Strong convergence of the gradients for \(p\)-Laplacian problems as \(p \rightarrow \infty \). (English) Zbl 1459.35219
Summary: 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 \(p \to \infty\) strongly in every \(L^q\) with \(1 \leq q < \infty\) 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 \(L^\infty \). 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).
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J25 | Boundary value problems for second-order elliptic equations |
References:
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