On the condition of tetrahedral polyconvexity, arising from calculus of variations

Kałamajska, Agnieszka; Kozarzewski, Piotr

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 475-495.

We study geometric conditions for integrand $f$ to define lower semicontinuous functional of the form $I_f\left(u\right){=}_{\Omega }^{\int }f\left(u\right)dx$, where $u$ satisfies certain conservation law. Of our particular interest is tetrahedral convexity condition introduced by the first author in 2003, which is the variant of maximum principle expressed on tetrahedrons, and the new condition which we call tetrahedral polyconvexity. We prove that second condition is sufficient but it is not necessary for lower semicontinuity of $I_f$, tetrahedral polyconvexity condition is non-local and both conditions are not equivalent. Problems we discuss are strongly connected with the rank-one conjecture of Morrey known in the multidimensional calculus of variations.

Classification : 49J10, 49J45
Mots clés : Quasiconvexity, compensated compactness, calculus of variations

J.-J. Alibert and B. Dacorogna, An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Ration. Mech. Anal. 117 (1992) 155–166. | DOI | MR | Zbl

N. Antonić and M. Lazar, Parabolic H-measures. J. Funct. Anal. 265 (2013) 1190–1239. | DOI | MR | Zbl

K. Astala, Analytic aspects of quasiconformality, Vol. II of Proceedings of the International Congress of Mathematicians. Berlin (1998) 617–626. | MR | Zbl

K. Astala, T. Iwaniec, I. Prause and E. Saksman, Burkholder integrals, Morrey’s problem and quasiconformal mappings. J. Amer. Math. Soc. 25 (2012) 507–531. | DOI | MR | Zbl

J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1978) 337–403. | DOI | MR | Zbl

J.M. Ball and F. Murat, Remarks on rank-one convexity and quasiconvexity. Ordinary and Partial Differential Equations, edited by B.D. Sleeman and R.J. Jarvis, Vol. III. Vol. 254 of Pitman Research Notes in Mathematics Series. Longman, New York (1991) 25–37. | MR | Zbl

A. Braides, I. Fonseca and G. Leoni, A-quasiconvexity: relaxation and homogenization. ESAIM: COCV 5 (2000) 539–577. | Numdam | MR | Zbl

E. Casadio-Tarabusi, An algebraic characterization of quasiconvex functions. Riserche Mat. 42 (1993) 11–24. | MR | Zbl

C. Carathéodory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32 (1911) 193–217. | DOI | JFM

K. Chełmiński and A. Kałamajska, New convexity conditions in the calculus of variations and compensated compactness theory. ESAIM: COCV 12 (2006) 64–92. | Numdam | MR | Zbl

B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals. Vol. 922 of Lect. Notes Math. Springer (1982). | MR | Zbl

B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin/Heidelberg (1989). | MR | Zbl

B. Dacorogna and J.-P. Haeberly, Some numerical methods for the study of the convexity notions arising in the calculus of variations. ESAIM: M2AN 32 (1998) 153–175. | DOI | Numdam | MR | Zbl

B. Dacorogna, J. Douchet, W. Gangbo and J. Rappaz, Some examples of rank-one convex functions in dimension two. Proc. R. Soc. Edinb. A 114 (1990) 135–150. | DOI | MR | Zbl

D. Faraco and X. Zhong, Quasiconvex functions and Hessian equations. Arch. Ration. Mech. Anal. 168 (2003) 245–252. | DOI | MR | Zbl

I. Fonseca and M. Kružík, Oscillations and concentrations generated by A-free mappings and weak lower semicontinuity of integral functionals. ESAIM: COCV 16 (2010) 472–502. | Numdam | MR | Zbl

I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30 (1999) 1355–1390. | DOI | MR | Zbl

S. Heinz, Quasiconvex functions can be approximated by quasiconvex polynomials. ESAIM: COCV 14 (2008) 795–801. | Numdam | MR | Zbl

T. Iwaniec, Nonlinear Cauchy-Riemann operators in $R^n$. Trans. Amer. Math. Soc. 354 (2002) 1961–1995. | DOI | MR | Zbl

J.-L. Joly, G. Métivier and J. Rauch, Trilinear compensated compactness and nonlinear geometric optics. Ann. Math. 142 (1995) 121–169. | DOI | MR | Zbl

J.-L. Joly, G. Métivier and J. Rauch, Diffractive nonlinear geometric optics with rectification. Indiana Univ. Math. J. 47 (1998) 1167–1241. | MR | Zbl

A. Kałamajska, On the condition of $\Lambda$-convexity in some problems of weak continuity and weak lower semicontinuity. Colloq. Math. 89 (2001), 43–78. | DOI | MR | Zbl

