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 & 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.  doi: 10.1007/BF01830667.  Google Scholar

[2]

V. Bangert, A uniqueness theorem for $Z$n-periodic variational problems,, Comment. Math. Helv., 62 (1987), 511.  doi: 10.1007/BF02564459.  Google Scholar

[3]

V. Bangert, On minimal laminations of the torus,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95.   Google Scholar

[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.   Google Scholar

[5]

L. Chierchia and C. Falcolini, A note on quasi-periodic solutions of some elliptic systems,, Z. Angew. Math. Phys., 47 (1996), 210.  doi: 10.1007/BF00916825.  Google Scholar

[6]

L. C. Evans, "Partial Differential Equations," volume 19 of "Graduate Studies in Mathematics,", American Mathematical Society, (1998).   Google Scholar

[7]

T. Kato, "Perturbation Theory for Linear Operators,", Die Grundlehren der mathematischen Wissenschaften, (1966).   Google Scholar

[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.   Google Scholar

[9]

M. Morse, "Variational Analysis: Critical Extremals and Sturmian Extensions,", Interscience Publishers [John Wiley & Sons, (1973).   Google Scholar

[10]

J. Moser, Minimal solutions of variational problems on a torus,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229.   Google Scholar

[11]

J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of "Lecture Notes in Mathematics,", Springer-Verlag, (2010).   Google Scholar

[12]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa(3), 13 (1959), 115.   Google Scholar

[13]

W. Senn, Strikte Konvexität fär Variationsprobleme auf dem $n$-dimensionalen Torus,, Manuscripta Math., 71 (1991), 45.  doi: 10.1007/BF02568393.  Google Scholar

[14]

W. M. Senn, Differentiability properties of the minimal average action,, Calc. Var. Partial Differential Equations, 3 (1995), 343.   Google Scholar

show all references

References:
[1]

V. Bangert, The existence of gaps in minimal foliations,, Aequationes Math., 34 (1987), 153.  doi: 10.1007/BF01830667.  Google Scholar

[2]

V. Bangert, A uniqueness theorem for $Z$n-periodic variational problems,, Comment. Math. Helv., 62 (1987), 511.  doi: 10.1007/BF02564459.  Google Scholar

[3]

V. Bangert, On minimal laminations of the torus,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95.   Google Scholar

[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.   Google Scholar

[5]

L. Chierchia and C. Falcolini, A note on quasi-periodic solutions of some elliptic systems,, Z. Angew. Math. Phys., 47 (1996), 210.  doi: 10.1007/BF00916825.  Google Scholar

[6]

L. C. Evans, "Partial Differential Equations," volume 19 of "Graduate Studies in Mathematics,", American Mathematical Society, (1998).   Google Scholar

[7]

T. Kato, "Perturbation Theory for Linear Operators,", Die Grundlehren der mathematischen Wissenschaften, (1966).   Google Scholar

[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.   Google Scholar

[9]

M. Morse, "Variational Analysis: Critical Extremals and Sturmian Extensions,", Interscience Publishers [John Wiley & Sons, (1973).   Google Scholar

[10]

J. Moser, Minimal solutions of variational problems on a torus,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229.   Google Scholar

[11]

J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of "Lecture Notes in Mathematics,", Springer-Verlag, (2010).   Google Scholar

[12]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa(3), 13 (1959), 115.   Google Scholar

[13]

W. Senn, Strikte Konvexität fär Variationsprobleme auf dem $n$-dimensionalen Torus,, Manuscripta Math., 71 (1991), 45.  doi: 10.1007/BF02568393.  Google Scholar

[14]

W. M. Senn, Differentiability properties of the minimal average action,, Calc. Var. Partial Differential Equations, 3 (1995), 343.   Google Scholar

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