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May 31 (final lecture): I showed that the p-system and the Euler equations are strictly hyperbolic systems (under natural assumptions). The solution of the Riemann problem for convex scalar conservation laws was derived. Finally, I discussed briefly the Kruzkov entropy condition and the correpsonding existence and uniqueness result.

Updated Syllabus/achievement requirements: Material covered in Evans' book: Chapter 8 (Calculus of variations) except 8.3, 8.4.4 (b), Chapter 9 (Nonvariational techniques) except 9.4, 9.5, Chapters 11 + 3 (Conservation laws): 11.1, 11.4.2, 11.4.3 (including Theorem 3 but not the proof), 3.4.2, 3.4.3, 3.4.5.

Written lecture notes on weak convergence methods available here

May 24: I continued with the theory of systems of conservation laws; The focus was on the notion of strict hyperbolicity and the derivation of some useful results about matrices that satisfy the strict hyperbolicity condition.

Next week: The plan is to solve the (scalar) Riemann problem.

May 10/11: This week I started with conservation laws. I introduced the equations, defined weak solutions and explained why we need them, and derived the Rankine-Hugoniot jump condition. Finally, I introduced the (traveling wave/vanishing viscosity) entropy condition for scalar conservation laws, which is a condition that encodes the missing physics in conservation laws and consequently restores the uniqueness of weak solutions.

No lectures next week (because of 17. mai!).

May 3/4: I finished with the nonlinear semigroup method, and mentioned an application of this method to the well-posedness of a class of nonlinear parabolic equations.

May 4: No Lecture (self-study/exercises).

Next week I will begin with the theory of shock waves for hyperbolic systems of conservation laws.

The date of (oral) examination is now fixed: Monday June 19.

April 26/27: I continued with nonlinear semigroup theory, introducing the concepts of nonlinear resolvent and Yosida approximation (of the subdifferential operator), and studying the properties of these operators. Moreover, I started with the proof that the subdifferential of a convex functional generates a semigroup of contractions, which eventually will yield the well-posedness of the corresponding abstract ODE.

Next week I will finish the section on the nonlinear semigroup method.

April 19/20: I prepared for the nonlinear semigroup method (for solving time-dependent PDEs) by discussing functional spaces involving time and by defining and studying the properties of sub-differentials of convex, proper, lower semicontinuous functionals.

Next week I will continue with the nonlinear semigroup method.

March 22/23: I proved uniqueness of weak solutions to a class of nonlinear elliptic equations satisfying a strict monotonicity condition, thereby completing the "well-posedness program" started last week. Additionally, I lectured about the Banach fixed point theorem and gave an application of it to the existence of a solution to some systems of reaction-diffusion equations.

Tomorrow: Self-study (prepare for exercises next week).

March 15/16: I proved existence of weak solutions to a class of nonlinear elliptic equations for which there is no variational formulation. The key point was a monotonicity condition on the nonlinearity, which allowed to use the Browder-Minty argument. I tried to convey an important strategy for proving existence of solutions to nonlinear PDEs without a variational structure, which can be summarized as follows: 1) Construct approximate solutions. 2) Compactness (convergence analysis). 3) Identify the limit from Step 2 as a solution to the (nonlinear) PDE.

The topics next week are uniqueness and fixed point theorems.

March 8/9: I proved the mountain pass theorem and used it to prove existence of at least one nontrivial weak solution to a class of semilinear elliptic PDEs.

Reminder: No lectures tomorrow (self-study).

March 29: Lectures will be given in room B71 (a workshop will take place in rom B1036).

March 1/2: I prepared for the Mountain Pass Theorem (MPT) by proving a deformation result. Next week is devoted to the proof of the MPT and an application to the existence of solutions to semilinear elliptic PDEs.

March 2 & 9: No Lecture (self-study).

Week 13 & 14: I will be out traveling; Siddhartha Mishra (CMA) will fill in for me; He will spend most of the time solving the exercises from Chapter 8.

Week 15: No lecture because of Easter.

February 22/23: I lectured on existence/uniqueness theory for some minimization problems with pointwise one-sided constraints (obstacle problems) and pointwise constraints (harmonic maps, Stokes equations), with particular emphasis on the corresponding Euler-Lagrange equations. The topic next week is a theory ensuring the existence of critical points for "abstract" functionals (not necessarily integral functionals).

February 15/16: Existence and uniqueness theory was developed for polyconvex integral functionals. Furthermore, minimization problems with integral constraints were analyzed, including proving existence of minimizers and deriving the corresponding Euler-Lagrange equation. The E-L equation took the form of a nonlinear eigenvalue problem with the eigenvalue being the Lagrange multiplier of the integral constraint. Next week's topics are minimization problems with pointwise one-sided constraints and pointwise constraints (the latter will be exemplified by the problem of harmonic maps into spheres and the incompressible Stokes' problem).

February 8/8: I proved existence and uniqueness of minimizers. The notion of a weak solution of the Euler-Lagrange equation was introduced. The result is that a minimizer of the integral function is such a weak solution, thereby revealing the (rigorous) connection between the minimization problem and the Euler-Lagrange PDE. The theory developed this week can be extended to convex systems of equations; I mentioned this and it is expected that you carry out the details yourself (which means essentially "chasing notations"). Next week's topics are polyconvex systems and minimization problems with constraints.

February 1/2: I lectured about weak convergence theory (lecture notes can be found on "Syllabus/achievement requirements) and how this fits into the existence theory of calculus of variation. We ended up proving that integral functionals are weakly lower semicontinuous provided the Lagrangian is bounded from below and convex. Next week our first goal is to use this result to prove existence of minimizers.

January 25/26: I lectured from pages 431-442 in Evans' book. Next week I will start with the existence theory of minimizers. To prepare for the existence theory you need to know something about weak convergence, and I will spend a good hour or so on this on Wednesday.

PS! Change of lecture times. Starting next week, the Thursday lecture will be from 9.15 to 11.00.

The lectures started today. I gave an overview of the course and provided an indication of the recommended prior knowledge (prerequisites). The next lecture will be on Wednesday January 25.