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Advanced Analysis (WiSe 22/23)
Goals
This course, starting from scratch, aims at providing with the basic tools for a rigorous understanding of some mathematical models in quantum mechanics. In particular, we will review L^p spaces, rearrangement inequalities, distributions, Fourier analysis, Sobolev spaces, introduction to the Calculus of variations, with examples from mathematical physics. The course will be of interest for graduate students, or motivated Bachelor students, who want to intensify their knowledge in analysis and partial differential equations as well as students who are particularly interested in mathematical physics.
Schedule
-Lectures: Wednesday 12:15-13:45 and Friday 08:30-10:00
-Exercises: Tuesday 08:30-10:00
Reference
E. H. Lieb, M. Loss, Analysis, Graduate studies in mathematics, AMS, 2001
Content and exercise sheets
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Week 0 (18 - 19 Oct): Chapter 1 (in the Lieb-Loss), in particular definition and properties of measure spaces, measurability, definition of the integral,
Monotone convergence theorem, Fatou Lemma, Dominated convergence theorem, Missing term in the Fatou Lemma, Almost everywhere convergence implies uniform convergence except on small sets
Exercise sheet -
Week 1 (25 - 26 Oct): End of Chapter 1 and beginning of Chapter 2 (Lp spaces), in particular product measures,
Fubini theorem, approximation by C infinity functions, definition of Lp spaces, Jensen's inequality, Hölder inequality
Exercise sheet -
Week 2 (02 - 03 Nov): Continuing Chapter 2, Minkowski inequality, Completeness of Lp spaces, dual space, weak convergence, Linear functionals separate, Weak lower semi-coninuity of norms.
Exercise sheet -
Week 3 (07 - 09 Nov): Finishing Chapter 2, Dual of Lp spaces (2.14) [Hanners inequality (2.5),
Differentiability of norms (2.6), Projection on convex sets (2.8)];
Banach Alaoglu (Bounded sequences have weak limits (2.18)) [Uniform boundedness principle (2.12),
Convolution (2.15), Approximation by smooth functions (2.16), Separability of Lp (2.17)]
Exercise sheet -
Week 4 (15 - 17 Nov): proof of Banach-Alaoglu. Rearragement inequalities (Chapter 3), 3.1-3.4 + 3.7: definition of
the symmetric decreasing rearrangement via the layer cake representation,
its properties, simple rearrangement inequality (its proof),
Riesz's rearrangement inequality (without proof). Integral inequalities (Chapter 4), 4.1-4.3 Young's inequality,
Hardy-Littlewood-Sobolev inequality.
Exercise sheet -
Week 5 (22 - 24 Nov): Chapter 5 (Fourier transform), definition in L1, basic properties, Fourier transform of a Gaussian,
Plancherel's theorem, inversion formula, extension to L2 and Lp for 1 \< p \< 2, convolutions.
Beginning of Chapter 6 (Distributions), definitions, examples with locally Lp functions, derivatives of distributions.
Exercise sheet -
Week 6 (29 Nov - 1 Dec): Chapter 6 (Distribution), definitions of the Sobolev spaces W^{1,p} and W^{1,p}_{loc},
interchanging convolutions and distributions, fundamental theorem of calculus for distributions,
equivalence of classical and distributional derivatives, multiplication and convolution of distributions by smooth functions
approximation of distributions by smooth functions, chain rule.
Exercise sheet -
Week 7 (6 - 8 Dec): End of Chapter 6 (Distribution), Chain rule and it's proof, derivative of the absolute value,
min/max of W^{1,p} functions, explicit computations for some distributions, L^1_loc functions are determined by D(\Omega),
distributions of zero derivative are constants.
Exercise sheet -
Week 8 (13 - 14 Dec): Chapter 7 (Sobolev space H1), 7.2 -- 7.7 (Definition, completeness, density of smooth functions,
Integration by part), 7.9 (Fourier characterization),
7.18 (dual of H1 and weak limits)
Exercise sheet -
Week 9 (20-22 Dec): End of Chapter 7 and beginning of Chatper 8 (Sobolev inequalities).
7.18 (Fourier characerization of H1 and of its dual, proofs), 6.21 Poisson's equation,
definition of the space D1, Sobolev's inequalities of H1, proof for the cases n>2 and n=1.
Exercise sheet -
Week 10 (8-12 Jan): Proof of the Sobolev inequality for n=2, weakly convergence (of the gradient) implies strong convergence on small sets,
weak convergence (of the gradient) implies convergence almost everywhere, generalizations to open domains: the cone condition
No exercises. -
Week 11 (17-19 Jan): Sobolev inequalities for W^{m,p} (8.8), Rellich-Kondrashov theorem (8.9),
Non-zero week convergence after translation (8.10) beginning of the proof.
Exercise sheet -
Week 12 (24-26 Jan): Non-zero week convergence after translation (8.10) end of the proof.
Symmetric decreasing rearrangement decreases kinetic energy (7.17), Poincaré inequality (finite dimensional case,
p=2 case on an interval, general version 8.11)
Exercise sheet -
Week 13 (31 Jan - 2 Feb): The Schrödinger equation, eigenvalue equation in finite dimension,
example with 1/|x|^2, V in L^{n/2} and V(x) = - 1/|x|^{5/2} for n=3, Domination of the potential by the kinetic energy,
Weak continuity of the potential, Euler-Lagrange equation, Existence of minimizers of the Schrödinger functional.
Exercise sheet