Young measures and relaxation of functionals for integrands \(f(x, u(x), u' (x))\)
type: Item
instance of: scholarly article
title: Young measures and relaxation of functionals for integrands \(f(x, u(x), u' (x))\)
zbMATH Open document ID: 0840.49001
author: Gilles Aubert
author: Rabah Tahraoui
published in: Differential and Integral Equations
publication date: 1996-06-12T00:00:00Z
review text: The authors consider the problem of the calculus of variations \[ \inf\Biggl\{\int^b_a f(x, u, u')dx\mid u\in W^{1,p}(a, b),\;u(a)= \alpha,\;u(b)= \beta\Biggr\}.\tag{P} \] Since the integrand \(\xi\to f(x, \eta, \xi)\) is not convex, (P) may have no solution. Under suitable classical assumptions the problem \[ \inf\Biggl\{\int^b_a f^{**}(x, u, u')dx\mid u\in W^{1,p}(a, b),\;u(a)= \alpha,\;u(b)= \beta\Biggr\},\tag{RP} \] (where \(f^{**}(x, \eta,.)\) is the biconjugate function of \(f(x, \eta, .)\)) admits solution and \(\inf(\text{P})= \inf(\text{RP})\). The authors first prove that (RP) admits Lipschitz solutions, solutions of (RP) are characterized thanks to Young measures. The main purpose of the paper is to give sufficient conditions for the existence of solutions for (P). In the case when \(f^{**}(x, \eta,.)\) is strictly convex on \(R\backslash K(x, \eta)\), with \[ K(x, \eta)= ]\tau_1(x, \eta), \tau_2(x, \eta)[= \{\xi\mid f^{**}(x, \eta, \xi)< f(x, \eta, \xi)\}, \] it is proved that if \[ \begin{multlined} f^{**}_{x\xi}(x, \eta, \theta \tau_1+ (1- \theta)\tau_2)+ (\theta\tau_1+ (1- \theta)\tau_2) f^{**}_{\eta\xi}(x, \eta, \theta\tau_1+ (1- \theta) \tau_2)-\\ \theta f^{**}_\eta(x, \eta, \theta\tau_1)- (1- \theta) f^{**}_\eta(x, \eta, \tau_2)\neq 0,\end{multlined} \] then every solution of (RP) is a solution of (P) and (P) admits solutions. This kind of result is closely related to conditions given by the reviewer [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 169-202 (1987; Zbl 0641.49012)]. Other similar results are also given for vector problems by the reviewer [J. Optimization Theory Appl. 82, No. 3, 571-592 (1994; Zbl 0808.49002)].
reviewed by: J-P Raymond
Mathematics Subject Classification ID: 49J05
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49K20
zbMATH DE Number: 840299
zbMATH Keywords: relaxed problems
zbMATH Keywords: calculus of variations
zbMATH Keywords: Lipschitz solutions
zbMATH Keywords: Young measures
zbMATH Keywords: existence of solutions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 840299