On some nonlinear partial differential equations involving the 1-Laplacian

Details On some nonlinear partial differential equations involving the 1-Laplacian On some nonlinear partial differential equations involving the 1-Laplacian

2007-03-16

arxiv.org/abs/math/0703497v1

Mathematics

Analysis of PDEs

16pages

Scientific paper

In this paper we present an approximation result concerning the first eigenvalue of the 1-Laplacian operator. More precisely, for $\Omega$ a bounded regular open domain, we consider a minimisation of the functional ${\ds \int_\Omega}|\nabla u|+n({\ds \int_\Omega} |u|-1)^2 $ over the space $W_0^{1,1}(\Omega)$. For $n$ large enough, the infimum is achieved in some sense on $BV(\Omega)$, and letting $n$ go to infinity this provides an approximation of the first eigenfunction for the first eigenvalue, since the term $n({\ds \int_\Omega} |u|^2-1)^2$ "tends" to the constraint $\|u\|_1=1$.

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