Partial Differential Equations

Description

This course introduced the basic properties of separable Hilbert spaces, taking the space of sequences l^2 as the reference example. The properties of the Hilbert cube have been analyzed. The problem of the existence of a basis in an infinite dimensional vector (respectively Banach, and Hilbert) space, in particular the Hamel basis (respectively the Schauder basis and the Hilbert basis), was also discussed. The first basic theorems of linear functional analysis in Banach spaces, such as the Hahn-Banach theorem (analytical and geometric versions), the Banach-Steinhaus theorem, the open mapping theorem, the closed graph theorem and their corollaries, were all studied. An other argument discussed in the course was distribution theory: the space of smooth functions with compact support was introduced, together with the notions of convergence required to introduce a distribution. Various properties and operations on distribution were studied, such as localization, support and convolution. Next, the Fourier transform and its main properties were carefully studied : the Fourier transform in L^1, in the space of rapidly decreasing functions, in L^2 and on tempered distributions. Some properties of Sobolev space (possibly with real exponent) have been analyzed, using mainly the Fourier transform. Applications of the Fourier transform to linear partial differential equations of elliptic, parabolic, and hyperbolic type, have been illustrated in the final part of the course. Several examples and exercises, as well as some counterexamples, have been illustrated during the lectures. -- NOTE: This course was recorded automatically in slots of one hour and processed without human intervention. Lectures are split between videos; their starting time may not coincide with the beginning of videos and intervals were not removed.