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Existence and partial regularity in the calculus of variations. (English) Zbl 0648.49008
In the first part of the paper the author considers integrals of the form $(1)\quad {\mathcal F}(u(\Omega)=\int_{\Omega}f(\cdot,u,Du)dx$ defined for vector valued functions $$u: {\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}^ N$$ and with the additional property that the integrand $$f(x,y,Q)$$ satisfies the growth condition $$f_ 0(Q)-g(x)\leq f(x,y,Q)\leq \lambda | Q|^ m+g(x)$$ with $$\lambda,g>0$$, $$m\geq 2$$ and $$f_ 0$$ quasiconvex with growth order m. He then introduces a concept of generalized quasi-minima $$u\in H^{1,m}(\Omega,{\mathbb{R}}^ N$$) of the functional (1) and shows $$Du\in L^{m+\epsilon}_{loc}(\Omega,{\mathbb{R}}^{nN})$$ for some small $$\epsilon >0$$. The proof uses a version of Caccioppoli’s inequality being valid for generalized quasi-minima.
A second chapter gives some existence theorems for minimizers of quasiconvex functionals (1), a final section is concerned with the partial regularity properties of these minimizers in a special case. Related results can be found for example in a paper by M. Giaquinta and G. Modica [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 185-208 (1986; Zbl 0594.49004)].
Reviewer: M.Fuchs

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 35D10 Regularity of generalized solutions of PDE (MSC2000) 49J20 Existence theories for optimal control problems involving partial differential equations 26B25 Convexity of real functions of several variables, generalizations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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##### References:
 [1] E.Acerbi - N.Fusco,Semicontinuity problems in the calculus of variations, to appear in Arch. Rat. Mech. Soc. Anal. · Zbl 0791.49041 [2] Ball, J. M., Convexity conditions and existence theorems in the non-linear elasticity, Arch. Rat. Mech. Soc. Anal., 63, 337-403 (1977) · Zbl 0368.73040 [3] Ball, J. M., Strict convexity, strong ellipticity, and regularity in the calculus of variations, Math. Proc. Camb. Phil. Soc., 87, 501-513 (1980) · Zbl 0451.35028 [4] Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc., 1, 443-474 (1979) · Zbl 0441.49011 [5] Fusco, N., Quasi-convessità e semicontinuità per integrali multipli di ordine superiors, Ricerche di Mat., 29, 307-323 (1980) · Zbl 0508.49012 [6] M.Giaquinta,Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math., Studies, t.134, Princeton University Press, 1983. · Zbl 0516.49003 [7] Giaquinta, M.; Giusti, E., Nonlinear elliptic systems with quadratic growth, Manuscripta Math., 24, 323-349 (1978) · Zbl 0378.35027 [8] Giaquinta, M.; Giusti, E., On the regularity of the minima of variational integrals, Acta. Math., 148, 31-46 (1982) · Zbl 0494.49031 [9] Giaquinta, M.; Giusti, E., On the regularity of the minima of non-differentiable functionals, Inventiones Math., 72, 79-107 (1983) [10] Giaquinta, M.; Giusti, E., Quasi-minima, Analyse non linéaire, Ann. Inst. Henri Poincaré, 1, n. 2, 79-107 (1984) · Zbl 0541.49008 [11] M.Giaquinta - G.Modica,Partial regularity of minimizers of quasiconvex integrals, preprint. · Zbl 0594.49004 [12] C. L.Evans,Quasiconvexity and partial regularity in the calculus of variations, to appear. · Zbl 0623.49008 [13] Morrey, C. B. Jr., Quasicovexity and lower semicontinuity of multiple integrals, Pac. J. Math., 2, 25-53 (1952) · Zbl 0046.10803 [14] Morrey, C. B. Jr., Multiple integrals in the calculus of variations (1966), New York: Spinger, New York [15] Marcellini, P.; Sbordone, C., On the existence of minima of multiple integral of the calculus of variations, J. Math. Pure Appl., 62, 1-9 (1983) · Zbl 0516.49011 [16] E.Giusti,Lecture Notes in Nankai Institute of Mathematics. [17] WuLancheng,Existence, uniqueness and regularity of the minimizer of a certain functional, to appear.
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