Lavrentiev Phenomenon Research Articles

The Lavrentiev phenomenon and the obstacle problem for the Dirichlet integral

It is shown that the Lavrentiev phenomenon can occur for the Dirichlet integral, when the obstacle problem is considered.

Interpretation of Lavrentiev phenomenon by relaxation: the higher order case

We consider integral functionals of the calculus of variations of the form \[ F ( u ) = ∫ 0 1 f ( x , u , u ′ , … , u ( n ) ) d x F(u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx} \] defined for u ∈ W n , ∞ ( 0 , 1 ) u \in {W^{n,\infty }}(0,1) , and we show that the relaxed functional F F with respect to the weak W loc n , 1 ( 0 , 1 ) W_{{\text {loc}}}^{n,1}(0,1) convergence can be written as \[ F ¯ ( u ) = ∫ 0 1 f ( x , u , u ′ , … , u ( n ) ) d x + L ( u ) , \overline F (u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx + L(u),} \] where the additional term L ( u ) L(u) , the Lavrentiev Gap, is explicitly identified in terms of F F .

On the Lavrentiev Phenomenon and the Validity of Euler-Lagrange Equations for a Class of Integral Functionals

We study the functional J(x)=∫ b aƒ(t,x(t),x′(t))dt and provide conditions on the integrand ƒ excluding the Lavrentiev Phenomenon and ensuring the validity of Euler-Lagrange Equations.

Interpretation of the Lavrentiev phenomenon by relaxation

We consider functionals of the calculus of variations of the form F( u)= ∝ 0 1 f( x, u, u′) dx defined for u ϵ W 1,∞(0, 1), and we show that the relaxed functional F with respect to weak W 1,1(0, 1) convergence can be written as F ̄ (u) = ∝ 0 1 f(x, u, u′) dx + L(u) , where the additional term L( u), called the Lavrentiev term, is explicitly identified in terms of F.

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

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.

Wellposedness and the Lavrentiev Phenomenon

The Lavrentiev phenomenon in the calculus of variations is viewed and handled as a value Hadamard illposedness problem. Regularization is obtained by a decoupling technique of Ball–Knowles via a variational convergence approach. In this way we extend known one-dimensional results, simplifying their proofs. The same approach yields regularization for problems involving multiple integrals. A criterion for Hadamard wellposedness of multiple integrals is presented, and a new sufficient condition for nonoccurrence of the one-dimensional phenomenon is obtained.

A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity

In this paper we describe and analyse a numerical method that detects singular minimizers and avoids the Lavrentiev phenomenon for three dimensional problems in nonlinear elasticity. This method extends to three dimensions the corresponding one dimensional method of Ball and Knowles.

A New Example of the Lavrentiev Phenomenon

This paper describes a new example of a calculus of variations problem with the Lavrentiev phenomenon, a problem with different value functions for different classes of admissible trajectories. Such problems provide examples of nonuniqueness for Hamilton-Jacobi equations since each value function is a solution. The authors describe these solutions and the corresponding optimal trajectories and focus on how a standard proof of optimality is adapted for these examples.

The Lavrentiev phenomenon for invariant variational problems

In this paper we consider the problem of minimizing $$\bar{I}\left[ y \right]: = \int\limits_{a}^{b} {\bar{f}} \left( {x,y\left( x \right),y'\left( x \right)} \right)dx$$ (1) in the set \(ar A\) of absolutely continuous functions y(·): [a, b] → ℝ satisfying the end conditions $$y(a) = lpha ,y(b) = 0,$$ (2) where α is a prescribed constant. In (1), [a, b] is a bounded interval, “′ ” denotes differentiation with respect to x, and the integrand \(ar f = ar f(x,y,p)\) is assumed to be smooth (C 3) and nonnegative. In addition, \(ar f\) is assumed to satisfy: $${ar f_{pp}} eqslant 0.$$ (3)

On The Lavrentiev Phenomenon

AbstractA new look at Manià's classical example of the Lavrentiev Phenomenon leads to several pertinent observations.

A numerical method for detecting singular minimizers

A numerical method for computing minimizers in one-dimensional problems of the calculus of variations is described. Such minimizers may have unbounded derivatives, even when the integrand is smooth and regular. In such cases, because of the Lavrentiev phenomenon, standard finite element methods may fail to converge to a minimizer. The scheme proposed is shown to converge to an absolute minimizer and is tested on an example. The effect of quadrature is analyzed. The implications for higher-dimensional problems, and in particular for fracture in nonlinear elasticity, are discussed.

On the Lavrentiev phenomenon

Additional assumptions, besides convexity and coercivity, are investigated to guarantee that the Lavrentiev phenomenon does not occur in a basic problem of the calculus of variations.