Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary conditions. Consider the semigroup $e^{tL}$, which is $L^2$-contractive by the Hille-Yosida theorem. Is it true that $e^{tL}f \to f$ as $t \to 0$ pointwise almost everywhere?
10
-
$\begingroup$ What are you assuming about $f$? $\endgroup$– Nate EldredgeCommented May 8, 2015 at 15:04
-
$\begingroup$ @NateEldredge Apriori I would want $f \in L^2(\Omega)$, if such a result is possible. $\endgroup$– anonymousCommented May 8, 2015 at 15:09
-
$\begingroup$ If $f \in L^2$ then $f$ is not well defined pointwise, so asking for pointwise convergence doesn't make sense. $\endgroup$– Nate EldredgeCommented May 8, 2015 at 15:28
-
1$\begingroup$ Presumably it should be pointwise almost everywhere. $\endgroup$– Robert IsraelCommented May 8, 2015 at 15:34
-
$\begingroup$ @RobertIsrael Thanks, that's what I had in mind, should have been more precise. $\endgroup$– anonymousCommented May 8, 2015 at 15:48
|
Show 5 more comments