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).