Sufficient conditions for strong local minima: the case of C 1 extremals

In this paper we settle a conjecture of Ball that uniform quasiconvexity and uniform positivity of the second variation are sufficient for a C 1 extremal to be a strong local minimizer. Our result holds for a class of variational functionals with a power law behavior at infinity. The proof is based on the decomposition of an arbitrary variation of the dependent variable into its purely strong and weak parts. We show that these two parts act independently on the functional. The action of the weak part can be described in terms of the second variation, whose uniform positivity prevents the weak part from decreasing the functional. The strong part "localizes", i.e. its action can be represented as a superposition of "Weierstrass needles", which cannot decrease the functional either, due to the uniform quasiconvexity conditions.

[1]  Michiel Hazewinkel,et al.  Cartesian currents in the calculus of variations , 2003 .

[2]  M. Giaquinta,et al.  Calculus of Variations I , 1995 .

[3]  Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations , 2003 .

[4]  J. Ball.  W^ quasiconvexity and variational problems for multiples integrals , 1984 .

[5]  R. J. Knops,et al.  Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity , 1984 .

[6]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[7]  Remarks on quasiconvexity and stability of equilibria for variational integrals , 1992 .

[8]  A. Taheri.  Local Minimizers and Quasiconvexity – the Impact of Topology , 2005 .

[9]  WEIERSTRASS CONDITION FOR THE GENERAL BASIC VARIATIONAL PROBLEM , 1995 .

[10]  Charles B. Morrey,et al.  QUASI-CONVEXITY AND THE LOWER SEMICONTINUITY OF MULTIPLE INTEGRALS , 1952 .

[11]  Irene Fonseca,et al.  Analysis of Concentration and Oscillation Effects Generated by Gradients , 1998 .

[12]  Bernard Dacorogna,et al.  Quasiconvexity and relaxation of nonconvex problems in the calculus of variations , 1982 .

[13]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[14]  J. Ball,et al.  W1,p-quasiconvexity and variational problems for multiple integrals , 1984 .

[15]  M. Giaquinta.  Cartesian currents in the calculus of variations , 1983 .

[16]  Magnus R. Hestenes,et al.  Sufficient Conditions for Multiple Integral Problems in the Calculus of Variations , 1948 .

[17]  R. Kohn,et al.  Optimal design and relaxation of variational problems, III , 1986 .

[18]  P. Pedregal.  Parametrized measures and variational principles , 1997 .

[19]  M. Hestenes.  Calculus of variations and optimal control theory , 1966 .

[20]  On homotopy conditions and the existence of multiple equilibria in finite elasticity , 1997 .

[21]  H. Simpson,et al.  On the positivity of the second variation in finite elasticity , 1987 .

[22]  J. Ball.  A version of the fundamental theorem for young measures , 1989 .

[23]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[24]  Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations , 2001, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[25]  Irene Fonseca,et al.  Lower semicontinuity of surface energies , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[26]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[27]  W. Rudin.  Real and complex analysis , 1968 .

[28]  Jan Kristensen,et al.  Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations , 2003 .

[29]  Lawrence C. Evans,et al.  Weak convergence methods for nonlinear partial differential equations , 1990 .

[30]  H. Fédérer.  Geometric Measure Theory , 1969 .

[31]  Théophile De Donder,et al.  Théorie invariantive du calcul des variations , 1936 .

[32]  Stefan Hildebrandt,et al.  The Lagrangian formalism , 1996 .

[33]  Robert V. Kohn,et al.  Local minimisers and singular perturbations , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[34]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[35]  Constantin Carathéodory,et al.  Calculus of variations and partial differential equations of the first order , 1965 .

[36]  A. Majda,et al.  Oscillations and concentrations in weak solutions of the incompressible fluid equations , 1987 .

[37]  Morton E. Gurtin,et al.  Two-phase deformations of elastic solids , 1983 .

[38]  Jr. László Székelyhidi.  The Regularity of Critical Points of Polyconvex Functionals , 2004 .

[39]  J. Ball,et al.  Hysteresis During Stress-Induced Variant Rearrangement , 1995 .

[40]  Paolo Marcellini,et al.  Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals , 1985 .

[41]  J. Ball.  The calculus of variations and materials science , 1998 .

[42]  Miroslav Šilhavý,et al.  The Mechanics and Thermodynamics of Continuous Media , 2002 .

[43]  W. Reid.  Sufficient Conditions by Expansion Methods for the Problem of Bolza in the Calculus of Variations , 1937 .

[44]  H. Schubert,et al.  O. D. Kellogg, Foundations of Potential Theory. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 31). X + 384 S. m. 30 Fig. Berlin/Heidelberg/New York 1967. Springer‐Verlag. Preis geb. DM 32,– , 1969 .

[45]  I. P. Natanson,et al.  Theory of Functions of a Real Variable , 1955 .

[46]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[47]  A. Taheri.  On Critical Points of Functionals with Polyconvex Integrands , 2002 .

[48]  Jerrold E. Marsden,et al.  Quasiconvexity at the boundary, positivity of the second variation and elastic stability , 1984 .

[49]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[50]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.

[51]  L. Young.  Lectures on the Calculus of Variations and Optimal Control Theory , 1980 .

[52]  Yury Grabovsky,et al.  Direct approach to the problem of strong local minima in calculus of variations , 2005, math/0509702.

[53]  G. Duclos.  New York 1987 , 2000 .

[54]  Gerhard Eduard Moritz Armsen.  GEODESIC FIELDS IN THE CALCULUS-OF-VARIATIONS FOR MULTIPLE-INTEGRALS , 1973 .

[55]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[56]  Hermann Weyl,et al.  Geodesic Fields in the Calculus of Variation for Multiple Integrals , 1935 .