To acquire a good knowledge of the general methods andÿclassical techniques necessary for the study ofÿweak solutions of partial differential equations.
Curriculum
teacher profile teaching materials
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.
Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle
Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle
Programme
Preliminaries- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.
Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle
Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle
Core Documentation
Functional analysis, H. Bre'zisType of delivery of the course
The course plans lectures.Attendance
Attendance is not required but strongly suggested.Type of evaluation
Seminar on a research paper. teacher profile teaching materials
Lawrence C. Evans - Partial Differential Equations. AMS
Programme
Definition and elementary properties of the Sobolev spaces. Extensions theorem. Sobolev inequalities. Trace operator, compactness. Duality. Fourier transform method. Second order elliptic equations: existence of weak solutions. Regularity:interior/boundary. Maximum principles. Arguments of evolutions problems: the wave equation.Core Documentation
Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.Lawrence C. Evans - Partial Differential Equations. AMS
Reference Bibliography
Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer. Lawrence C. Evans - Partial Differential Equations. AMS. R. Adams - Sobolev Spaces.Type of delivery of the course
Lectures on theory and exercises will be held.Type of evaluation
An oral test will also be held to evaluate the knowledge of the course topics. teacher profile teaching materials
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.
Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle
Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle
Mutuazione: 20410518 AM420 - SPAZI DI SOBOLEV ED EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM-40 HAUS EMANUELE, FEOLA ROBERTO
Programme
Preliminaries- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.
Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle
Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle
Core Documentation
Functional analysis, H. Bre'zisReference Bibliography
Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer. Lawrence C. Evans - Partial Differential Equations. AMS. R. Adams - Sobolev Spaces.Type of delivery of the course
The course plans lectures.Attendance
Attendance is not required but strongly suggested.Type of evaluation
Seminar on a research paper. teacher profile teaching materials
Lawrence C. Evans - Partial Differential Equations. AMS
Mutuazione: 20410518 AM420 - SPAZI DI SOBOLEV ED EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM-40 HAUS EMANUELE, FEOLA ROBERTO
Programme
Definition and elementary properties of the Sobolev spaces. Extensions theorem. Sobolev inequalities. Trace operator, compactness. Duality. Fourier transform method. Second order elliptic equations: existence of weak solutions. Regularity:interior/boundary. Maximum principles. Arguments of evolutions problems: the wave equation.Core Documentation
Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.Lawrence C. Evans - Partial Differential Equations. AMS
Reference Bibliography
Haim Breziz - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer. Lawrence C. Evans - Partial Differential Equations. AMS. R. Adams - Sobolev Spaces.Type of delivery of the course
Lectures on theory and exercises will be held.Type of evaluation
An oral test will also be held to evaluate the knowledge of the course topics.