Menù principale
B018755 - HIGHER ANALYSIS
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Further information
Type of Assessment
Course program
Academic Year 2023-24
Coorte 2023 - Second Cycle Degree in MATHEMATICS
Course year
First year - Second Semester
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/05 - MATHEMATICAL ANALYSIS
Credits
9
Teaching Hours
72
Teaching Term
19/02/2024 ⇒ 14/06/2024
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Mutuality
Course teached as:
B018755 - ANALISI SUPERIORE
Second Cycle Degree in MATHEMATICS
Curriculum APPLICATIVO
B018755 - ANALISI SUPERIORE
Second Cycle Degree in MATHEMATICS
Curriculum APPLICATIVO
Teaching Language
Italian
Course Content
- Some results and methods for the study of the stability for inverse problems. Boundary regularity.
Carleman estimates unique continuation properties
- Construction of topological degree theory in Euclidean spaces, with applications to some problems of differential equattions. Axiomatic approach, extensions of the theory to more general spaces.
Carleman estimates unique continuation properties
- Construction of topological degree theory in Euclidean spaces, with applications to some problems of differential equattions. Axiomatic approach, extensions of the theory to more general spaces.
Suggested readings (Search our library's catalogue)
- L.C. Evans, Partial differential equations, Second edition.
Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp.
- F. John, Partial differential equations. Reprint of the fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1991. x+249 pp.
- G. Dinca, J. Mawhin. Brouwer Degree: The Core of Nonlinear Analysis. Birkhauser 2021
- N. G. Lloyd. Degree Theory. Cambridge tracts in Mathematics n. 73
In addition: notes by the instructors; online materials (freely available) suggested by the instructors
Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp.
- F. John, Partial differential equations. Reprint of the fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1991. x+249 pp.
- G. Dinca, J. Mawhin. Brouwer Degree: The Core of Nonlinear Analysis. Birkhauser 2021
- N. G. Lloyd. Degree Theory. Cambridge tracts in Mathematics n. 73
In addition: notes by the instructors; online materials (freely available) suggested by the instructors
Learning Objectives
Knowledge objectives: acquisition of knowledge of the principal topics and questions of the course (see "Extended program")
Competence objectives: understanding of language and methods for the study of inverse problems and of unique continuation properties for partial differential equations.
Understanding the foundations of the topological degree theory.
Skills acquired at the end of the course: ability to understand and/or formalize simple inverse and control problems, to propose methods for their resolution, to interpret the relative results. Ability to use the topological degree as a tool to investigate problems in differential equation theory. Being able to access the scientific literature for in-depth study.
Competence objectives: understanding of language and methods for the study of inverse problems and of unique continuation properties for partial differential equations.
Understanding the foundations of the topological degree theory.
Skills acquired at the end of the course: ability to understand and/or formalize simple inverse and control problems, to propose methods for their resolution, to interpret the relative results. Ability to use the topological degree as a tool to investigate problems in differential equation theory. Being able to access the scientific literature for in-depth study.
Prerequisites
Fundamentals of the theory of Ordinary Differential Equations. Functional Analysis. Normed spaces and continuous linear maps. Banach spaces. Hilbert spaces. Differential calculus in R^n. Theory of Lebesgue measure. L^p spaces. Holder spaces. Convolution. Holomorphic functions
Teaching Methods
Lectures and discussion of exercises/problems in the classroom. Seminar activity on material provided by the instructors
Further information
9 CFU, that means: 225 hours (students' total workload), 72 hours (classes)
Type of Assessment
Oral examination: more specifically, a seminar on a topic agreed with each and every student
Course program
1. EXAMPLES OF INVERSE PROBLEMS. Tomography. Backward problem for the heat equation. The inverse conductivity problem: formulation of the problem. The Dirichlet-to-Neumann map: definition, its main properties.
2. SOBOLEV SPACES AND BOUNDARY VALUE PROBLEMS. Definition of Sobolev spaces with integer exponent. Density Theorems. The dual space of H_0^1. Notes about Sobolev spaces with noninteger exponent. Sobolev spaces and Fourier trasform. Trace. Embedding Sobolev Theorems. Morrey inequality. Compactness theorems. Variational formulation of some boundary value problems for second order elliptic partial differential equations. Dirichlet and Neumann problem. Lax - Milgram Theorem. L^2 regularity theory for weak solutions: (i) interior regularity, (ii) boundary regularity for Dirichlet boundary value problem. Dirichlet to Neumann map. Inclusion inverse problem.
3. UNIQUE CONTINUATION PROPERTIES AND CAUCHY PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS.
a) Equations in the analytical field. Cauchy Kovalevski Theorem. Holmgren Theorem.
b) Hadamard’s definition of well-posed problem
c) Uniqueness and continuous dependence of solutions to equations with nonanalytic coefficients. Stability estimates for elliptic equations of second order. Introduction to the method of Carleman estimates. Three sphere inequality.
2. THE CAUCHY PROBLEM FOR PDE.
a) Analytic solutions. The Cauchy-Kovalevskaya theorem. Holmgren's theorem.
b) Well-posed problems in the sense of Hadamard.
c) Uniqueness and continuous dependence for equations with non-analytic coefficients. Stability estimates for second order elliptic equations. Introduction to the method of Carleman estimates. Three spheres inequality.
4. Construction of topological degree (Brouwer) in Euclidean spaces through the differential approach. Fundamental properties. Alternate constructions based on volume integration and on surface integration. Extensions of topological degree theory (degree of maps between differentiable manifolds, fixed point index, degree of tangent vector fields, Leray-Schauder degree).
5. Applications of degree theory to boundary value problems for differential equations.
2. SOBOLEV SPACES AND BOUNDARY VALUE PROBLEMS. Definition of Sobolev spaces with integer exponent. Density Theorems. The dual space of H_0^1. Notes about Sobolev spaces with noninteger exponent. Sobolev spaces and Fourier trasform. Trace. Embedding Sobolev Theorems. Morrey inequality. Compactness theorems. Variational formulation of some boundary value problems for second order elliptic partial differential equations. Dirichlet and Neumann problem. Lax - Milgram Theorem. L^2 regularity theory for weak solutions: (i) interior regularity, (ii) boundary regularity for Dirichlet boundary value problem. Dirichlet to Neumann map. Inclusion inverse problem.
3. UNIQUE CONTINUATION PROPERTIES AND CAUCHY PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS.
a) Equations in the analytical field. Cauchy Kovalevski Theorem. Holmgren Theorem.
b) Hadamard’s definition of well-posed problem
c) Uniqueness and continuous dependence of solutions to equations with nonanalytic coefficients. Stability estimates for elliptic equations of second order. Introduction to the method of Carleman estimates. Three sphere inequality.
2. THE CAUCHY PROBLEM FOR PDE.
a) Analytic solutions. The Cauchy-Kovalevskaya theorem. Holmgren's theorem.
b) Well-posed problems in the sense of Hadamard.
c) Uniqueness and continuous dependence for equations with non-analytic coefficients. Stability estimates for second order elliptic equations. Introduction to the method of Carleman estimates. Three spheres inequality.
4. Construction of topological degree (Brouwer) in Euclidean spaces through the differential approach. Fundamental properties. Alternate constructions based on volume integration and on surface integration. Extensions of topological degree theory (degree of maps between differentiable manifolds, fixed point index, degree of tangent vector fields, Leray-Schauder degree).
5. Applications of degree theory to boundary value problems for differential equations.