Solvability estimates for the Dirichlet problem

Below you find my hand-written lecture notes from a series of five seminars that I gave in Gothenburg April-June 2025.
Relevant papers are listed on p.33-34.

Lecture notes

Part A (p.1-5)

Motivation for studying second order divergence form elliptic equations with non-smooth non-symmetric coefficients.
Local estimates De Giorgi, Nash, Moser.
Changes of variables.

Part B (p.5-12)

Examples of solvability estimates:
Energy estimates.
Maximum principle, L_infty solvability.
Estimates for smooth coefficients.
Elliptic measure
Estimates of the non-tangential maximal function.
Estimates of the square function, BMO solvability.

Part C (p.13-23)

L_2 theory for conjugate systems.
The perturbed Cauchy-Riemann generator BD.
Transformation A->B of accretive coefficients.
Bounded holomorhpic functional calculus of BD, Hardy subspaces.
Quadratic estimates for BD.
Dirichlet solvability and boundedness of the Hilbert transform.
Estimate of the NT max function by the square function.

Part D (p.23-32)

p-solvability estimates.
The Rellich identity for hermitean coefficients.
Boundedness of the Hilbert transform for triangular coefficients.
The Koch counterexample to p-solvability.
BMO solvability for general real non-symmetric coefficients.
L_infty->Carleson estimate by change of variables to triangular coefficients.
Stopping time argument for construction of good covers.