Timeline for Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds

7 events

when toggle format	what		by	license	comment
Jun 2, 2020 at 22:20	comment	added	Ramiro Lafuente		That’s very helpful, thanks!
Jun 2, 2020 at 10:16	comment	added	Jochen Glueck		[continuation] (v) Apply the Krein-Rutman theorem to $L^{-1}$ in order to see that its spectral radius is in the spectrum and that it has a positive eigenvector. (vi) Use the spectral mapping theorem for resolvents to get back to the operator $L$.
Jun 2, 2020 at 10:16	comment	added	Jochen Glueck		Here's an outline of the general Krein-Rutman based strategy that was suggested by @leomonsaingeon: (i) Choose a space to work on - for instance, $L^2$ over the manifold. (ii) Show that all spectral value of $L$ have real part $\ge \varepsilon$ for some $\varepsilon > 0$. (iii) Show compactness of the resolvent (for instance by showing that the domain of $L$ embeds compactly into $L^2$) (iv) Use the maximum principle to show that $L^{-1}$ a positive operator. [to be continued]
Jun 2, 2020 at 9:10	comment	added	leo monsaingeon		Oooops, indeed I meant Krein-Rutman, sorry. I don't have any reference for manifolds, but I'm pretty sure the standard proof should carry through
Jun 2, 2020 at 9:08	comment	added	Ramiro Lafuente		Do you mean the Krein-Rutman theorem perhaps? do you know a reference for the above result in the closed manifolds case?
Jun 2, 2020 at 8:35	comment	added	leo monsaingeon		Yes, this is a consequence of the Krein-Milman theorem.
Jun 2, 2020 at 5:42	history	asked	Ramiro Lafuente	CC BY-SA 4.0