Strong convergence of the gradients for p-Laplacian problems as p → ∞

In this paper we prove that the gradients of solutions to the Dirichlet problem for -Delta(p)u(p) = f, with f > 0, converge as p -> infinity strongly in every L-q with 1 <= q < infinity to the gradient of the limit function. This convergence is sharp since a simple example in 1-d shows that there is no convergence in L-infinity. For a nonnegative f we obtain the same strong convergence inside the support of f. The same kind of result also holds true for the eigenvalue problem for a suitable class of domains (as balls or stadiums). (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

In this study, it is proven that the gradients of solutions to the Dirichlet problem for -Delta(p)u(p) = f, with f > 0, converge strongly to the gradient of the limit function as p tends to infinity in every L-q space with 1 <= q < infinity. The sharpness of this convergence is demonstrated with a simple 1-dimensional example showing no convergence in L-infinity. Additionally, the same strong convergence is obtained within the support of nonnegative f, and similar results hold true for the eigenvalue problem for a certain class of domains, such as balls or stadiums.