Qualitative Behavior of Local Minimizers of Singular Perturbed Variational Problems
Source abstract
Abstract
We consider a non-convex variational problem (P) and the corresponding singular perturbed problem (P ε ). The qualitative behavior of stable critical points of (P ε ) depending on ε and a lower order term is discussed and we prove compactness of a sequence of stable critical points as ε ↘ 0. Moreover we show whether this limit is the global minimizer of (P). Furthermore uniform convergence is considered as well as the convergence rate depending on ε.
Source: expertdata · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Differential Equations and Numerical Methods · Nonlinear Partial Differential Equations · Contact Mechanics and Variational Inequalities
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (24)
Show source-provided references
1. Alikakos, N.D., Bates, P.W.: On the singular limit in a phase field model of phase transitions. Ann. Inst. Henri Poincare, Anal. non lineaire 5, 141–178 (1988)10.1016/s0294-1449(16)30349-3 ↗
2. Alikakos, N.D., Shaing, K.C.: On the singular limit for a class of problems modelling phase transitions. SIAM J. Math. Anal. 18, 1453–1462 (1987)10.1137/0518105 ↗
3. Alikakos, N.D., Simpson, H.C.: A variational approach for a class of singular perturbation problems and applications. Proc. Roy. Soc. Edinburgh., Sect A 107, 27–42 (1987)10.1017/S0308210500029334 ↗
4. Angenent, S.B., Mallet-Paret, J., Peletier, L.A.: Stable transition layers in a semilinear bondary value problem. J. Differential Equations 67, 212–242 (1987)10.1016/0022-0396(87)90147-1 ↗
5. Carr, J., Gurtin, M.E., Slemrod, M.: One-dimensional structured phase transitions under prescribed loads. J. Elasticity 15, 133–142 (1985)10.1007/BF00041988 ↗
6. Carr, J., Gurtin, M.E., Slemrod, M.: Structured phase transitions on a finite interval. Arch. Rational Mech. Anal. 86, 317–351 (1984)10.1007/BF00280031 ↗
7. Cellina, A., Colombo, G.: On a classical problem of the calculus of variations without convexity assumptions. Ann. Inst. Henri Poincare, Anal. non lineaire 7, 97–106 (1990)10.1016/s0294-1449(16)30306-7 ↗
8. Dacorogna, B.: Direct Methods in the Calculus of Variations. Springer, Berlin Heidelberg New York (1989)10.1007/978-3-642-51440-1 ↗
9. Ericksen, J.L.: Equilibrium of bars. J. Elasticity 5, 191–201 (1975)10.1007/BF00126984 ↗
10. Healey, T.J., Kielhöfer, H.: Global continuation via higher-gradient regularization and singular limits in forced one dimensional phase transitions. SIAM J. Math. Anal. 31, 1307–1331 (2000)10.1137/S0036141098340065 ↗
11. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840. Springer, Berlin Heidelberg New York (1981)10.1007/BFb0089647 ↗
12. James, R.D.: Co-existent phases in the one-dimensional static theory of elastic bars. Arch. Rational Mech. Anal. 72, 99–140 (1979/80)10.1007/BF00249360 ↗
13. Kielhöfer, H.: Pattern formation of the stationary Cahn-Hilliard equation model. Proc. Roy. Soc. Edinburgh, Sect A 127, 1219–1243 (1997)10.1017/S0308210500027037 ↗
14. Kielhöfer, H.: Minimizing sequences selected via singular perturbations, and their pattern formation. Arch. Rational Mech. Anal. 155, 261–276 (2000)10.1007/s002050000097 ↗
15. Kielhöfer, H.: Critical points of nonconvex and noncoercive functionals. Calc. Var. Partial Differ. Equ. 16, 243–272 (2003)10.1007/s005260100149 ↗
16. Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Commun. Pure Appl. Math. 47, 405–435 (1994)10.1002/cpa.3160470402 ↗
17. Müller, S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differential Equations 1, 169–204 (1993)10.1007/BF01191616 ↗
18. Lilli, M.: The Effect of a Singular Perturbation to a 1-d Nonconvex Variational Problem, Augsburger Schriften zur Mathematik, Physik und Informatik, Logos, Berlin (2005)
19. Lilli, M.: Classical Solutions for Non-Elliptic Euler-Lagrange Equations via Continuation (in press)
20. Lilli, M., Healey, T.J., Kielhöfer, H.: Singular perturbation as a selection criterion for young-measure solutions. SIAM J. Math. Anal. (to appear)
21. Modica, L., Mortola, S.: Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. 14-A, 526–529 (1977)
22. Natanson, I.P.: Theory of Functions of a Real Variable. F Ungar, New York (1955)
23. Protter, M.H., Weinberger H.F.: Maximum Principles in Differential Equations. Springer, Berlin, Heidelberg New York (1967)
24. Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, Berlin Heidelberg New York (2002)10.1007/978-1-4757-5037-9 ↗