A. Kałamajska, On $\Lambda$-convexity conditions in the theory of lower semicontinuous functionals. J. Convex. Anal. 10 (2003) 419–436. | MR | Zbl

A. Kałamajska, On new geometric conditions for some weakly lower semicontinuous functionals with applications to the rank-one conjecture of Morrey. Proc. R. Soc. Edinb. A 133 (2003) 1361–1377. | DOI | MR | Zbl

C.-F. Kreiner and J. Zimmer, Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices. ESAIM: COCV 12 (2006) 253–270. | Numdam | MR | Zbl

J. Kristensen, On condition for polyconvexity. Proc. Amer. Math. Soc. 128 (2000) 1793–1797. | DOI | MR | Zbl

M. Kružík, On the composition of quasiconvex functions and the transposition. J. Convex. Anal. 6 (1999) 207–213. | MR | Zbl

P. Marcellini, Quasiconvex quadratic forms in two dimensions. Appl. Math. Optim. 11 (1984) 183–189. | DOI | MR | Zbl

A. Mielke, Necessary and sufficient conditions for polyconvexity of isotropic functions. J. Convex Anal. 12 (2005) 291–314. | MR | Zbl

C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952) 25–53. | DOI | MR | Zbl

C.B. Morrey, Multiple integrals in the Calculus of Variations. Springer-Verlag, Berlin (1966). | MR | Zbl

F. Murat, A survey on compensated compactness, Contributions to modern calculus of variations, edited by L. Cesari, Vol. 148 of Pitman Research Notes in Mathematics Series. Longman, Harlow (1987) 145–183. | MR

S. Müller, Variational models for microstructure and phase transitions, Collection: Calculus of variations and geometric evolution problems (Cetraro 1996). Lect. Notes Math. Springer, Berlin (1999) 85–210. | MR | Zbl

S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 20 (1999) 1087–1095. | DOI | MR | Zbl

G.P. Parry, On the planar rank-one convexity condition. Proc. R. Soc. Edinb. A 125 (1995) 247–264. | DOI | MR | Zbl

P. Pedregal, Weak continuity and weak lower semicontinuity for some compensation operators. Proc. R. Soc. Edinb. A 113 (1989) 267–279. | DOI | MR | Zbl

P. Pedregal, Some remarks on quasiconvexity and rank-one convexity. Proc. R. Soc. Edinb. A 126 (1996) 1055–1065. | DOI | MR | Zbl

P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser (1997). | MR | Zbl

P. Pedregal and V. Šverák, A note on Quasiconvexity and rank-one Convexity for $2\times 2$ Matrices. J. Convex. Anal. 5 (1998) 107–117. | MR | Zbl

F. Rindler, Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms. Arch. Ration. Mech. Anal. 215 (2015) 1–63. | DOI | MR | Zbl

J. Sivaloganathan, Implications of rank one convexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5 (1988) 99–118. | DOI | Numdam | MR | Zbl

V. Šverák, Examples of rank-one convex functions. Proc. R. Soc. Edinb. A 114 (1990) 237–242. | DOI | MR | Zbl

V. Šverák, Quasiconvex functions with subquadratic growth. Proc. R. Soc. London Ser. A 433 (1991) 723–725. | DOI | MR | Zbl

V. Šverák, rank-one property does not imply quasiconvexity. Proc. R. Soc. Edinb. A 120 (1992) 185–189. | DOI | MR | Zbl

V. Šverák, Lower semicontinuity of variational integrals and compensated compactness, Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994. Birkhäuser Verlag, Basel, Switzerland (1995) 1153–1158. | MR | Zbl

L. Tartar, The Compensated Compactness Method Applied to Systems of Conservation Laws. In Systems of Nonlinear Partial Differential Equations, edited by J.M. Ball. D. Riedel Publ. Company (1983) 263–285. | MR | Zbl

L. Tartar, Mathematical tools for studying oscillations and concentrations: From Young measures to H-measures and their variants, in Multiscale problems in science and technology, Challenges to mathematical analysis and perspectives, edited by N. Antonič, C.J. Van Duijin and W. Jager. Proceedings of the conference on multiscale problems in science and technology, held in Dubrovnik, Croatia, September 3–9, 2000. Springer, Berlin (2002). | MR | Zbl

B. Yan, On rank-one convex and polyconvex conformal energy functions with slow growth. Proc. R. Soc. Edinb. A 127 (1997) 651–663. | DOI | MR | Zbl

K.W. Zhang, A construction of quasiconvex function with linear growth at infinity. Ann. Sc. Norm. Super. Pisa Cl. Sci. Serie IV XIX (1992) 313–326. | Numdam | MR | Zbl