American Institute of Mathematical Sciences

June  2011, 6(2): 241-255. doi: 10.3934/nhm.2011.6.241

Perturbation and numerical methods for computing the minimal average energy

 1 Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, United States 2 Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257

Received  January 2011 Revised  April 2011 Published  May 2011

We investigate the differentiability of minimal average energy associated to the functionals $S_\epsilon (u) = \int_{\mathbb{R}^d} \frac{1}{2}|\nabla u|^2 + \epsilon V(x,u)\, dx$, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter $\epsilon$, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.
Citation: Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks and Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241
References:
 [1] V. Bangert, The existence of gaps in minimal foliations, Aequationes Math., 34 (1987), 153-166. doi: 10.1007/BF01830667. [2] V. Bangert, A uniqueness theorem for $Z$n-periodic variational problems, Comment. Math. Helv., 62 (1987), 511-531. doi: 10.1007/BF02564459. [3] V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95-138. [4] T. Blass, R. de la Llave and E. Valdinoci, A comparison principle for a Sobolev gradient semi-flow, Commun. Pure Appl. Anal., 10 (2011), 69-91. [5] L. Chierchia and C. Falcolini, A note on quasi-periodic solutions of some elliptic systems, Z. Angew. Math. Phys., 47 (1996), 210-220. doi: 10.1007/BF00916825. [6] L. C. Evans, "Partial Differential Equations," volume 19 of "Graduate Studies in Mathematics," American Mathematical Society, Providence, RI, 1998. [7] T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, 1966. [8] R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344. [9] M. Morse, "Variational Analysis: Critical Extremals and Sturmian Extensions," Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1973. [10] J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272. [11] J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of "Lecture Notes in Mathematics," Springer-Verlag, Berlin, second edition, 2010. [12] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa(3), 13 (1959), 115-162. [13] W. Senn, Strikte Konvexität fär Variationsprobleme auf dem $n$-dimensionalen Torus, Manuscripta Math., 71 (1991), 45-65. doi: 10.1007/BF02568393. [14] W. M. Senn, Differentiability properties of the minimal average action, Calc. Var. Partial Differential Equations, 3 (1995), 343-384.

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References:
 [1] V. Bangert, The existence of gaps in minimal foliations, Aequationes Math., 34 (1987), 153-166. doi: 10.1007/BF01830667. [2] V. Bangert, A uniqueness theorem for $Z$n-periodic variational problems, Comment. Math. Helv., 62 (1987), 511-531. doi: 10.1007/BF02564459. [3] V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95-138. [4] T. Blass, R. de la Llave and E. Valdinoci, A comparison principle for a Sobolev gradient semi-flow, Commun. Pure Appl. Anal., 10 (2011), 69-91. [5] L. Chierchia and C. Falcolini, A note on quasi-periodic solutions of some elliptic systems, Z. Angew. Math. Phys., 47 (1996), 210-220. doi: 10.1007/BF00916825. [6] L. C. Evans, "Partial Differential Equations," volume 19 of "Graduate Studies in Mathematics," American Mathematical Society, Providence, RI, 1998. [7] T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, 1966. [8] R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344. [9] M. Morse, "Variational Analysis: Critical Extremals and Sturmian Extensions," Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1973. [10] J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272. [11] J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of "Lecture Notes in Mathematics," Springer-Verlag, Berlin, second edition, 2010. [12] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa(3), 13 (1959), 115-162. [13] W. Senn, Strikte Konvexität fär Variationsprobleme auf dem $n$-dimensionalen Torus, Manuscripta Math., 71 (1991), 45-65. doi: 10.1007/BF02568393. [14] W. M. Senn, Differentiability properties of the minimal average action, Calc. Var. Partial Differential Equations, 3 (1995), 343-384.
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