The Approximation of Higher-Order Integrals of the Calculus of Variations and the Lavrentiev Phenomenon

2005 ◽  
Vol 44 (1) ◽  
pp. 99-110 ◽  
Author(s):  
Alessandro Ferriero
Keyword(s):  
Calculus Of Variations ◽  
Higher Order ◽  
Lavrentiev Phenomenon

scholarly journals Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

2020 ◽  
Vol 18 (1) ◽  
pp. 1-9
Author(s):  
Carlo Mariconda ◽  
Giulia Treu
Keyword(s):  
Convex Function ◽  
Calculus Of Variations ◽  
Open Subset ◽  
Lipschitz Function ◽  
Lavrentiev Phenomenon ◽  
Smooth Functions ◽  
Bounded Open Subset ◽  
Admissible Functions

Abstract We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

scholarly journals Optimality conditions for the calculus of variations with higher-order delta derivatives

2011 ◽  
Vol 24 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Rui A.C. Ferreira ◽  
Agnieszka B. Malinowska ◽  
Delfim F.M. Torres
Keyword(s):  
Optimality Conditions ◽  
Calculus Of Variations ◽  
Higher Order ◽  
Delta Derivatives

scholarly journals Composition functionals in higher order calculus of variations and Noether's theorem

2021 ◽  
pp. 1-18
Author(s):  
Gastão S. F. Frederico ◽  
J. Vanterler da C. Sousa ◽  
Ricardo Almeida
Keyword(s):  
Calculus Of Variations ◽  
Higher Order ◽  
Noether’S Theorem ◽  
Noether's Theorem

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.

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