This paper gives some answers to the following problem: Minimize $F(C,u)=\lambda\int_{\Omega-C}|u-g|^p\,dx+H^{n-1}(C\cap\Omega)$, where $\Omega$ is an open subset of $\bold R^n$, $\lambda>0$, $p\geq 1$, $g\in L^p(\Omega)\cap L^\infty(\Omega)$, $C$ is closed in $\bold R^n$, $u\in C^1(\Omega-C)$ satisfies in addition $\nabla u=0$ in $\Omega-C$ and $H^{n-1}$ denotes the Hausdorff $(n-1)$-dimensional measure in $\bold R^n$. The main answer: Denote by $(K,w)$ a minimizing pair for the functional $F$ so that $F(K,w)\leq F(C,u)$. Then we have: (1) for every $x\in\Omega$ a suitable $r>0$ exists such that $w$ takes on only a finite number of values in the ball $B_{x,r}-K$; (2) $K\cap\Omega=K_{\operatorname{reg}}\cup K_{\operatorname{sing}}$, where $K_{\operatorname{reg}}$ is a smooth hypersurface of class $C^{1,1/2}$ in $\Omega$ and $K_{\operatorname{sing}}$ is a closed subset with $H^{n-1}(K_{\operatorname{sing}})=0$; (3) there exists a new pair $(K_1,w_1)$ minimizing $F$ which satisfies $K_1\subset K$, $H^{n-1}((K-K_1)\cap\Omega)=0$, $K_1=\overline{K_1\cap\Omega}$, $w_1=w$ in $\Omega-K$ and $\lim_{r\to\infty} r^{1-n}H^{n-1}(K_1\cap B_{x,r})\geq\omega_{n-1}$ for all $x\in K_1\cap\Omega$.'' The roots of the preceding theorem are found in a paper of \n E. De Giorgi, G. Congedo and Tamanini\en [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), no. 4, 673--678].

### Regularity properties of optimal segmentations

#### Abstract

This paper gives some answers to the following problem: Minimize $F(C,u)=\lambda\int_{\Omega-C}|u-g|^p\,dx+H^{n-1}(C\cap\Omega)$, where $\Omega$ is an open subset of $\bold R^n$, $\lambda>0$, $p\geq 1$, $g\in L^p(\Omega)\cap L^\infty(\Omega)$, $C$ is closed in $\bold R^n$, $u\in C^1(\Omega-C)$ satisfies in addition $\nabla u=0$ in $\Omega-C$ and $H^{n-1}$ denotes the Hausdorff $(n-1)$-dimensional measure in $\bold R^n$. The main answer: Denote by $(K,w)$ a minimizing pair for the functional $F$ so that $F(K,w)\leq F(C,u)$. Then we have: (1) for every $x\in\Omega$ a suitable $r>0$ exists such that $w$ takes on only a finite number of values in the ball $B_{x,r}-K$; (2) $K\cap\Omega=K_{\operatorname{reg}}\cup K_{\operatorname{sing}}$, where $K_{\operatorname{reg}}$ is a smooth hypersurface of class $C^{1,1/2}$ in $\Omega$ and $K_{\operatorname{sing}}$ is a closed subset with $H^{n-1}(K_{\operatorname{sing}})=0$; (3) there exists a new pair $(K_1,w_1)$ minimizing $F$ which satisfies $K_1\subset K$, $H^{n-1}((K-K_1)\cap\Omega)=0$, $K_1=\overline{K_1\cap\Omega}$, $w_1=w$ in $\Omega-K$ and $\lim_{r\to\infty} r^{1-n}H^{n-1}(K_1\cap B_{x,r})\geq\omega_{n-1}$ for all $x\in K_1\cap\Omega$.'' The roots of the preceding theorem are found in a paper of \n E. De Giorgi, G. Congedo and Tamanini\en [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), no. 4, 673--678].
##### Scheda breve Scheda completa Scheda completa (DC)
1991
Massari, Umberto; Tamanini, I.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/534447
##### Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

• ND
• 19
• 13