Young measures and relaxation of functionals for integrands \(f(x, u(x), u' (x))\)
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)].
scientific article; zbMATH DE number 840299