Partial Differential Equations II (Summer 2017)
Prof. Phan Thanh Nam, Dr. Sergey Morozov, Hongshuo Chen
Lecture Notes by Martin Peev
Homework Sheets
General Information
Goal: We study the existence and regularity of weak solutions in Sobolev spaces. We will focus on second order elliptic equations and time-dependent Schrödinger equations.Audience: Master students of Mathematics and Physics, TMP-Master.
Time and place:
- Lectures: Tuesday, Thursday 8-10 in B132
- Exercises: Monday 16-18 in B132
- Tutorials: Friday 12-14 in B251
References: Some excellent textbooks are
- Lieb-Loss: Analysis, Amer. Math. Soc. 2001
- Evans: Partial Differential Equations, Amer. Math. Soc. 2010
- Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext 2011
Exercises and Turotials: Every week a homework sheet will be posted here. You can handle your solutions to Dr. Morozov in the exercise section (Monday) or bring them to Hongshuo Chen (corrector) at his office (B403). You are strongly encouraged to do the homework as it is the best way to prepare for the final exam.
The tutorial section (Friday) is provided to answer your questions and to discuss complementary materials.
Grade: Final grade is determined by your performance on homework exercises, midterm exam and final exam.
- You can get up to 100 points from the final exam.
- You can get a bonus of 10 points if you solve at least half of problems in homework sheets.
- You can get a bonus of 10 points if you solve at least half of problems in midterm exam.
Exams: You can bring your notes, including the homework sheets and their solutions. Electronic devices are not allowed. You can use all results stated in the lectures and the homework sheets. Please bring your identity and your student card.
Brief contents of lectures
25/4/2017: Chapter 1: Measure space. Measurable functions. Lebesgue integral. Dominated convergence. Fatou's lemma and the Brezis-Lieb refinement.27/4: Theorem of Fubini-Tonelli. L^p spaces. Hölder's and Minkowsky's inequalities. Completeness of L^p. Convolution. Young's inequality. Density of smooth functions of compact support in L^p. Separability of L^p. Bounded linear functionals on L^p.
2/5: Uniqueness of weak limits. Hanner's inequality. Uniform convexity of L^p. Lower semicontinuity of norms. Uniform boundedness principle. The dual to L^p. Banach-Alaoglu theorem.
4/5: Chapter 2: Test functions and distributions. Locally integrable functions are determined by distributions (Fundamental lemma of calculus of variations). Derivatives of distributions. Equivalence of classical and distributional derivatives. Distributions with zero gradient are constants.
9/5: Chapter 3: Fourier transform. Fourier transform of differential operators and convolutions. Plancherel identity. Inverse formula. Hausdorff-Young inequality. Riesz-Thorin interpolation Theorem. Fourier transform of Gaussian. Heat equation. Heat kernel.
11/5: Fourier transform of 1/|x|^s. Poisson's equation. Green function of Laplacian. Yukawa potential. Chapter 4: Sobolev spaces H^m(R^d). Completeness of H^m. Test functions are dense in H^m.
16/5: Chain rule. Derivative of the absolute value. Diamagnetic inequality. Fourier characterization of H^m(R^d). Chapter 5: Sobolev inequalities. Uncertainty principle. Scaling argument. Sobolev inequality for gradient.
18/5: Sobolev inequalities in low dimensions. Sobolev continuous embedding. Sobolev compact embedding in bounded sets. Weak convergence in H^1 implies pointwise convergence.
23/5: Sobolev inequatilies for W^{m,p}(R^d). Chapter 6: Ground states for Schrödinger operators. Minimizers are weak solutions to eigenvalue equations. Existence of minimizers with trapping potentials.
30/5: Stability with potentials V in L^p+L^infty. Existence of minimizers with potentials vanishing at infinity. Perron-Frobenius principle. Explicit ground state of hydrogen atom.
1/6: Relation between hydrogen inequality and Heisenberg principle. Hardy's inequality. Chapter 7: Harmonic functions. Mean-value theorem. Harmonic functions are smooth. Harnack's inequality. Semi-bounded harmonic functions on R^d are constants. Newton's theorem (statement).
8/6: Newton's theorem. Positive distributions. Sub- and super-harmonic functions. Mean-value inequality. Strong maximum principles. Lower bound for (-\Delta +m^2)f >= 0.
13/6: Uniqueness of Schrödinger minimizer. Chapter 8: Smoothness of weak solutions. Regularity theorem for Poisson's equation. Application to Schrödinger's equation.
22/6: Chapter 9: Concentration-compactness method. Hartree functional: existence and non-existence of minimizers.
27/6: A general functional with external and interaction potentials in L^p+L^q. Binding inequality and existence of minimizers.
29/6: Translation-invariant functionals. Description of vanishing sequences. Binding inequality and existence of minimizers for translation-invariant functionals.
4/7: Existence of minimizers in various models: Choquard-Pekar problem, Gagliardo-Nirenberg interpolation inequality, Thomas-Fermi theory.
6/7: Thomas-Fermi equation and non-existence of negative ions. Chapter 10: Boundary value problems. Sobolev spaces in bounded domains H^m(U). Extension by reflection.
11/7: Extension theorem for H^1(U) with C^1 boundary. Sobolev embedding for H^1(U). Density of C^infty(U) in H^1(U). Space H_0^1 (U) as closure of C_c^infty(U).
13/7: Trace operator. Compact embedding for trace. An equivalent description of H_0^1(U) by trace. Dirichlet and Neumann problems.
18/7: Regularity of weak solutions. Translation method. Green formula.
20/7: Examples in 1D. Chapter 11: Schroedinger dynamics. Self-adjointness and existence of linear dynamics.
27/7/2017: Nonlinear Schroedinger equation. Defocusing, cubic nonlinearity and connection to the Bose-Einstein condensation. Duhamel 's formular. Local well-posedness in d=1,2,3. Conservation laws. Log Sobolev inequality in d=2. Global existence in d=1,2.