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Beyond the Diffusion Equation and Miscellaneous Topics

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Homogenization Theory for Multiscale Problems

Part of the book series: MS&A ((MS&A,volume 21))

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Abstract

We present in this final chapter a selection of topics in homogenization theory. We consider other equations than the diffusion equation 1, or, for this very equation, techniques different in nature.

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References

  1. Scott N. Armstrong and Pierre Cardaliaguet. Quantitative stochastic homogenization of viscous Hamilton-Jacobi equations. Comm. Partial Differential Equations, 40(3):540–600, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  2. Scott N. Armstrong, Pierre Cardaliaguet, and Panagiotis E. Souganidis. Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations. J. Amer. Math. Soc., 27(2):479–540, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  3. Marco Avellaneda and Fang Hua Lin. Lp bounds on singular integrals in homogenization. Commun. Pure Appl. Math., 44(8–9):897–910, 1991.

    Google Scholar 

  4. Grégoire Allaire. Shape optimization by the homogenization method., volume 146. New York, NY: Springer, 2002.

    MATH  Google Scholar 

  5. Scott N. Armstrong and Charles K. Smart. Quantitative stochastic homogenization of elliptic equations in nondivergence form. Arch. Ration. Mech. Anal., 214(3):867–911, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  6. Andrea Braides and Anneliese Defranceschi. Homogenization of multiple integrals, volume 12 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998.

    Google Scholar 

  7. Xavier Blanc, Marc Josien, and Claude Le Bris. Precised approximations in elliptic homogenization beyond the periodic setting. Asymptotic Analysis, 116(2):93–137, 2020.

    Article  MathSciNet  MATH  Google Scholar 

  8. Xavier Blanc, Claude Le Bris, and Pierre-Louis Lions. Erratum to the article “On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diffusion”. https://www.ljll.math.upmc.fr/~blanc/erratum_jmpa.pdf, 2020.

  9. Xavier Blanc, Claude Le Bris, and Pierre-Louis Lions. On correctors for linear elliptic homogenization in the presence of local defects. Commun. Partial Differ. Equations, 43(6):965–997, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  10. Xavier Blanc, Claude Le Bris, and Pierre-Louis Lions. On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diffusion. J. Math. Pures Appl. (9), 124:106–122, 2019.

    Article  MathSciNet  MATH  Google Scholar 

  11. Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou. Asymptotic analysis for periodic structures. Reprint of the 1978 original with corrections and bibliographical additions. Providence, RI: AMS Chelsea Publishing, 2011.

    Google Scholar 

  12. Andrea Braides. Γ-convergence for beginners., volume 22. Oxford: Oxford University Press, 2002.

    Book  Google Scholar 

  13. Luis A. Caffarelli and Panagiotis E. Souganidis. Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Invent. Math., 180(2):301–360, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  14. Pierre Cardaliaguet and Panagiotis E. Souganidis. On the existence of correctors for the stochastic homogenization of viscous Hamilton-Jacobi equations. C. R. Math. Acad. Sci. Paris, 355(7):786–794, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  15. Luis A. Caffarelli, Panagiotis E. Souganidis, and L. Wang. Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math., 58(3):319–361, 2005.

    Google Scholar 

  16. Gianni Dal Maso and Luciano Modica. Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math., 368:28–42, 1986.

    Google Scholar 

  17. Lawrence C. Evans. Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A, 120(3–4):245–265, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  18. Lawrence C. Evans. Partial differential equations. 2nd ed., volume 19. Providence, RI: American Mathematical Society (AMS), 2010.

    Google Scholar 

  19. William M. Feldman, Jean-Baptiste Fermanian, and Bruno Ziliotto. An example of failure of stochastic homogenization for viscous Hamilton-Jacobi equations without convexity. J. Differential Equations, 280:464–476, 2021.

    Article  MathSciNet  MATH  Google Scholar 

  20. William M. Feldman and Panagiotis E. Souganidis. Homogenization and non-homogenization of certain non-convex Hamilton-Jacobi equations. J. Math. Pures Appl. (9), 108(5):751–782, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  21. Elena Kosygina, Fraydoun Rezakhanlou, and S. R. S. Varadhan. Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Comm. Pure Appl. Math., 59(10):1489–1521, 2006.

    Google Scholar 

  22. Ioannis Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus. 2nd ed., volume 113. New York etc.: Springer-Verlag, 1991.

    Google Scholar 

  23. Claude Le Bris. Systèmes multi-échelles. Modélisation et simulation., volume 47. Berlin: Springer, 2005.

    Book  MATH  Google Scholar 

  24. Jean-François Le Gall. Mouvement brownien, martingales et calcul stochastique., volume 71. Paris: Springer, 2013.

    Book  MATH  Google Scholar 

  25. Pierre-Louis Lions, George Papanicolaou, and S. R. Srinivasa Varadhan. Homogenization of Hamilton-Jacobi equations. Unpublished, 1996.

    Google Scholar 

  26. Pierre-Louis Lions and Panagiotis E. Souganidis. Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Appl. Math., 56(10):1501–1524, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  27. Pierre-Louis Lions and Panagiotis E. Souganidis. Homogenization of “viscous” Hamilton-Jacobi equations in stationary ergodic media. Comm. Partial Differential Equations, 30(1–3):335–375, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  28. Bernt Øksendal. Stochastic differential equations. Universitext. Springer-Verlag, Berlin, sixth edition, 2003.

    Google Scholar 

  29. Fraydoun Rezakhanlou and James E. Tarver. Homogenization for stochastic Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 151(4):277–309, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  30. Leonard C. G. Rogers and David Williams. Diffusions, Markov processes and martingales. Vol. 1: Foundations. 2nd ed. Cambridge: Cambridge University Press, 2000.

    Google Scholar 

  31. Leonard C. G. Rogers and David Williams. Diffusions, Markov processes, and martingales. Vol. 2: Itô calculus. 2nd ed. Cambridge: Cambridge University Press, 2000.

    Google Scholar 

  32. Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, third edition, 1999.

    Google Scholar 

  33. Vadim V. Yurinskii. On the averaging of non-divergent equations of second order with random coefficients. Sib. Mat. Zh., 23(2):176–188, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  34. Bruno Ziliotto. Stochastic homogenization of nonconvex Hamilton-Jacobi equations: a counterexample. Comm. Pure Appl. Math., 70(9):1798–1809, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  35. Vasilii V. Zhikov, Sergei M. Kozlov, and Olga A. Olejnik. Homogenization of differential operators and integral functionals. Berlin: Springer-Verlag, 1994.

    MATH  Google Scholar 

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Blanc, X., Le Bris, C. (2023). Beyond the Diffusion Equation and Miscellaneous Topics. In: Homogenization Theory for Multiscale Problems. MS&A, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-031-21833-0_6

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