| Abstract: |
| We will discuss the Neumann Green function and scale invariant regularity estimates for the equation $-div(A\nabla u+bu)+c\nabla u+du=-div f+g$ with Neumann data in Lipschitz domains $\Omega\subseteq\mathbb R^n$.
Under the assumption that $A$ is elliptic and bounded, we will see a necessary structural condition on the lower order coefficients that guarantees at most one dimensional kernels, as well as boundedness close to the boundary. Under the optimal assumptions $b,c\in L^n$ and $d\in L^{n/2}$, we will then show estimates for the $L^2$ theory that are scale invariant: that is, they depend only on the norms of the coefficients and the Lipschitz character of $\Omega$. One difficulty will be the existence of nontrivial kernels, which will differentiate the theorems considered, but this will be identified by a specific integral of the coefficient $d$.
We will also discuss the analogue of Green`s function for the Neumann problem, called the Neumann Green function, and show that it satisfies scale invariant estimates in appropriate weak $L^p$ spaces. These estimates will lead to scale invariant local boundedness, under specific assumptions for the Neumann data in Lorentz spaces that are both necessary and optimal. |
|