The Lavrentiev phenomenon for invariant variational problems

1988 ◽  
Vol 102 (1) ◽  
pp. 57-93 ◽  
Author(s):  
A. C. Heinricher ◽  
V. J. Mizel
Keyword(s):  
Variational Problems ◽  
Lavrentiev Phenomenon

Element removal method for singular minimizers in variational problems involving Lavrentiev phenomenon

1992 ◽  
Vol 439 (1905) ◽  
pp. 131-137 ◽  
Keyword(s):  
Numerical Methods ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Variational Problems ◽  
Lavrentiev Phenomenon ◽  
Removal Method ◽  
Element Removal Method ◽  
Element Removal ◽  
Singular Minimizers

A numerical method called element removal method is designed to calculate singular minimizers which cannot be approximated by simple applications of standard numerical methods because of the so-called Lavrentiev phenomenon. The convergence of the method is proved. The results of numerical experiments show that the method is effective.

scholarly journals On the Lavrentiev phenomenon for multiple integral scalar variational problems

2014 ◽  
Vol 266 (9) ◽  
pp. 5921-5954 ◽  
Author(s):  
Pierre Bousquet ◽  
Carlo Mariconda ◽  
Giulia Treu
Keyword(s):  
Multiple Integral ◽  
Variational Problems ◽  
Lavrentiev Phenomenon

scholarly journals A TRUNCATION METHOD FOR DETECTING SINGULAR MINIMIZERS INVOLVING THE LAVRENTIEV PHENOMENON

2006 ◽  
Vol 16 (06) ◽  
pp. 847-867 ◽  
Author(s):  
YU BAI ◽  
ZHI-PING LI
Keyword(s):  
Numerical Method ◽  
Variational Problems ◽  
Numerical Results ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Minimizing Sequences ◽  
Lavrentiev Phenomenon ◽  
Absolute Minimizers ◽  
Sobolev Exponent ◽  
Singular Minimizers

A numerical method using the truncation technique on the integrand is developed for computing singular minimizers or singular minimizing sequences in variational problems involving the Lavrentiev phenomenon. It is proved that the method can detect absolute minimizers with various singularities whether the Lavrentiev phenomenon is involved or not. It is also proved that, when the absolute infimum is not attainable, the method can produce minimizing sequences. Numerical results on Manià's example and a two-dimensional problem involving the Lavrentiev phenomenon with continuous Sobolev exponent dependence, are given to show the efficiency of the method.

scholarly journals Analysis of a Class of Penalty Methods for Computing Singular Minimizers

2010 ◽  
Vol 10 (2) ◽  
pp. 137-163 ◽  
Author(s):  
C. Carstensen ◽  
C. Ortner
Keyword(s):  
Finite Element ◽  
Calculus Of Variations ◽  
Nonlinear Partial Differential Equations ◽  
Variational Problems ◽  
General Strategy ◽  
Penalty Parameter ◽  
Lavrentiev Phenomenon ◽  
Convergence Results ◽  
Conforming Finite Element Method ◽  
Singular Minimizers

AbstractAmongst the more exciting phenomena in the field of nonlinear partial differential equations is the Lavrentiev phenomenon which occurs in the calculus of variations. We prove that a conforming finite element method fails if and only if the Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods have to be designed for the detection of the Lavrentiev phenomenon in the computational calculus of variations. We formulate and analyze a general strategy for solving variational problems in the presence of the Lavrentiev phenomenon based on a splitting and penalization strategy. We establish convergence results under mild conditions on the stored energy function. Moreover, we present practical strategies for the solution of the discretized problems and for the choice of the penalty parameter.

scholarly journals Cosmic Analogues of Classic Variational Problems

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 71 ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Relativistic Particle ◽  
Friedmann Equation ◽  
Variational Calculus ◽  
Variational Problems ◽  
Particle Mechanics ◽  
One Dimensional ◽  
Spatially Homogeneous

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

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