Title

Biting convergence of null-Lagrangians

Date of Award

2003

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Tadeusz Iwaniec

Keywords

Biting, Convergence, Null-Lagrangians, Euler-Lagrange equations, Calculus

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

In the calculus of variations it is essential to work with weakly sequentially compact spaces. Due to the lack of reflexivity of the space [Special characters omitted.] , given any bounded sequence { f j } ⊂ [Special characters omitted.] , we cannot guarantee the existence of a subsequence which converges weakly. Biting convergence comes to the rescue when the only available information about a sequence in [Special characters omitted.] is its boundedness. This notion of convergence has been designed mainly to deal with bounded sequences in [Special characters omitted.] that fail to be equi-integrable. We give a new proof of the theorem of K. Zhang [ Z ] on biting convergence of Jacobian determinants for mappings of Sobolev class [Special characters omitted.] . The theorem states that given a bounded sequence of functions { f j } ⊂ [Special characters omitted.] , there exists a subsequence {[Special characters omitted.] } such that [Special characters omitted.] converges in the biting sense to J ( x , f ). The novelty of our approach is in using [Special characters omitted.] -estimates with the exponents 1 [Special characters omitted.] p < n , as developed in [ IS1, IL, I1 ]. These rather strong estimates compensate for lack of equi-integrability.

We extend this result to more general wedge products of differential forms and then to null-Lagrangians. Null-Lagrangians are characterized by Euler-Lagrange equations. If the value of the energy integral [Special characters omitted.] [ f ] = [Special characters omitted.] does not change by adding to f a function that vanishes on the boundary of [Special characters omitted.] , then the function E is called a null-Lagrangian. Elliptic complexes, Hodge decomposition and non-linear commutators play a vital role in our proof.

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