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EnglishLet $A=\sum_{i,j=1}^Na_{ij}(x)D_{ij}+\sum_{i=1}^Nb_i(x)D_i$ be an elliptic differential operator with unbounded coefficients on $\R^N$ and assume that the associated Feller semigroup $(T(t))_{t\geq 0}$ has an invariant measure $\mu$. Then $(T(t))_{t\geq 0}$ extends to a strongly continuous semigroup $(T_p(t))_{t\geq 0}$ on $L^p(\mu)=L^p(\R^N,\mu)$ for every $1\leq p\leq \infty$. We prove that, under mild conditions on the coefficients of $A$, the space of test functions $C_c^\infty(\R^N)$ is a core for the generator $(A_p,D_p)$ of $(T_p(t))_{t\geq 0}$ in $L^p(\mu)$ for $1\leq p<\infty$.
| Titolo: | Cores for Feller semigroups with an invariant measure | |
| Autori: | ||
| Data di pubblicazione: | 2006 | |
| Rivista: | ||
| Abstract: | Let $A=\sum_{i,j=1}^Na_{ij}(x)D_{ij}+\sum_{i=1}^Nb_i(x)D_i$ be an elliptic differential operator with unbounded coefficients on $\R^N$ and assume that the associated Feller semigroup $(T(t))_{t\geq 0}$ has an invariant measure $\mu$. Then $(T(t))_{t\geq 0}$ extends to a strongly continuous semigroup $(T_p(t))_{t\geq 0}$ on $L^p(\mu)=L^p(\R^N,\mu)$ for every $1\leq p\leq \infty$. We prove that, under mild conditions on the coefficients of $A$, the space of test functions $C_c^\infty(\R^N)$ is a core for the generator $(A_p,D_p)$ of $(T_p(t))_{t\geq 0}$ in $L^p(\mu)$ for $1\leq p<\infty$. | |
| Handle: | http://hdl.handle.net/11587/102082 | |
| Appare nelle tipologie: | Articolo pubblicato su Rivista |
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