1
$\begingroup$

Let's consider the following Poisson equation with Neumann boundary condition

\begin{align*} -\Delta u &= f \quad \text{in } \Omega \\ \partial_n u&=0 \quad \text{on } \partial \Omega, \end{align*} where $\Omega \subset \mathbb{R}^3$ is bounded domain with smooth boundary. It is obvious that \begin{align*} \int_\Omega f \, dx =0 \end{align*} to ensure existence of solutions to the Poisson equation.

Recently, I have read famous paper "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I,II" (Comm. Pure Appl. Math., 1959,1964) written by Agmon,Douglis, and Nirenberg. In this paper, they introduced the complementing boundary condition which is very algebraic. I checked that the Poisson equation with Neumann boundary condition satisfies the complementing boundary condition.

The weird thing here is that it seems to satisfy the complementing boundary condition regardless of values $\int f$. I think I may have missed something in Agmon's argument. I would appreciate it if you could provide me with an answer. Thank you.

Improve this question
$\endgroup$
2
  • $\begingroup$ Note the proper use of \text{} in MathJax. And also I changed $\displaystyle \int f dx$ to $\displaystyle \int f\, dx. \qquad$ $\endgroup$
    – Michael Hardy
    Commented Mar 14, 2023 at 21:29
  • 1
    $\begingroup$ Yes, the complementing condition is independent of any conditions of f. But nowhere is it claimed that the complementing condition guarantees existence. $\endgroup$
    – Michael Renardy
    Commented Mar 15, 2023 at 2:54

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .