Lipschitz properties in variable exponent problems via relative rearrangement (Q619656)

Let \(\Omega\) be an open bounded set and \(u_*\) (resp., \(u^*\)) be the decreasing (resp., increasing) rearrangement of \(u:\Omega\to{\mathbb R}\). For a bounded measurable function \(p:\Omega\to[1,+\infty)\), put \(\Phi_p(u):=\int_\Omega|u(x)|^{p(x)}\) and define the variable Lebesgue space \(L^{p(\cdot)}(\Omega)\) as the set of all measurable functions \(u:\Omega\to\mathbb R\) satisfying \(\Phi_p(u)<\infty\). It is well known that this is a Banach space when equipped with the norm \(\|u\|_{p(\cdot)}:=\inf\{\lambda>0:\Phi_p(u/\lambda)\leq 1\}\). The first main result of the paper complements those obtained by \textit{A. Fiorenza} and \textit{J. M. Rakotoson} [J. Math. Pures Appl. (9) 88, No.~6, 506--521 (2007; Zbl 1137.46016)]. It says that, if \(p:\Omega\to [1,+\infty)\) is a bounded measurable function such that its increasing rearrangement \(p^*\) satisfies \(p^*(0)>1\) and \(|p^*(t)-p^*(\sigma)|\,|\log|t-\sigma||=O(1)\) near zero, then \(\|[(u+v)_*-u_*]_{**}\|_{p^*(\cdot)}=O(\|v\|_{p(\cdot)})\) for all \(u\in L^1(\Omega)\) and \(v\in L_+^{p(\cdot)}(\Omega)\), where \(L_+^{p(\cdot)}(\Omega):=\{f\in L^{p(\cdot)}(\Omega): f\geq 0\}\) and \(f_{**}(s)=(1/s)\int_0^s g_*(\sigma)\,d\sigma\) for \(g\in L^1(\Omega)\). This result is applied to establish Lipschitz properties for quasilinear problems with a variable exponent when the right-hand side is in some dual spaces of suitable Sobolev spaces associated to the variable exponent.