MAT1061CourseOutline

From PDEwiki

Students in MAT1061 are encouraged to contribute course related content to the wiki. In particular, major concepts can be defined and interlinked here. As the content develops, we may transfer it over to the Dispersive PDE Wiki and/or the PDE pages on Wikipedia.

Contents 1 Fourier Transform 2 Function Spaces on 3 Operators and Estimates 4 Calculus of Variations

Fourier Transform

• Definition, phase plane, Schwartz class of test functions.

• Algebraic properties of the Fourier transform: translation/modulation; dilation; product/convolution. Gaussian wave packets.

• Uncertainty principle; phase space localization; Bernstein's inequality.

• Mapping properties: Plancherel and Hausdorff-Young Theorems, Riemann-Lebesgue Lemma.

• Solving some PDEs.

• Local solvability of constant coefficient PDEs; Malgrange-Ehrenpreiss Theorem, Levy's Example, Greiner-Kohn-Stein.

Function Spaces on

• Littlewood-Paley decomposition; size of Littlewood-Paley pieces; etc.

• (Integer) Sobolev spaces W^k,p and embedding estimates.

• Littlewood-Paley square function estimate.

• Hormander-Mikhlin Multiplier Theorem.

• Fractional Sobolev spaces W^s,p and embedding estimates.

• Other function spaces.

Operators and Estimates

• Banach algebra property; low-low, low-high, high-low, high-high frequency interactions.

• Multilinear Fourier multiplier operators.

• Initial value problem for KdV equation; low regularity global well-posedness.

• Multilinear corrections term algorithm; almost conservation laws for KdV.

• Coifman-Meyer multilinear multiplier theorem.

• Abstract bootstrap principle and an example.

Calculus of Variations

• Principle of least action; first variation; Euler-Lagrange equation.

• Systems of Euler-Lagrange Equations; Null Lagrangians; Brouwer's fixed point theorem; Ginzburg-Landau Vortices.

• Existence of minimizers; direct methods.

• Minimizers are weak solutions.

• Regularity of weak solutions of scalar elliptic Euler-Lagrange equations.