Course: Partial Differential Equations II (SoSe2017)

Unofficial lecture notes by Martin Peev

Brief contents of lectures (regularly updated)

25/4: 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: 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.