It is known that when $g \in L^n(\Omega)$ ($\Omega$ open and bounded in $R^n$, with regular boundary $\partial \Omega$), any minimizer of the functional $F(C,u) = H^{n-1}(C) + \lambda \int_{\Omega \setminus C} |u(x) - g(x)| dx$ among relatively closed subsets $C$ of $\Omega$ and piecewise-constant functions $u$ on $\Omega \setminus C$, gives rise to a finite decomposition of $\Omega \setminus C$. Here we exhibit a piecewise-constant function $g$ on the unit disk $D$ of $R^2$, with radial symmetry, such that $g\in L^q(\Omega)$ for all $1\leq q < 2$ and the unique minimizer of $F$ has infinitely many components. We also fill a gap occurred in the proof of Proposition 5.2 of [8].

On minimizing partitions with infinitely many components / Leonardi, Gian Paolo; Tamanini, Italo. - In: ANNALI DELL'UNIVERSITÀ DI FERRARA. SEZIONE 7: SCIENZE MATEMATICHE. - ISSN 0430-3202. - STAMPA. - XLIV:(1998), pp. 41-57.

On minimizing partitions with infinitely many components

LEONARDI, Gian Paolo;
1998

Abstract

It is known that when $g \in L^n(\Omega)$ ($\Omega$ open and bounded in $R^n$, with regular boundary $\partial \Omega$), any minimizer of the functional $F(C,u) = H^{n-1}(C) + \lambda \int_{\Omega \setminus C} |u(x) - g(x)| dx$ among relatively closed subsets $C$ of $\Omega$ and piecewise-constant functions $u$ on $\Omega \setminus C$, gives rise to a finite decomposition of $\Omega \setminus C$. Here we exhibit a piecewise-constant function $g$ on the unit disk $D$ of $R^2$, with radial symmetry, such that $g\in L^q(\Omega)$ for all $1\leq q < 2$ and the unique minimizer of $F$ has infinitely many components. We also fill a gap occurred in the proof of Proposition 5.2 of [8].
XLIV
41
57
On minimizing partitions with infinitely many components / Leonardi, Gian Paolo; Tamanini, Italo. - In: ANNALI DELL'UNIVERSITÀ DI FERRARA. SEZIONE 7: SCIENZE MATEMATICHE. - ISSN 0430-3202. - STAMPA. - XLIV:(1998), pp. 41-57.
Leonardi, Gian Paolo; Tamanini, Italo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/454162
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