The authors consider the problem $\min F(C,u)$, where the functional $F$ is defined by $F(C,u)=\lambda \int_{\Omega\sbs C}|u-g|^pdx+H^{n-1}(C\cap \Omega)$ on the admissible pairs $(C,u)$ with $C$ closed in $R^n$ and $u$ constant on the (open, connected) components of $\Omega\sbs C$. Here, $\Omega$ denotes an open subset of $R^n$, $\lambda$ and $p$ are positive real numbers with $p\geq 1$, $g\in L^p(\Omega)$, and $H^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure in $R^n$. The following main result is proved. If $\Omega$ is open and bounded in $R^n (n\geq 2)$, with boundary locally of class $C^1$, if $g\in L^{np}(\Omega)$ and $(K,w)$ is a minimizer of $F$, then the family of connected components of $\Omega\sbs K$ is finite.
On the finiteness of optimal partitions
MASSARI, Umberto;
1993
Abstract
The authors consider the problem $\min F(C,u)$, where the functional $F$ is defined by $F(C,u)=\lambda \int_{\Omega\sbs C}|u-g|^pdx+H^{n-1}(C\cap \Omega)$ on the admissible pairs $(C,u)$ with $C$ closed in $R^n$ and $u$ constant on the (open, connected) components of $\Omega\sbs C$. Here, $\Omega$ denotes an open subset of $R^n$, $\lambda$ and $p$ are positive real numbers with $p\geq 1$, $g\in L^p(\Omega)$, and $H^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure in $R^n$. The following main result is proved. If $\Omega$ is open and bounded in $R^n (n\geq 2)$, with boundary locally of class $C^1$, if $g\in L^{np}(\Omega)$ and $(K,w)$ is a minimizer of $F$, then the family of connected components of $\Omega\sbs K$ is finite.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
