The Mixed Boundary Value Problem for Elliptic Operators

We consider solutions to the elliptic PDE $Lu = \mathrm{div}(A\nabla u) = 0$ with a variable coefficient matrix $A$ on a Lipschitz domain $\Omega$. The best-studied boundary value problems for these operators are the pure Dirichlet (or Dirichlet-regularity) and Neumann problems, where one prescribes boundary data in $L^p$ or $W^{1,p}$. Although not completely resolved, the theory surrounding these purely posed problems has seen significant progress since the 1990s.

In this talk, we study the setting where the boundary of the domain is divided into two parts, prescribing Dirichlet-regularity data on one part and Neumann data on the other. For the Laplacian, this is classically known as the Zaremba problem. This problem drew particular interest after the observation in the mid-20th century that there exist harmonic functions on the upper half-plane with vanishing Dirichlet and Neumann data that are not in $W^{1,2}$.To overcome these classical obstacles for operators with variable, merely bounded, and measurable coefficients, we establish nontangential maximal function estimates in $L^p$ for the gradient of the solution. This allows us to prove the existence of solutions to the mixed boundary value problem, as well as uniqueness under slightly more restrictive conditions.This is joint work with Hongjie Dong.