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
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Webinar Speaker
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