Variational integrals of nearly linear growth. (English) Zbl 0889.35026

From the authors’ abstract: We study variational integrals and related equations whose integrand grows almost linearly with respect to the gradient. A prototype of such functionals is \[ I[u] = \int _{\Omega } {|}\nabla u{|} A ({|}\nabla u{|}) dx \] 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 \[ \int {|}\nabla u{|} A^{1\pm \varepsilon } ({|}\nabla u{|}) dx \leq C \int _{\Omega } {|}\nabla v{|} A^{1\pm \varepsilon }({|}\nabla v{|}) dx \] at least for some small \(\varepsilon >0\).
Reviewer: P.Drábek (Plzeň)


35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
49N60 Regularity of solutions in optimal control