Editing Calculus of variations (section)

== Lavrentiev phenomenon ==

Hilbert was the first to give good conditions for the Euler–Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler–Lagrange equations in the interior.

However [[Mikhail Lavrentyev|Lavrentiev]] in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions. For instance the following problem, presented by Manià in 1934:<ref>{{Cite journal|last=Manià|first=Bernard|date=1934|title=Sopra un esempio di Lavrentieff| journal=Bollenttino dell'Unione Matematica Italiana|volume=13|pages=147–153}}</ref>
<math display="block">L[x] = \int_0^1 (x^3-t)^2 x'^6,</math>
<math display="block">{A} = \{x \in W^{1,1}(0,1) : x(0)=0,\ x(1)=1\}.</math>

Clearly, <math>x(t) = t^{\frac{1}{3}}</math>minimizes the functional, but we find any function <math>x \in W^{1, \infty}</math> gives a value bounded away from the infimum.

Examples (in one-dimension) are traditionally manifested across <math>W^{1,1}</math> and <math>W^{1,\infty},</math> but Ball and Mizel<ref>{{Cite journal|last=Ball & Mizel|date=1985|title=One-dimensional Variational problems whose Minimizers do not satisfy the Euler-Lagrange equation.|journal=Archive for Rational Mechanics and Analysis|volume=90|issue=4|pages=325–388| doi=10.1007/BF00276295|bibcode=1985ArRMA..90..325B|s2cid=55005550}}</ref> procured the first functional that displayed Lavrentiev's Phenomenon across <math>W^{1,p}</math> and <math>W^{1,q}</math> for <math>1 \leq p < q < \infty.</math> There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals. 

Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.<ref>{{Cite journal|last=Ferriero|first=Alessandro|date=2007|title=The Weak Repulsion property | journal=Journal de Mathématiques Pures et Appliquées|volume=88|issue=4|pages=378–388| doi=10.1016/j.matpur.2007.06.002 | doi-access=}}</ref>