Final Report Summary - EMBEDDIRICHLET (Embeddings of weighted Sobolev spaces and applications to Dirichlet problems)
The project is concerned with a problem of pure mathematics, more precisely, in the domains of analysis and partial differential equations. The study of elliptic partial differential equations is a topic of current and wide interest, at the interface between functional analysis and partial differential equations. The aspect of the topic that concerns functional analysis is related to the abstract properties of certain suitable Banach spaces, mainly the Sobolev spaces. The other aspects of the subject concern techniques adapted to the study of partial differential equations (variational methods, minimax principles, sub- and supersolution method, etc.). Up to now, these abstract setting and adapted techniques have been mainly developed in the classical situation of an elliptic equation, under certain boundary condition (Dirichlet, Neumann, or other – in this project, we mainly focus on Dirichlet boundary conditions), involving a nondegenerate differential operator like the Laplacian or the p-Laplacian. Then the analytic properties of the classical Sobolev spaces (embedding properties, regularity theory) can be invoked for studying the existence, multiplicity, and qualitative properties of solutions of a given equation.
One purpose of the project is to contribute to the classical theory by proposing new techniques and/or improving existing techniques for the study of certain elliptic partial differential equations. The main concern of the project is to go beyond the classical setting by studying degenerate partial differential equations, involving degenerate differential operators like the weighted p-Laplacian. The project is then motivated by the lack of appropriate abstract foundation for this study. Then, the precise aims of the project are as follows. The first, basic aim is to build an abstract setting on which one can then rely for the concrete study of degenerate partial differential equations. Namely, the first purpose is to define suitable weighted Sobolev spaces satisfying suitable analytic properties, mainly embedding properties, and adapted to the study of elliptic problems involving Dirichlet boundary conditions.
The next aim of the project is to adapt the classical techniques of partial differential equations to the study of degenerate problems. To this purpose, it is necessary to develop new, more adapted techniques, or to improve existing techniques.
This project was carried out in the period February 15, 2012–February 14, 2014 within the Department of Mathematics of the Ben Gurion University of the Negev, under my supervision.
One purpose of the project is to contribute to the classical theory by proposing new techniques and/or improving existing techniques for the study of certain elliptic partial differential equations. The main concern of the project is to go beyond the classical setting by studying degenerate partial differential equations, involving degenerate differential operators like the weighted p-Laplacian. The project is then motivated by the lack of appropriate abstract foundation for this study. Then, the precise aims of the project are as follows. The first, basic aim is to build an abstract setting on which one can then rely for the concrete study of degenerate partial differential equations. Namely, the first purpose is to define suitable weighted Sobolev spaces satisfying suitable analytic properties, mainly embedding properties, and adapted to the study of elliptic problems involving Dirichlet boundary conditions.
The next aim of the project is to adapt the classical techniques of partial differential equations to the study of degenerate problems. To this purpose, it is necessary to develop new, more adapted techniques, or to improve existing techniques.
This project was carried out in the period February 15, 2012–February 14, 2014 within the Department of Mathematics of the Ben Gurion University of the Negev, under my supervision.