On a class of elliptic operators with unbounded coefficients in convex domains

Da Prato, Giuseppe, and Lunardi, Alessandra. "On a class of elliptic operators with unbounded coefficients in convex domains." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 315-326. <http://eudml.org/doc/252303>.

@article{DaPrato2004,
abstract = {We study the realization $A$ of the operator $\mathcal\{A\} =\frac\{1\}\{2\} \triangle - (DU, D\cdot)$ in $L^\{2\}(\Omega, \mu)$, where $\Omega$ is a possibly unbounded convex open set in $\mathbb\{R\}^\{N\}$, $U$ is a convex unbounded function such that $\lim_\{x \rightarrow \partial \Omega, \, x \in \Omega\} U(x) = + \infty$ and $\lim_\{|x| \rightarrow + \infty, \, x \in \Omega\} U(x) = + \infty$, $DU(x)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu(dx) = c \exp (-2 U(x)) dx$ is a probability measure, infinitesimally invariant for $\mathcal\{A\}$. We show that $A$, with domain $D(A) = \\{u \in H^\{2\}(\Omega,\mu): (DU, Du) \in L^\{2\}(\Omega,\mu)\\}$ is a dissipative self-adjoint operator in $L^\{2\}(\Omega,\mu)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by $A$.},
author = {Da Prato, Giuseppe, Lunardi, Alessandra},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Kolmogorov operators; Unbounded coefficients; Convex domains},
language = {eng},
month = {12},
number = {3-4},
pages = {315-326},
publisher = {Accademia Nazionale dei Lincei},
title = {On a class of elliptic operators with unbounded coefficients in convex domains},
url = {http://eudml.org/doc/252303},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Da Prato, Giuseppe
AU - Lunardi, Alessandra
TI - On a class of elliptic operators with unbounded coefficients in convex domains
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/12//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 3-4
SP - 315
EP - 326
AB - We study the realization $A$ of the operator $\mathcal{A} =\frac{1}{2} \triangle - (DU, D\cdot)$ in $L^{2}(\Omega, \mu)$, where $\Omega$ is a possibly unbounded convex open set in $\mathbb{R}^{N}$, $U$ is a convex unbounded function such that $\lim_{x \rightarrow \partial \Omega, \, x \in \Omega} U(x) = + \infty$ and $\lim_{|x| \rightarrow + \infty, \, x \in \Omega} U(x) = + \infty$, $DU(x)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu(dx) = c \exp (-2 U(x)) dx$ is a probability measure, infinitesimally invariant for $\mathcal{A}$. We show that $A$, with domain $D(A) = \{u \in H^{2}(\Omega,\mu): (DU, Du) \in L^{2}(\Omega,\mu)\}$ is a dissipative self-adjoint operator in $L^{2}(\Omega,\mu)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by $A$.
LA - eng
KW - Kolmogorov operators; Unbounded coefficients; Convex domains
UR - http://eudml.org/doc/252303
ER -