- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- BIRS Workshop Lecture Videos /
- The non-local mean-field equation on an interval.
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
The non-local mean-field equation on an interval. De la Torre, Azahara
Description
We study the quantization properties for a non-local mean-field equation and give a necessary and sufficient condition for the existence of solution for a ``Mean Field''-type equation in an interval with Dirichlet-type boundary condition. We restrict the study to the $1$-dimensional case and consider the fractional mean-field equation on the interval $I = (-1, 1)$ $$(-\Delta)^\frac{1}{2} u=\rho \frac{e^{u}}{\int_I e^{u}dx},$$ subject to Dirichlet boundary conditions. As in the $2-$dimensional case, it is expected that for a sequence of solutions to our equation, either we get a $\mathcal{C}^{\infty}$ limiting solution or, after a suitable rescaling, we obtain convergence to the Liouville equation. Then, we can reduce the problem to the study of the non-local Liouville's equation. One of the key points here is to find an appropriate Pohozaev identity. We prove that existence holds if and only if $\rho < 2\pi$. This requires the study of blowing-up sequences of solutions. In particular, we provide a series of tools which can be used (and extended) to higher-order mean field equations of non-local type. We provide a completely non-local method for this study, since we do not use the localization through the extension method. Instead, we use the study of blowing-up sequences of solutions. This is a work done in collaboration with with A. Hyder, Y. Sire and L. Martinazzi.
Item Metadata
Title |
The non-local mean-field equation on an interval.
|
Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
|
Date Issued |
2019-05-08T14:46
|
Description |
We study the quantization properties for a non-local mean-field equation and give a necessary and sufficient condition for the existence of solution for a ``Mean Field''-type equation in an interval with Dirichlet-type boundary condition. We restrict the study to the $1$-dimensional case and consider the fractional mean-field equation on the interval $I = (-1, 1)$
$$(-\Delta)^\frac{1}{2} u=\rho \frac{e^{u}}{\int_I e^{u}dx},$$
subject to Dirichlet boundary conditions. As in the $2-$dimensional case, it is expected that for a sequence of solutions to our equation, either we get a $\mathcal{C}^{\infty}$ limiting solution or, after a suitable rescaling, we obtain convergence to the Liouville equation. Then, we can reduce the problem to the study of the non-local Liouville's equation. One of the key points here is to find an appropriate Pohozaev identity.
We prove that existence holds if and only if $\rho < 2\pi$. This requires the study of blowing-up sequences of solutions. In particular, we provide a series of tools which can be used (and extended) to higher-order mean field equations of non-local type.
We provide a completely non-local method for this study, since we do not use the localization through the extension method. Instead, we use the study of blowing-up sequences of solutions.
This is a work done in collaboration with with A. Hyder, Y. Sire and L. Martinazzi.
|
Extent |
33.0 minutes
|
Subject | |
Type | |
File Format |
video/mp4
|
Language |
eng
|
Notes |
Author affiliation: University of Freiburg
|
Series | |
Date Available |
2019-11-05
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0384926
|
URI | |
Affiliation | |
Peer Review Status |
Unreviewed
|
Scholarly Level |
Postdoctoral
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International