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Moritz Egert

   Moritz Egert


Équipe Analyse Harmonique


Mathématiques
Bât. 425
Université Paris-Sud
91405 Orsay Cedex
France


Email:
Office : 232
Phone : (+33) 1 69 15 57 55


Research interests

I am interested in the harmonic analysis and operator theory of differential operators, in particular elliptic and parabolic systems in divergence form.

In my PhD Thesis, supervised by Robert Haller-Dintelmann at TU Darmstadt, I solved the Kato Square Root Problem for systems with mixed Dirichlet/Neumann boundary conditions posed on rough domains beyond the Lipschitz class. Besides harmonic analysis and operator theory, this required to bring into play a third, exciting flavor: geometric measure theory.

More recently, I worked on classical elliptic and parabolic boundary value problems on the upper half space and on non-autonomous maximal regularity questions.

Other research interests of mine lie in potential theory, in particular trace and extension theorems for Sobolev functions and Hardy's inequality. I have also worked on numerical approximation schemes for strongly continuous semigroups. My full scientific CV can be found here (in French).


Publications

1. Non-local self-improving properties: A functional analytic approach. with P. Auscher and S. Bortz and O. Saari, 18 pages, submitted 2017.

2. Non-local Gehring lemmas. with P. Auscher and S. Bortz and O. Saari, 46 pages, submitted 2017.

3. On regularity of weak solutions to parabolic systems. with P. Auscher and S. Bortz and O. Saari, 19 pages, submitted 2017.

4. On uniqueness results for Dirichlet problems of elliptic systems without DeGiorgi-Nash-Moser regularity. with P. Auscher, 22 pages, submitted 2017.

5. The Dirichlet problem for second order parabolic operators in divergence form. with P. Auscher and K. Nyström, 26 pages, submitted 2016.

6. L2 well-posedness of boundary value problems for parabolic systems with measurable coefficients, with P. Auscher and K. Nyström, 83 pages, submitted 2017.

7. Characterizations of Sobolev functions that vanish on a part of the boundary, with P. Tolksdorf, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 4, 729-743.

8. On non-autonomous maximal regularity for elliptic operators in divergence form, with P. Auscher, Arch. Math. 107 (2016), no. 3, 271–284.

9. Mixed boundary value problems on cylindrical domains, with P. Auscher, Adv. Differential Equ. 22 (2017), no.~1/2, 101-168

10. Hardy's inequality for functions vanishing on a part of the boundary, with Haller-Dintelmann and J. Rehberg, Potential Anal. 43 (2015), no.1, 49-78.

11. The Kato Square Root Problem for Mixed Boundary Conditions, with R. Haller-Dintelmann and P. Tolksdorf, J. Funct. Anal. 267 (2014), no.5, 1419-1461.

12. The Kato Square Root Problem follows from an Extrapolation Property of the Laplacian, with R. Haller-Dintelmann and P. Tolksdorf, Publ. Math. 61 (2016), no. 2, 451-483.

13. Convergence of subdiagonal Padé approximations to C0-semigroups, with J. Rozendaal, J. Evol. Equ. 13 (2013), no.4, 875-895.


Thesis

On Kato's conjecture and mixed boundary conditions, PhD thesis, Sierke Verlag, Göttingen, 2015, ISBN: 978-3-86844-719-4. (Ask me for a copy, if you are interested!)


Teaching experience

I worked as an undergraduate teaching assistant for several analysis courses at TU Darmstadt permanently between 2008 and 2011 and as a graduate teaching assistant between 2012 and 2014.

I have been a coordinator for projects of the Internet Seminar on Evolution Equations:  In 2014 I have offered the project The Dirichlet Laplacian as generator on spaces of continuous functions and in 2015 the project Fractional powers and Kato's conjecture (both jointly with Robert Haller-Dintelmann).


Invited talks

1. Équations paraboliques non-autonomes par un système de Cauchy-Riemann, Le séminaire d’EDP-physique mathématique, Université Bordeaux, 9 may 2017
2. Cauchy-Riemann system for non-autonomous parabolic PDEs, Operator semigroups in analysis: modern developments, Bedlewo, Pologne, 24 avril
3. Systèmes de Cauchy-Riemann et EDPs du second ordre,Séminaire EDP-Analyse Lyon, Université Lyon 1, 4 april 2017
4. Cauchy-Riemann systems for second order partial differential equations, Postdoc seminar, MSRI Berkeley, 14 march 2017
5. Non-autonomous maximal regularity for elliptic operators in divergence form, Journée d'Analyse Non Linéaire, Université Paris-Sud, 6 June 2016
6. Non-autonomous maximal regularity for divergence-form operators, Analysis Seminar, Birmingham School of Mathematics, 21 March 2016
7. Hardy's inequality for functions vanishing on a part of the boundary, Analysis and Stochastics Seminar, Uppsala Universitet, 16 February 2016
8. On Hardy's inequality for functions with partially vanishing trace, Harmonic Analysis Seminar, Université Paris-Sud, 19 January 2016
9. On Kato's Square Root Problem, WIAS Berlin, 11 February 2015
10. On an elliptic mixed boundary value problem, TU Delft, 13 November 2014
11. Square Roots of Elliptic Systems, Harmonic Analysis Seminar of the Louisiana State University, Baton Rouge, USA, 3 October 2012


Contributed talks

1. Mixed boundary value problems on cylindrical domains, PDE 2015, WIAS Berlin, 2 December 2015
2. Polynomial Decay of Bounded C0-semigroups on Hilbert Spaces, Final workshop of the 16th Internet Seminar on Evolution Equations, Blaubeuren, Germany, 10 June 2013
3. The Kato Square Root Problem for Mixed Boundary Conditions, Symposium "Operator Semigroups meet Complex Analysis, Harmonic Analysis and Mathematical Physics", Herrnhut, Germany, 3 June 2013
4. Rational Approximations of Semigroups without Scaling and Squaring, Final workshop of the 15th Internet Seminar on Evolution Equations, Blaubeuren, Germany, 8 June 2012
5. Feedback Stabilizability of Delay Systems in Banach Spaces, Final workshop of the 14th Internet Seminar on Evolution Equations, Blaubeuren, Germany, 10 June 2011
6. Barenblatt's solution to the Porous Medium Equation, Final workshop of the 13th Internet Seminar on Evolution Equations, Kacov, Czech Republic, 18 June 2010


Things of interest

Follow my "career" as a racing cyclist here (in German).

The current edition of the Internet Seminar on Evolution Equations deals with parabolic operators with bounded and unbounded coefficients.

Here, it only takes you a minute to write a mathematical paper (Claim: 5 is less than 7. Proof: One direction is obvious. So, let us prove the converse...)