Seminar

Second-order equations of the form $\nabla\cdot A\nabla u=0$, with $A$ a uniformly elliptic matrix, have many applications and have been studied extensively. A well-known foundational result of the theory is that, if the coefficients $A$ are real-valued, symmetric, and constant along the vertical coordinate (and merely bounded measurable in the horizontal coordinates), then the Dirichlet problem with boundary data in $L^2$ or $\dot W^2_1$ and the Neumann problem with boundary data in $L^2$ are well-posed in the upper half-space.



The theory of higher-order elliptic equations of the form $\nabla^m \cdot A\nabla^m u=0$ is far less well understood. In this talk we will generalize well-posedness of the $L^2$ Neumann problem in the half-space to the case of higher-order equations with real symmetric vertically constant coefficients.