Gradient estimates in Orlicz spaces for quasilinear elliptic equation
The authors obtain gradient estimates in Orlicz spaces for the weak solutions of quasilinear elliptic equations of \(p\)-Laplacian type. More precisely, let \(u\) be a local weak solution of the equation \[ \text{div\,}\big(| \nabla u| ^{p-2}\nabla u\big) = \text{div\,}\big(| f| ^{p-2}f\big)\;\text{in}\;\Omega\subset \mathbb R^n \] with \(p>1.\) Suppose further \(\phi\colon \mathbb R\to[0,\infty)\) is a convex and even function which is nondecreasing on \([0,\infty)\) and satisfies \[ A_1(s/t)^{\alpha_1}\leq \phi(s)/\phi(t)\leq A_2(s/t)^{\alpha_2}\;\text{for}\;0 < s \leq t, \] where \(A_2\geq A_1 > 0\) and \(\alpha_1\geq\alpha_2>1.\) Define the Orlicz class \(K^\phi(\Omega)\) consisting of the measurable functions \(g\) for which \[ \int_\Omega \phi(| g| )dx<\infty, \] and let \(L^\phi(\Omega)\) be the Orlicz space obtained as the linear hull of \(K^\phi(\Omega)\) given by the measurable functions \(g\) for which \(\mu g\in K^\phi(\Omega)\) for any constant \(\mu > 0.\) The main result of the paper asserts that for all \(f\) with \(| f| ^p\in L^\phi_{\text{loc}}(\Omega)\) one has \(| \nabla u| ^p\in L^\phi_{\text{loc}}(\Omega)\) and \[ \int_{B_{R/2}(x_0)} \phi(| \nabla u| ^p)dx \leq C(n,p,\phi)\left[ \phi\left({1\over{R^p}}\int_{B_{R}(x_0)} | u| ^pdx\right) +\int_{B_{R}(x_0)} \phi(| f| ^p)dx \right] \] for each ball \(B_R(x_0)\subset\Omega.\)
scientific article; zbMATH DE number 5355003
scientific article