## Semicontinuity problems in the calculus of variations.(English)Zbl 0565.49010

The authors give some semicontinuity and relaxation results for integrals of the calculus of variations. The following is one of the most interesting theorems proved in the paper. It is a very deep result: Let $$\Omega$$ be an open set in $${\mathbb{R}}^ n$$. Let us assume that $$f=f(x,s,\xi)$$ is a real Carathéodory function defined in $$\Omega \times {\mathbb{R}}^ m\times {\mathbb{R}}^{nm},$$ quasiconvex with respect to $$\xi$$ in Morrey’s sense, and such that $$0\leq f(x,s,\xi)\leq a(x)+c(| s|^ p+| \xi |^ p)$$ for a.e. $$x\in \Omega$$, and for every $$s\in {\mathbb{R}}^ m$$, $$\xi \in {\mathbb{R}}^{nm}$$, where c is a positive constant, $$p\geq 1$$ and $$a\in L^ 1_{loc}(\Omega)$$. Then the functional $u\in W^{1,p}(\Omega;{\mathbb{R}}^ m)\to \int_{\Omega}f(x,u(x),Du(x))\quad dx$ is sequentially lower semicontinuous in the weak topology of $$W^{1,p}(\Omega;{\mathbb{R}}^ m)$$. A result of existence of minima by the reviewer and C. Sbordone [J. Math. Pures Appl. 62, 1-9 (1983; Zbl 0516.49011)] and a semicontinuity theorem by the reviewer [Manuscripta Math. 51, 1-28 (1985)] are related to the quoted semicontinuity theorem. Moreover, the book by B. Dacorogna [”Weak continuity and weak lower semi-continuity of non-linear functionals”, Lect. Notes Math. 922 (1982; Zbl 0484.46041)] is related to the relaxation results.
Reviewer: P.Marcellini

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 26B25 Convexity of real functions of several variables, generalizations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 54C08 Weak and generalized continuity 49J10 Existence theories for free problems in two or more independent variables

### Keywords:

semicontinuity; relaxation

### Citations:

Zbl 0516.49011; Zbl 0484.46041
Full Text:

### References:

  Adams, R. A.: *Sobolev spaces, Academic Press, New York, 1975. · Zbl 0314.46030  Ball, J. M.: On the calculus of variations and sequentially weakly continuous maps, Ordinary and partial differential equations (Proc. Fourth Conf., Univ. Dundee, Dundee 1976), pp. 13-25. Lecture Notes in Math., Vol. 564, Springer, Berlin, Heidelberg, New York, 1976.  Ball, J. M.: Constitutive inequalities and existence theorems in nonlinear elastostatics, Nonlinear analysis and mechanics: Heriot-Watt Symposium (Edinburgh, 1976), Vol. I, pp. 187-241. Res. Notes in Math., No. 17, Pitman, London, 1977.  Ball, J. M.: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal, Vol. 63 (1977), 337-403. · Zbl 0368.73040  Ball, J. M.; Currie, J. C.; Olver, P. J.: Null lagrangians, weak continuity, and variational problems of any order, J. Funct. Anal., 41 (1981), 135-174. · Zbl 0459.35020  Dacorogna, B.: A relaxation theorem and its application to the equilibrium of gases, Arch. Rational Mech. Anal, 77 (1981), 359-386. · Zbl 0492.49002  Eisen, G.: A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals, Manuscripta Math., 27 (1979), 73-79. · Zbl 0404.28004  Ekeland, I.; Temam, R.: *Convex analysis and variational problems, Nortt Holland, Amsterdam, 1976. · Zbl 0322.90046  Fusco, N.: Quasi-convessità e semicontinuità per integrali multipli di ordine superiore, Ricerche Mat., 29 (1980), 307-323. · Zbl 0508.49012  Liu, F.-C.: A Luzin type property of Sobolev functions, Indiana Univ. Math. J., 26 (1977), 645-651. · Zbl 0368.46036  Marcellini, P.; Sbordone, C.: Semicontinuity problems in the calculus of variations, Nonlinear Anal., 4 (1980), 241-257. · Zbl 0537.49002  Meyers, N. G.: Quasi-convexity and lower Semicontinuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965), 125-149. · Zbl 0166.38501  Morrey, C. B.: Quasi-convexity and the Semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25-53. · Zbl 0046.10803  Morrey, C. B.: *Multiple integrals in the calculus of variations. Springer Berlin, Heidelberg, New York 1966. · Zbl 0142.38701  Serrin, J.: On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc. 101 (1961), 139-167. · Zbl 0102.04601  Stein, E. M.: *Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, 1970. · Zbl 0207.13501  Tonelli, L.: La semicontinuita nel calcolo delle variazioni, Rend. Circ. Matem. Palermo 44 (1920), 167-249. · JFM 47.0472.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.