The speaker: Professor Alexander Nazarov (St. Petersburg Department of Steklov Institute of Mathematics (POMI) and St. Petersburg State University, Russia)
Title of the talk: “The Hopf-Oleinik Lemma for the divergence-type equations”
Time: Thursday, 27 of January, 15:00 - 16:00
Abstract: The Hopf-Oleinik lemma, known also as the “normal derivative lemma”, is one of the important tools in qualitative analysis of partial differential equations.This lemma states that a supersolution of a partial differential equation with a minimum value at a boundary point, must increase linearly away from its boundary minimum provided the boundary is smooth enough. A major part of all known results on the normal derivative lemma concerns equations with nondivergence structure and strong solutions (see [1] and [2] for some recent results and the comprehensive historical review).
The case of the divergence-type equations is less studied. It is well known that the normal derivative lemma fails for uniformly elliptic equations in divergence form with bounded and even continuous leading coefficients. Thus, one has to require more smoothness of the leading coefficients.
For the parabolic divergence-type equations, the normal derivative lemma can be also extracted from the lower bound estimates of the Green function for the corresponding operator.
We present a version of the Hopf-Oleinik lemma for general elliptic and parabolic equations in divergence form under the sharp requirements on the coefficients of equations and on the boundary of a domain. All our assumptions are significantly weakened in comparison with the previous works. In fact, our requirements are close to the necessary ones.
The talk is based on the paper [3].
References
[1] A.I. Nazarov, A centennial of the Zaremba-Hopf-Oleinik lemma, SIAM J. Math. Anal. 44(2012), no. 1, 437–453.
[2] D.E. Apushkinskaya, A.I. Nazarov, A counterexample to the Hopf-Oleinik lemma (elliptic case), Anal. PDE 9(2016), no. 2, 439–458.
[3] D.E. Apushkinskaya, A.I. Nazarov, On the Boundary Point Principle fordivergence-type equations, Rend. Lincei Mat. Appl. 30(2019), 677–699.
ZOOM link:
https://us02web.zoom.us/j/83673138944?pwd=azQyc1BwUjJFY3VjMStLbFR5Z2VwUT09
ZOOM Id: 836 7313 8944
Passcode: 808818
We are pleased to inform you that the information on previous talks is available on the seminar web page
https://researchseminars.org/seminar/Cafe_Analysis_and_Fluid
see PastTalks section
Organizing Committee: Professors Zhen Lei, Mikhail Korobkov, Adele Ferone, Remigio Russo, and Konstantin Pileckas
The seminar is organized by Fudan University (Shanghai, China) with the collaboration of Universita degli studi della Campania "Luigi Vanvitelli" (Caserta, Italy) and Vilnius University (Vilnius, Lithuania). This seminar is meant to gather renowned experts working on Geometric and Real Analysis with applications to PDEs and Mathematical Physics and represents an opportunity for presenting high-impact research, as well as facilitating the exchange of ideas between researchers.
2022-01-24