Differential and Integral Equations

Variational integrals of nearly linear growth

Luigi Greco, Tadeusz Iwaniec, and Carlo Sbordone

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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.

Article information

Differential Integral Equations, Volume 10, Number 4 (1997), 687-716.

First available in Project Euclid: 1 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J85
Secondary: 49J40: Variational methods including variational inequalities [See also 47J20]


Greco, Luigi; Iwaniec, Tadeusz; Sbordone, Carlo. Variational integrals of nearly linear growth. Differential Integral Equations 10 (1997), no. 4, 687--716. https://projecteuclid.org/euclid.die/1367438637

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