Strong convergence for Neumann p-Laplacian problems with spatial dependence as p goes to infinity

Abstract

This paper investigates the asymptotic behavior of solutions $u_p$ to p-Laplacian type equations as p goes to ∞, under a homogeneous Neumann boundary condition given by$\{\begin{array}{cc}-\mathrm{div}\left(\frac{\left|\nabla u\right(x\left){|}^{p-2}\nabla u\right(x)}{k_p{\left(x\right)}^p}\right)=f\left(x\right)& \text{ in }\Omega \\ k_p{\left(x\right)}^{-p}\left|\nabla u\right(x){|}^{p-2}\frac{\partial u}{\partial \eta }=0& \text{ on }\partial \Omega ,\end{array}$ where $k_p\left(x\right)$ are space-dependent diffusion functions. Under suitable conditions on the diffusion functions $k_p$, particularly the uniform convergence $\underset{p\to \infty }{\lim }{k}_p\left(x\right)=k\left(x\right)$ on $\bar{\Omega }$, Mazon, Rossi and Toledo [17] show that, along a subsequence, the sequence of solutions $\left\{u_p\right\}$ converges uniformly to a limit function $u_{\infty }$ and the sequence of gradients $\left\{\nabla u_p\right\}$ converges weakly to $\nabla u_{\infty }$ in Lebesgue spaces as p goes to ∞. Among other results, the present paper proves that the sequence of gradients $\left\{\nabla u_p\right\}$ actually converges strongly on the so-called transport set as p goes to ∞. This strong convergence is useful information for optimal transport problems. As a consequence, we also obtain the strong convergence of gradients $\nabla u_p$ on the support of the sign-changing source function f as p goes to ∞.