Quasi-minima. (English) Zbl 0541.49008

A notion of quasi-minimum (Q-minimum) is defined for the functional of calculus of variations \(F(u;\Omega)=\int_{\Gamma}f(x,u(x),Du(x))dx,\) where \(\Omega\) is a bounded domain in \(R^ n\), \(u=(u^ 1,...,u^ N)\quad(N\geq 1)\) is a function defined on \(\Omega\) and f(x,u,p) is a Carathéodory function. Then it is shown that this notion includes among others the minima of variational integrals, the solutions of elliptic differential equations and of variational inequalities, the quasi-regular mappings.
The problem of the regularity of Q-minima is also discussed, and several results concerning the regularity for Q-minima in \(L^ p\) and \(C^{0\alpha}\)-spaces are obtained. In the case where \(N=1\) some qualitative properties for Q-minima like the weak maximum principle or the Liouville property are proved. A stability result for Q-minima with respect to \(\Gamma\)-convergence of a sequence of functionals is also given.
Reviewer: Z.Denkowski


49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables
35B65 Smoothness and regularity of solutions to PDEs
49J40 Variational inequalities
35J50 Variational methods for elliptic systems
49K10 Optimality conditions for free problems in two or more independent variables
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[1] E. Acerbi, N. Fusco, Semicontinuity Problems in the Calculus of Variations. Arch. Rat. Mech. Anal., to appear. · Zbl 0565.49010
[2] Boyarskii, B. V., Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk SSR, t. 102, 661-664, (1955)
[3] Boyarskii, B. V., Generalized solutions of a system of differential equations of the first order of elliptic type with discontinuous coefficients, Mat. Sbornik, t. 43, 451-503, (1957)
[4] Giorgi, E. D.E., Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. torino cl. Sci. Fis. Mat. Nat., t. 3, 3, 25-43, (1957) · Zbl 0084.31901
[5] E. De Giorgi, Generalized limits in Calculus of Variations, in « Topics in Functional Analysis 1980-1981 » Quaderni Scuola Norm. Sup. Pisa, 1981.
[6] Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc., t. 1, 443-474, (1979) · Zbl 0441.49011
[7] Frehse, J., A discontinuous solution of a mildly nonlinear elliptic system, Math. Z., t. 134, 229-230, (1973) · Zbl 0267.35038
[8] Fusco, N., Quasi-convessità e semicontinuità per integrali multipli di ordine superiore, Ricerche di Mat., t. 29, 307-323, (1980) · Zbl 0508.49012
[9] Gehring, F. W., The L^{p}-integrability of the partial derivatives of a quasi conformal mapping, Acta Math., t. 130, 265-277, (1973) · Zbl 0258.30021
[10] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Studies, t. 105, (1983), Princeton University Press · Zbl 0516.49003
[11] M. Giaquinta, On the differentiability of the extremals of variational integrals, in « Nonlinear Analysis, Function Spaces and Application 2 », Teubner Texte zur Mathem., Leipzig, 1982. · Zbl 0494.49032
[12] Giaquinta, M.; Iusti, E. G., On the regularity of the minima of variational integrals, Acta Math., t. 148, 31-46, (1982) · Zbl 0494.49031
[13] Giaquinta, M.; Iusti, E. G., Differentiability of minima of non-differentiable functionals, Inventiones Math., t. 72, 285-298, (1983) · Zbl 0513.49003
[14] Giaquinta, M.; Modica, G., Regularity results for some classes of higher order nonlinear elliptic systems, J. für reine u. angew. Math., t. 311-312, 145-169, (1979) · Zbl 0409.35015
[15] Hildebrandt, S.; Widman, K.-O., Some regularity results for quasilinear elliptic systems of second order, Math. Z., t. 142, 67-86, (1975) · Zbl 0317.35040
[16] Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and quasilinear elliptic equations, (1968), Academic Press New York · Zbl 0164.13002
[17] Marcellini, P.; Sbordone, C., On the existence of minima of multiple integrals of the calculus of variations, J. Math. pures et appl., t. 62, 1-9, (1983) · Zbl 0516.49011
[18] Meyers, N. G., An L^{p}-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa, t. 17, 3, 189-206, (1963) · Zbl 0127.31904
[19] Meyers, N. G.; Elcrat, A., Some results on regularity for solutions of nonlinear elliptic systems and quasiregular functions, Duke Math. J., t. 42, 121-136, (1975) · Zbl 0347.35039
[20] Necas, J., LES Méthodes directes en théorie des équations elliptiques, (1967), Praha, Akademia · Zbl 1225.35003
[21] Reshetnyak, Y. G., Extremal properties of mappings with bounded distortion, Siberian Math. J., t. 10, 962-969, (1969), transl. of Sibirsk Mat. Z. · Zbl 0203.38901
[22] Sbordone, C., Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Sc. Norm. Sup. Pisa, t. 2, 4, 617-638, (1975) · Zbl 0317.49012
[23] Serrin, J., Local behaviour of solutions of quasilinear equations, Acta Math., t. 111, 247-302, (1964) · Zbl 0128.09101
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