. P. Gillespie
Keyword(s):  
Calculus Of Variations ◽  
General Problem ◽  
Necessary Condition ◽  
Double Integral ◽  
Integral Problem ◽  
Image Position

In a previous paper in these Proceedings the problem of the double integralwas discussed when the function F had the formwhereIt is proposed in the present paper to extend the method to the general problem, where F may have any form provided only that it satisfies the necessary condition of being homogeneous of the first degree in A, B, C.

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