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On the behaviour of the solutions to \(p\)-Laplacian equations as \(p\) goes to 1. (English) Zbl 1158.35024

The authors study the behavior of the weak solutions \(u_p\) as \(p\) goes to \(1\) of the problem \[ -\text{div}(| \nabla u_p| ^{p-2}\nabla u_p) = f\;\text{ in}\;\Omega, \]
\[ u_p = 0 \;\text{ on}\;\partial\Omega, \] where \(\Omega\) is an open set in \(\mathbb{R}^n\) with Lipschitz boundary and \(p>1\). The authors analyze several cases with different assumptions on \(f\); the most general assumption considered is \(f\in W^{-1,\infty}(\Omega)\). The result are illustrated by several remarks and examples.

MSC:

35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
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