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 −Δpup=f, with f>0, converge as p→∞ strongly in every Lq with 1≤q<∞ 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∞. 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).