A direct proof of the Tonelli's partial regularity result

Alessandro Ferriero

doi:10.3934/dcds.2012.32.2089

Article Contents

2012, Volume 32, Issue 6: 2089-2099. Doi: 10.3934/dcds.2012.32.2089

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A direct proof of the Tonelli's partial regularity result

Alessandro Ferriero^1,

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco,28049 Madrid

Received: March 31, 2011

Revised: June 30, 2011

Published: May 2012

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Abstract

The aim of this work is to give a simple proof of the Tonelli's partial regularity result which states that any absolutely continuous solution to the variational problem $$\min\left\{\int_a^b L(t,u(t),\dot u(t))dt: u\in{\bf W}_0^{1,1}(a,b)\right\}$$ has extended-values continuous derivative if the Lagrangian function $L(t,u,\xi)$ is strictly convex in $\xi$ and Lipschitz continuous in $u$, locally uniformly in $\xi$ (but not in $t$). Our assumption is weaker than the one used in [2, 4, 5, 6, 13] since we do not require the Lipschitz continuity of $L$ in $u$ to be locally uniform in $t$, and it is optimal as shown by the example in [12].

Keywords:

Variational problems,
regularity,
quasi-linear elliptic differential equations.

Mathematics Subject Classification: Primary: 49B05, 49A05, 49C05, 35J20.

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References

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