Gradient rearrangement estimates for singular and degenerate anisotropic $p$-Laplace equations

Abstract

In this paper, we prove global pointwise estimates on the decreasing rearrangement \(u^{\ast}\) and \([H(\nabla u)]^{\ast}\) of SOLAs \(u\) of an elliptic Poisson equation driven by the anisotropic \(p\)-Laplace operator \(Δ_p^H\) for given \(L^1\)-data and equipped with either homogeneous Dirichlet or Neumann boundary conditions. Here, we assume that the dimension \(n\ge 2\), \(1<p<n\) and \(H\) is norm on \(\R^n\). In the degenerate range \(2 \leq p <n\), this gives an anisotropic extension and a full-interval completion of the rearrangement theory of Cianchi-Maz'ya [JEMS, 2014]. In particular, we obtain a new rearrangement estimate for gradients by decomposing the data for singular range \(1<p<2\), where the nonlinear interpolation method aren't available. Moreover, the endpoint weak-type gradient estimate is obtained for all \(1<p<n\). In the range \(p_*(n)<p<n\) with \(p_*(n):=\frac{3n-2}{2n-1}\), we prove a stability estimate for differences of gradients. These results extends the known ones in the literature to the singular range \(p_*(n)<p<2\).

arXiv commentKeywords: anisotropic p-Laplacian, singular elliptic equation, SOLA, gradient rearrangement, Lorentz space, Dirichlet problem, Neumann problem