Sparse domination for boundary value problems

This project concerns the fundamental Dirichlet and Neumann boundary value problems for linear divergence form elliptic partial differential equations, with $L_p$ boundary data. Following harmonic analysis tradition since the 1970s, with the pioneering work of B.

Dahlberg, we seek to prove non-tangential maximal function estimates of solutions, under minimal regularity assumptions on the coefficients and the boundary.

The goal is to establish boundedness of certain generalized singular integral operators on function spaces defined by the Carleson functional. For classical singular integrals, such bounds have recently been proved in joint work with T. Hytönen, using the state-of-the-art technique of sparse domination.

This opens up the possibility of proving solvability estimates for the Neumann problem for non-symmetric non-smooth coefficients.

Unlike available techniques, which are limited to the Dirichlet problem and real coefficients, the singular integral approach that we propose appears to generalize to more general systems of partial differential equations and is not limited to the above Neumann problem.In the effort to establish these important estimates, it is likely that I will involve suitable colleagues in the fields of harmonic analysis and non-smooth boundary value problems with expert knowledge for the project.

The timing for the project is right: positive inital results have been obtained and the needed techniques from harmonic analysis appear to be available.