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A construction of quasiconvex functions with linear growth at infinity. (English) Zbl 0778.49015
We develop a method for constructing nontrivial quasiconvex functions with $$p$$-th growth at infinity from known quasiconvex functions.
Definition. A continuous function $$f: M^{N\times n}\to\mathbb{R}$$ is quasiconvex in the sense of Morrey if $$\int_ U f(P+D\phi(x))dx\geq f(P) \text{meas}(U)$$ for every $$P\in M^{N\times n}$$, $$\phi\in C^ 1_ 0(U; R^ N)$$, and every open bounded subset $$U\subset\mathbb{R}^ n$$.
The main result is the following:
Theorem. Suppose that the continuous function $$f: M^{N\times n}\to\mathbb{R}$$ is quasiconvex and that for some real constant $$\alpha$$, the level set $$K_ \alpha:=\bigl\{P\in M^{N\times n}: f(P)\leq\alpha\bigr\}$$ is compact. Then, for every $$1\leq q<+\infty$$, there is a continuous quasiconvex function $$g_ q\geq 0$$, such that $$-C_ 1+c| P|^ q\leq g_ q(P)\leq C_ 1+C_ 2| P|^ q$$ and $$K_ \alpha=\bigl\{P\in M^{N\times n}: g_ q(P)=0\bigr\}$$, where $$C_ 1\geq 0$$, $$c>0$$, $$C_ 2>0$$ are constants.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation
##### Keywords:
$$p$$-th growth at infinity; quasiconvex functions
Full Text:
##### References:
 [1] E. Acerbi - N. Fusco , Semicontinuity problems in the calculus of variations , Arch. Rational Mech. Anal. , 86 ( 1984 ) 125 - 145 . MR 751305 | Zbl 0565.49010 · Zbl 0565.49010 [2] E.J. Balder , A general approach to lower semicontinuity and lower closure in optimal control theory , SIAM J. Control Optim. , 22 ( 1984 ) 570 - 597 . MR 747970 | Zbl 0549.49005 · Zbl 0549.49005 [3] J.M. Ball , Convexity conditions and existence theorems in nonlinear elasticity , Arch. Rational Mech. Anal. , 63 ( 1977 ) 337 - 403 . MR 475169 | Zbl 0368.73040 · Zbl 0368.73040 [4] J.M. Ball , Constitutive inequalities and existence theorems in nonlinear elasticity, in ”Nonlinear Analysis and Mechanics: Heriot-Watt Symposium” . Vol. 1 (edited by R.J. Knops), Pitman , London , 1977 . MR 478899 | Zbl 0377.73043 · Zbl 0377.73043 [5] J.M. Ball , Sets of gradients with no rank-one connections , Preprint, 1988 . MR 1070479 [6] J.M. Ball , A version of the fundamental theorem of Young measures, to appear in Proceedings of Conference on ”Partial Differential Equations and Continuum Models of Phase Transitions” , Nice , 1988 (edited by D. Serre), Springer . MR 1036070 | Zbl 0991.49500 · Zbl 0991.49500 [7] J.M. Ball - J.C. Currie - P.J. Olver , Null Lagrangians, weak continuity, and variational problems of arbitrary order , J. Funct. Anal. , 41 ( 1981 ) 135 - 174 . MR 615159 | Zbl 0459.35020 · Zbl 0459.35020 [8] J.M. Ball - R.D. James , Fine phase mixture as minimizers of energy , Arch. Rational Mech. Anal. , 100 ( 1987 ) 13 - 52 . MR 906132 | Zbl 0629.49020 · Zbl 0629.49020 [9] J.M. Ball - Kewei Zhang , Lower semicontinuity of multiple integrals and the biting lemma , Proc. Roy. Soc. Edinburgh , 114A ( 1990 ) 367 - 379 . MR 1055554 | Zbl 0716.49011 · Zbl 0716.49011 [10] H. Berliocchi - J.M. Lasry , Intégrandes normales et mesures paramétrées en calcul des variations , Bull. Soc. Math. France , 101 ( 1973 ) 129 - 184 . Numdam | MR 344980 | Zbl 0282.49041 · Zbl 0282.49041 [11] B. Dacorogna , Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals , Lecture Notes in Math. Springer-Verlag , Berlin . Vol. 922 ( 1980 ). MR 658130 | Zbl 0484.46041 · Zbl 0484.46041 [12] I. Diestel - J.J. Uhl JR. , Vector Measures , American Mathematical Society, Mathematics Surveys , No. 15 , Providence , 1977 . MR 453964 | Zbl 0369.46039 · Zbl 0369.46039 [13] I. Ekeland - R. Temam , Convex Analysis and Variational Problems , North-Holland , 1976 . MR 463994 | Zbl 0322.90046 · Zbl 0322.90046 [14] Ionescu Tulcea , Topics in the Theory of Lifting , Springer , New York , 1969 . MR 276438 | Zbl 0179.46303 · Zbl 0179.46303 [15] D. Kinderlehrer , Remarks about equilibrium configurations of crystals , in ” Material Instabilities in Continuum Mechanics ” (ed. J.M. Ball), Oxford University Press , ( 1988 ) 217 - 241 . MR 970527 | Zbl 0850.73037 · Zbl 0850.73037 [16] R. Kohn , personal communication . [17] C.B. Morrey , Multiple Integrals in the Calculus of Variations , Springer-Verlag , New York , 1966 . MR 202511 | Zbl 0142.38701 · Zbl 0142.38701 [18] Y.G. Reshetnyak , On the stability of conformal mappings in multidimensional spaces , Siberian Math. J. , 8 ( 1967 ) 69 - 85 . Zbl 0172.37801 · Zbl 0172.37801 [19] Y.G. Reshetnyak , Stability theorems for mappings with bounded excursion , Siberian Math. J. , 9 ( 1968 ) 499 - 512 . Zbl 0176.03503 · Zbl 0176.03503 [20] E.M. Stein , Singular Integrals and Differentiability Properties of Functions , Princeton University Press , Princeton ( 1970 ). MR 290095 | Zbl 0207.13501 · Zbl 0207.13501 [21] V Šverák , Quasiconvex functions with subquadratic growth , Preprint, 1990 . MR 1116970 · Zbl 0741.49016 [22] L. Tartar , Compensated compactness and applications to partial differential equations, in ”Nonlinear Analysis and Mechanics: Heriot-Watt Symposium” , Vol. IV (edited by R.J. Knops), Pitman 1979 . MR 584398 | Zbl 0437.35004 · Zbl 0437.35004 [23] L. Tartar , The compensated compactness method applied to system of conservation laws, in ”Systems of Nonlinear Partial Differential Equations” , NATO ASI Series , Vol. C111 (edited by J.M. Ball), Reidel , 1982 , 263 - 285 . MR 725524 | Zbl 0536.35003 · Zbl 0536.35003
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