American Institute of Mathematical Sciences

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

Perturbation and numerical methods for computing the minimal average energy

Timothy Blass 1, and  Rafael de la Llave 2,

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.
Keywords: Lindstedt series, Minimal average energy, Sobolev gradient descent, Plane-like minimizers, Cell problem, quasiperiodic solutions of PDE..
Mathematics Subject Classification: Primary: 35B27, 58E15; Secondary: 41A58, 65M9.
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), 153-166. doi: 10.1007/BF01830667.  Google Scholar

[2]

V. Bangert, A uniqueness theorem for $Z$n-periodic variational problems, Comment. Math. Helv., 62 (1987), 511-531. doi: 10.1007/BF02564459.  Google Scholar

[3]

V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95-138.  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), 69-91.  Google Scholar

[5]

L. Chierchia and C. Falcolini, A note on quasi-periodic solutions of some elliptic systems, Z. Angew. Math. Phys., 47 (1996), 210-220. doi: 10.1007/BF00916825.  Google Scholar

[6]

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

[7]

T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, 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), 1309-1344.  Google Scholar

[9]

M. Morse, "Variational Analysis: Critical Extremals and Sturmian Extensions," Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1973.  Google Scholar

[10]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272.  Google Scholar

[11]

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

[12]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa(3), 13 (1959), 115-162.  Google Scholar

[13]

W. Senn, Strikte Konvexität fär Variationsprobleme auf dem $n$-dimensionalen Torus, Manuscripta Math., 71 (1991), 45-65. doi: 10.1007/BF02568393.  Google Scholar

[14]

W. M. Senn, Differentiability properties of the minimal average action, Calc. Var. Partial Differential Equations, 3 (1995), 343-384.  Google Scholar

show all references

References:
[1]

V. Bangert, The existence of gaps in minimal foliations, Aequationes Math., 34 (1987), 153-166. doi: 10.1007/BF01830667.  Google Scholar

[2]

V. Bangert, A uniqueness theorem for $Z$n-periodic variational problems, Comment. Math. Helv., 62 (1987), 511-531. doi: 10.1007/BF02564459.  Google Scholar

[3]

V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95-138.  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), 69-91.  Google Scholar

[5]

L. Chierchia and C. Falcolini, A note on quasi-periodic solutions of some elliptic systems, Z. Angew. Math. Phys., 47 (1996), 210-220. doi: 10.1007/BF00916825.  Google Scholar

[6]

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

[7]

T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, 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), 1309-1344.  Google Scholar

[9]

M. Morse, "Variational Analysis: Critical Extremals and Sturmian Extensions," Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1973.  Google Scholar

[10]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272.  Google Scholar

[11]

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

[12]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa(3), 13 (1959), 115-162.  Google Scholar

[13]

W. Senn, Strikte Konvexität fär Variationsprobleme auf dem $n$-dimensionalen Torus, Manuscripta Math., 71 (1991), 45-65. doi: 10.1007/BF02568393.  Google Scholar

[14]

W. M. Senn, Differentiability properties of the minimal average action, Calc. Var. Partial Differential Equations, 3 (1995), 343-384.  Google Scholar

[1]

Antonin Chambolle, Gilles Thouroude. Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Networks & Heterogeneous Media, 2009, 4 (1) : 127-152. doi: 10.3934/nhm.2009.4.127
[2]

V. Mastropietro, Michela Procesi. Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. Communications on Pure & Applied Analysis, 2006, 5 (1) : 1-28. doi: 10.3934/cpaa.2006.5.1
[3]

Tien-Tsan Shieh. From gradient theory of phase transition to a generalized minimal interface problem with a contact energy. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2729-2755. doi: 10.3934/dcds.2016.36.2729
[4]

G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509
[5]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[6]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187
[7]

Peter Takáč. Stabilization of positive solutions for analytic gradient-like systems. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 947-973. doi: 10.3934/dcds.2000.6.947
[8]

Herbert Gajewski, Jens A. Griepentrog. A descent method for the free energy of multicomponent systems. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 505-528. doi: 10.3934/dcds.2006.15.505
[9]

Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311
[10]

Martí Prats. Beltrami equations in the plane and Sobolev regularity. Communications on Pure & Applied Analysis, 2018, 17 (2) : 319-332. doi: 10.3934/cpaa.2018018
[11]

Alessio Figalli, Vito Mandorino. Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1325-1346. doi: 10.3934/dcds.2011.31.1325
[12]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983
[13]

Armands Gritsans, Felix Sadyrbaev. The Nehari solutions and asymmetric minimizers. Conference Publications, 2015, 2015 (special) : 562-568. doi: 10.3934/proc.2015.0562
[14]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[15]

Dominique Lecomte. Hurewicz-like tests for Borel subsets of the plane. Electronic Research Announcements, 2005, 11: 95-102.

[16]

Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036
[17]

Peter A. Hästö. On the existance of minimizers of the variable exponent Dirichlet energy integral. Communications on Pure & Applied Analysis, 2006, 5 (3) : 415-422. doi: 10.3934/cpaa.2006.5.415
[18]

Xiaming Chen. Kernel-based online gradient descent using distributed approach. Mathematical Foundations of Computing, 2019, 2 (1) : 1-9. doi: 10.3934/mfc.2019001
[19]

Ting Hu. Kernel-based maximum correntropy criterion with gradient descent method. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4159-4177. doi: 10.3934/cpaa.2020186
[20]

Feng Bao, Thomas Maier. Stochastic gradient descent algorithm for stochastic optimization in solving analytic continuation problems. Foundations of Data Science, 2020, 2 (1) : 1-17. doi: 10.3934/fods.2020001

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences