# Partial Differential Equations

## Learning outcomes of the course unit

The aim of the course is both to present some classical results for elliptic partial differential equations, and to describe some general ideas (Galerkin finite-dimensional approximation, a priori estimates, perturbative methods) and important elements and abstract harmonic analysis results (weak Lebesgue spaces, Fourier transform, Marcinkiewicz interpolation theorem, Calderón-Zygmund decomposition, singular operators) that see, as a practical application, the results on the PDEs mentioned above. In particular, at the end of the course:

The student must have acquired the knowledge of the linear and nonlinear elliptic equations (of the p-Laplacian type) described in detail in the extended program; he must be able to relate the results presented in the course, highlighting similarities and differences. He will also have to master the abstract results on which the contrite results for PDEs are based. (knowledge and ability to understand)

The student must be able to manipulate the weak formulation of the treated equations to deduce various types of information; apply the described abstract results to problems similar to the practical ones treated in the classroom.

(ability to apply knowledge and understanding)

The student must be able to communicate mathematical contents related to the program in a clear and precise way; he will have to acquire an appropriate scientific lexicon. (communication skills)

The student must be able to evaluate the consistency and correctness of the results obtained by him or others and the relationships / differences between similar results for different differential operators. (autonomy of judgment)

The student will be able to independently deepen his knowledge in the course topics, starting from the basic and fundamental knowledge provided by the course. This ability will be subject to the evaluation test and will require the student to be able to consult specialist texts independently and, possibly with help, research articles, even outside the topics covered in detail during the lessons. (learning ability).

## Prerequisites

The student must be familiar with the differential calculus of Analysis 1 and Analysis 2. Furthermore, some basic concepts of Functional Analysis (convergence in law, weak convergence, separability, reflexivity, etc.), of measure theory (Lebesgue spaces, limit-integral exchange theorems) and of advanced analysis (Sobolev spaces).

## Course contents summary

The course will include basic results for elliptic partial differential equations, of linear and nonlinear type. Both the problems of existence of weak solutions and regularity of them will be treated. For these purposes, important tools and classical results of harmonic analysis will be presented, such as weak Lebesgue spaces, maximal operators, Marcinkiewicz interpolation theorem, theory of singular operators. Particular attention will be paid to the presentation of the results as special cases and practical applications of more general ideas and methods (a priori estimates, perturbative methods, for instance) that can be implemented, possibly, also on other problems. As far as possible, we will try to treat some results both from the linear and thenon-linear point of view, in order to highlight the necessity for substantially different approaches.

## Course contents

Harmonic functions, weak formulation and Weyl Lemma.

Linear equations with variable coefficients: existence via Lax-Milgram (sketch), Schauder theory in the case of Holder data, De Giorgi theory (Holder regularity of solutions for measurable coefficients, sketch), Harnack inequalities and Gehring theory (higher integrability of the gradient).

Linear equations with non-homogeneous variable coefficients: W^{1, q} estimates for continuous coefficients. Marcinkiewicz interpolation theorem.

Laplace equation: estimates for the second derivatives. Calderon-Zygmund decomposition, singular operators.

p-Laplacian equation: existence via monotonicity methods, gradient regularity via a priori estimates, W^{1,q} estimates.

## Recommended readings

L. Ambrosio, A. Carlotto, A. Massaccesi: Lectures on Elliptic Partial Differential Equations, Appunti. Scuola Normale Superiore di Pisa.

M. Giaquinta: Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics, ETH Zürich.

M. Giaquinta, L. Martinazzi: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs (second edition). Appunti. Scuola Normale Superiore di Pisa.

R. E. Showalter: Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs, 49.

E. M. Stein: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press.

E. DiBenedetto: C^{1+\alpha} local regularity of weak solutions of degenerate elliptic equations, Nonlin. Analysis 1983.

## Teaching methods

Lessons will be held at the blackboard, leaving some exercises as home exercises to the students.

## Assessment methods and criteria

Seminar on a chosen topic of the program.

## Other informations

If part of the students has already seen some results proposed in other courses, the teacher reserves the possibility to slightly differentiate the program for this part of the class.