Poincaré inequalities on domains and elliptic PDE

DATES: 25 - 29 November 2019 (5 sessions)
TIME: 09:00 - 11:00 (a total of 10 hours)
LOCATION: BCAM Seminar room

ABSTRACT:
Poincaré inequalities transfer infinitesimal information encoded in the derivative to larger scales. They are among the most important tools in the theory of partial differential equations. The aim of the 10 hour minicourse is to illustrate this general idea with concrete applications to the p-Laplace equation, for finite p > 1, which is a nonlinear generalisation of the classical Laplace equation. The p-Laplace equation arises naturally when we consider minimisers of a p-energy functional with prescribed Sobolev boundary values in a given open set. Solutions of p-Laplace equation have various stability properties: we will show higher integrability for gradients of solutions. Poincaré inequalities and harmonic analytical techniques play a key role in the proof. Lectures are aimed at advanced undergraduate and graduate students, and they are based on a forthcoming joint book of the lecturer with J. Kinnunen and J. Lehrbäck. Other references are given below.

PROGRAMME:
- Poincaré inequalities on Euclidean cubes and domains
- Reverse Hölder inequalities and their improvement
- p-Laplace equation and weak solutions on Euclidean open sets - Energy estimates for weak solutions
- Higher integrability for gradients of weak solutions
- Other stability results


PREREQUISITES: Real analysis.

REFERENCES:
1. Hajlasz, Piotr & Koskela, Pekka. (2000). Sobolev Met Poincaré. Mem. Amer. Math. Soc. 145.
2. Jost, Jürgen. (2002). Partial Differential Equations. Graduate Texts in Mathematics, 214. Springer- Verlag.
3. Kinnunen, Juha. (1994). Higher Integrability With Weights. Ann. Acad. Sci. Fenn. Ser. A I Math.
4. Kilpeläinen, Tero & Heinonen, Juha & Martio, Olli. (2006). Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover publications.
5. Lindqvist, Peter. (2007). Notes on the p-Laplace equation (second edition). Report. University of Jyväskylä Department of Mathematics and Statistics. 161.


*Registration is free, but mandatory before November 21st.
To sign-up go to https://forms.gle/hCyGLkAWf9fXhQ8o9 and fill the registration form.

Student grants are available. Please, let us know if you need support for travel and accommodation expenses in the previous form before October 27th.

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