Partial Differential Equations
Code
11634
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Oleksiy Karlovych
Weekly hours
6
Teaching language
Português
Objectives
At the end of this course students will have acquired basic knowledge and skills in the area of partial differential equations of in order to:
- Understand advanced contents in the area;
- Being able to start research on a topic in the area.
Subject matter
1. Regular and singular distributions. Distributional derivative. Spaces L1loc(W), Lp(W), Hm(W), Hm0(W). The Hahn-Banach theorem.
2. Poincaré and Korn inequalities. Lipschitz domains. Boundary values and normal derivative. Regularity and variational formulation. The Lax-Milgram theorem.
3. Spaces H½ (∂Ω), Hk-1/2(∂Ω), H-k+1/2(∂Ω). Existence, uniquiness and regularity of the solution of the non-homogeneous Dirichlet problem.
4. Spaces H-m(Ω). Green’s formulas. Space H∆( Ω). Generalized Green’s formulas.
5. Elliptic, parabolic and hyperbolic partial differential equations.
6. Semigrups theory (the Yosida-Phillips theorem).
Bibliography
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.
L.C. Evans, Partial Differential Equations, American Mathematical Society, 2nd edition, 2010.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2011.
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985.
J.-L. Lions and E. Magenes, Nonhomogeneous boundary value problems and Applications, vol.I a III. Springer, 1971-1973.