NMMA406 -- Dalibor Pražák, spring 2015/16
1. Vector valued functions
Bochner integral: measurability, weak measurability,
integrability. Pettis and Bochner theorem. Lebesgue point, Lebesgue theorem.
Convolution kernels. Definition and properties of L^p(I;X) spaces.
Density, approximation by convolution, separability.
Absolutely continuous vector functions.
L01+L02 (24.2.2016)
Existence of derivative almost
everywhere and Newton-Leibniz formula. Weak time derivative: equivalent
definitions. Space W^{1,p}(I;X).
L03 (2.3.2016)
More on the spaces L^p(I;X) -- uniform convexity, reflexivity, dual space.
More on weakly differentiable vector-valued functions --
Extension theorem, density of smooth functions.
Gelfand triple. Continuous representatives.
L04+L05 (9.3.2016)
Embedding into continuous functions.
Ehrling's lemma. Compact embedding (Aubin-Lions lemma).
L06 (16.3.2016)
2. Parabolic 2nd order equation
Formulation of the problem. Initial condition, Dirichlet boundary condition.
Assumptions on the nonlinearities. Sobolev spaces W^{1,2}, W^{1,2}_0 and W^{-1,2} and their properties -- density, embeddings, duality, reflexivity. Poincaré inequality. Nonlinear elliptic operator. Definition of weak solution (w.s.); equivalent formulations and basic properties: weak differentiability, L^2-continuous representatives. Uniqueness of w.s. Gronwall lemma. Compactness of w.s. Monotone operator, hemicontinuous operator. Minty's trick.
L07+L08 (23.3.2016)
Compactness of w.s. finished. Existence of w.s. via Galerkin approximation.
L09 (30.3.2016)
Lemma: chain rule for weak derivatives.
Maximum principle for the w.s.
Strong solution for the heat equation.
3. Hyperbolic 2nd order equation
Formulation of the problem. Definition of the weak solution.
Remarks concerning the continuity of w.s.
Uniqueness of w.s. (formal proof only).
L10+L11 (6.4.2016)
Remarks on weak solutions, formal a priori estimates and admissible
test functions. Uniqueness of weak solutions: correct proof for
hyperbolic equation. Lemma on testing wave equation with time
derivative. Corollary: alternative proof of uniqueness.
L12 (13.4.2016)
Hyperbolic equation: existence of solutions.
Strong solutions for more regular data.
Corollary: (strong) continuity of weak solutions.
Remarks on (non)improving of regularity in time:
parabolic vs. hyperbolic. Reversing time in the
wave equation. Wave principle and finite speed of propagation.
L13+L14 (20.4.2016)
4. Theory of semigroups
Definition of c_0-semigroup. Exponential bound and continuity.
Generator of semigroup. Properties of the generator; density
of domain and closedness. Semigroup as a classical solution
to an abstract linear equation.
L15 (27.4.2016)
Unicity of semigroup of given generator.
Resolvent of unbounded operator. Resolvent and laplace
transform of semigroup. Semigroup of contractions.
Hille-Yosida theorem for contractions.
L16 (11.5.2016)
General version of Hille-Yosida theorem.
Lumer-Phillips theorem. Nonhomogeneous abstract linear equation.
Classical solution, strong solution. Hille's theorem on
characterization of integrability of graph norm.
Mild solution. Properties of convolution with the semigroup.
Equivalent definition of mild solution.
Remarks: strong implies mild, but not vice-versa.
Uniqueness of classical and strong solution.
Approximations of mild solutions by classical solutions via convolution.
Regularity of mild solution for better data.
L17+18 (18.5.2016)