\(C^1\)-regularity for minima of functionals with \(p(x)\)-growth (Q1684881)

The author studies the regularity of the local minima to the functional of the form \[ F(w)= \int_\Omega f(x,w,Dw)+g(x)\, w(x) \, dx. \] where \(f\) satisfies a \(p(x)\)-growth condition. Under some suitable structure conditions and by assuming that \(g\) belongs to the Lorentz space \(L^{n,1}\) and \(p(\cdot)\) and \(f(\cdot,z,\xi)f(\cdot,z,\xi)\) satisfy Dini type continuity property, the author proves that the local minimizer \( u \in W^{1,p(x)} (\Omega)\) is in \(C^1(\Omega)\). It is interesting to observe that the arguments of the proofs are not an extension of the case \(p(x)=p\).