New homogenization results for convex integral functionals and their Euler–Lagrange equations
Tóm tắt
We study stochastic homogenization for convex integral functionals
$$\begin{aligned} u\mapsto \int _D W(\omega ,\tfrac{x}{\varepsilon },\nabla u)\,\textrm{d}x,\quad \text{ where }\quad u:D\subset {\mathbb {R}}^d\rightarrow {\mathbb {R}}^m, \end{aligned}$$
defined on Sobolev spaces. Assuming only stochastic integrability of the map
$$\omega \mapsto W(\omega ,0,\xi )$$
, we prove homogenization results under two different sets of assumptions, namely
Condition
$$\bullet _2$$
directly improves upon earlier results, where p-coercivity with
$$p>d$$
is assumed and
$$\bullet _1$$
provides an alternative condition under very weak coercivity assumptions and additional structure conditions on the integrand. We also study the corresponding Euler–Lagrange equations in the setting of Sobolev-Orlicz spaces. In particular, if
$$W(\omega ,x,\xi )$$
is comparable to
$$W(\omega ,x,-\xi )$$
in a suitable sense, we show that the homogenized integrand is differentiable.
Tài liệu tham khảo
Akcoglu, M.A., Krengel, U.: Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323, 53–67 (1981)
Anza Hafsa, O., Mandallena, J.: Homogenization of nonconvex integrals with convex growth. J. Math. Pures Appl. 96, 167–189 (2011)
Bella, P., Schäffner, M.: Local boundedness and Harnack inequality for solutions of linear nonuniformly elliptic equations. Commun. Pure Appl. Math. 74(3), 453–477 (2021)
Bonfanti, G., Cellina, A., Mazzola, M.: The higher integrability and validity of the Euler–Lagrange equation for solutions to variational problems. SIAM J. Control Optim. 50, 888–899 (2012)
Braides, A., Defranceschi, A.: Homogenization of multiple integrals. Oxford Lecture Series in Mathematics and its Applications, 12. The Clarendon Press, Oxford University Press, New York (1998). xiv+298 pp
Briane, M., Casado Díaz, J.: A new div-curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian. J. Differ. Equ. 260(7), 5678–5725 (2016)
Briane, M., Casado-Díaz, J., Luna-Laynez, M., Pallares-Martín, A.: \(\Gamma \)-convergence of equi-coercive nonlinear energies defined on vector-valued functions, with non-uniformly bounded coefficients. Nonlinear Anal. 151, 187–207 (2017)
Carozza, M., Kristensen, J., Passarelli Di Napoli, A.: On the validity of the Euler–Lagrange system. Commun. Pure Appl. Anal. 14, 51–62 (2015)
Chlebicka, I., Gwiazda, P., Świerczewska-Gwiazda, A., Wróblewska-Kamińska, A.: Partial Differential Equations in Anisotropic Musielak–Orlicz Spaces. Springer Monographs in Mathematics. Springer, Cham (2021)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Progress in Nonlinear Differential Equations and Their Applications, vol. 8. Birkhäuser Boston Inc., Boston (1993)
Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368, 28–42 (1986)
Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland, Amsterdam (1978)
Desch, W., Grimmer, R.: On the wellposedness of constitutive laws involving disspation potentials. Trans. Am. Math. Soc. 353, 5095–5120 (2001)
D’Onofrio, C., Zeppieri, C.I.: \(\Gamma \)-convergence and stochastic homogenisation of degenerate integral functionals in weighted Sobolev spaces. Proc. Edinb. Math. Soc. (2) 153, 491–544 (2023)
Duerinckx, M., Gloria, A.: Stochastic homogenization of nonconvex unbounded integral functionals with convex growth. Arch. Ration. Mech. Anal. 221, 1511–1584 (2016)
Dunford, N., Schwartz, J.T.: with the assistance of W.G. Bade and R.G. Bartle. Linear Operators, Part I: General Theory, Pure and Applied Mathematics, vol. 7. Interscience Publishers, Inc., New York (1958)
Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Studies in Mathematics and its Applications, vol. 1. North-Holland, Amsterdam (1976)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^p\) Spaces. Springer, New York (2007)
Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)
Koch, L., Ruf, M., Schäffner, M.: On the Lavrentiev gap for convex, vectorial integral functionals, Preprint (2023) arXiv:2305.19934
Kozek, A.: Orlicz spaces of functions with values in Banach spaces. Comment. Mat. 19, 259–288 (1977)
Kozek, A.: Convex integral functionals on Orlicz spaces. Comment. Mat. 21, 109–135 (1979)
Krengel, U.: Ergodic Theorems. De Gruyter Studies in Mathematics 6. De Gruyter, Berlin (1985)
Messaoudi, K., Michaille, G.: Stochastic homogenization of nonconvex integral functionals. ESAIM Math. Model. Numer. Anal. 28, 329–356 (1994)
Lions, J.-L.: Quelques méthodes de résolution des problémes aux limites non linéaires. (French) Dunod, Paris; Gauthier-Villars, Paris (1969) xx+554 pp
Mordukhovitch, B.S., Nam, N.M., Rector, R.B., Tran, T.: Variational geometric approach to generalized differential and conjugate calculi in convex analysis. Set-Valued Var. Anal. 25, 731–755 (2017)
Müller, S.: Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Ration. Mech. Anal. 99, 189–212 (1987)
Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(3), 489–507 (1978)
Neukamm, S., Schäffner, M., Schlömerkemper, A.: Stochastic homogenization of nonconvex discrete energies with degenerate growth. SIAM J. Math. Anal. 49, 1761–1809 (2017)
Neukamm, S., Schäffner, M.: Quantitative homogenization in nonlinear elasticity for small loads. Arch. Ration. Mech. Anal. 230, 343–396 (2018)
Ruf, M., Ruf, T.: Stochastic homogenization of degenerate integral functionals and their Euler-Lagrange equations. J. Éc. Polytech. Math. 10, 253–303 (2023)
Ruf, M., Zeppieri, C.I.: Stochastic homogenization of degenerate integral functionals with linear growth. Calc. Var. Partial Differ. Equ. 62, 138 (2023)
Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, pp. 136–212, Res. Notes in Math., 39, Pitman, Boston, Mass.-London (1979)
Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50(4), 675–710, 877 (1986)