Calculus of Variations and Geometric Measure Theory
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A. Ferriero

The weak repulsion property

created by ferrieroa on 19 Apr 2007
modified on 03 Dec 2007


Published Paper

Inserted: 19 apr 2007
Last Updated: 3 dec 2007

Journal: J. Math. Pures Appl.
Year: 2007


In 1926 M. Lavrentiev \cite{La} proposed an example of a variational problem whose infimum over the Sobolev space $*W*^{1,p}$, for some values of $p\geq1$, is strictly lower than the infimum over $*W*^{1,\infty}$. This energy gap is known since then as the Lavrentiev phenomenon.

The aim of this paper is to provide a deeper insight into this phenomenon by shedding light on an unnoticed feature. Any energy that presents the Lavrentiev gap phenomenon is unbounded in any neighbourhood of any minimizer in $*W*^{1,p}$.

We also show a finer result in case of regular minimizers and the repulsion property (observed by J. Ball and V. Mizel \cite{BM}) for any power $\alpha>1$ of a Lagrangian that exhibits the Lavrentiev gap phenomenon.


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