Type: Seminar

Modified: Thu 6 May 2021 1143

Distribution: World

Expiry: 10 May 2021

CalTitle1: Blow-up of solutions of critical elliptic equations in three dimensions

Auth: dhauer@73.70.50.210.sta.wbroadband.net.au (dhauer) in SMS-SAML

Dear friends and colleagues,

on **Monday, 10 May 2021 **at** 1 PM**,
** Professor Rupert Frank ** (Caltech University, United States, and
@ Ludwig Maximilian University of Munich, Germany) is giving a talk in our
Asia-Pacific Analysis and PDE Seminar on

**
Blow-up of solutions of critical elliptic equations in three dimensions
**.

**Abstract:**

We describe the asymptotic behavior of positive solutions \(u_\varepsilon\) of the equation \[-\Delta u + au = 3\,u^{5-\varepsilon}\qquad \textrm{in \(\Omega\subseteq\,\mathbb{R}^3\)}\] with a homogeneous Dirichlet boundary condition. The function \(a\) is assumed to be critical in the sense of Hebey and Vaugon and the functions \(u_\varepsilon\) are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation \[-\Delta u + (a+\varepsilon V) u = 3\,u^5\qquad\textrm{ in \(\Omega\).}\] For the variational problem corresponding to the latter problem we also obtain precise energy asymptotics and a detailed description of the blow-up behavior of almost minimizers (but not necessarily minimizers or solutions).

More information and how to attend this talk can be found at the seminar webpage .

Best wishes,

Daniel

------

Rupert Frank

Professor @ Caltech University, United States, and @ Ludwig Maximilian University of Munich, Germany.

He defended his PhD thesis in 2007 @ the Royal Institute of Technology in Stockholm under the supervision
of Ari Laptev. After a post-doctoral internship and assistant professorship @ Princeton, in 2013 he
became professor @ Caltech and in 2016 at LMU Munich.