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1997 Variational integrals of nearly linear growth
Luigi Greco, Tadeusz Iwaniec, Carlo Sbordone
Differential Integral Equations 10(4): 687-716 (1997).

Abstract

We study variational integrals and related equations whose integrand grows almost linearly with respect to the gradient. A prototype of such functionals is $$ I[u] =\int |\nabla u|\,A\big(|\nabla u|\big)\,dx, $$ where $A$ is slowly increasing to $\infty $. For instance, $A(t)=\log^\alpha(1+t)$, $\alpha >0$, or $A(t)=\log\log (e+t)$, etc. We show that the minimizer $u$ subject to the Dirichlet data $v$ satisfies the estimate $$ \int |\nabla u|\,A^{1\pm\epsilon}\big(|\nabla u|\big)\,dx\le C\int|\nabla v|\, A^{1\pm\epsilon}\big(|\nabla v|\big)\,dx $$ at least for some small $\epsilon >0$. This extends previous results [12], [16] on integrals with power growth.

Citation

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Luigi Greco. Tadeusz Iwaniec. Carlo Sbordone. "Variational integrals of nearly linear growth." Differential Integral Equations 10 (4) 687 - 716, 1997.

Information

Published: 1997
First available in Project Euclid: 1 May 2013

zbMATH: 0889.35026
MathSciNet: MR1741768

Subjects:
Primary: 35J85
Secondary: 49J40

Rights: Copyright © 1997 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.10 • No. 4 • 1997
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