1
$\begingroup$

Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator.

My question is about the intergal operator $T(v)(x)=\int_{\Omega} (\triangle_y g_N(x,y))v(y) dy$.

Is it true that $T$ is reasonably nice? say $T:L^2(\Omega) \to L^2(\Omega) $

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Please insert parentheses - is $v(y)$ subject to differentiation? $\endgroup$
    – Sebastian Goette
    Commented Jun 14, 2017 at 6:40
  • $\begingroup$ @SebastianGoette I have clarified the expression. Thanks $\endgroup$
    – Ali
    Commented Jun 14, 2017 at 17:01
  • 1
    $\begingroup$ Why would it not be true that $T(v) = v$? The Neumann (as well as Dirichlet) boundary condition is self-adjoint, so you should expect $g_N(x,y) = g_N(y,x)$, meaning that $\Delta_y g_N(x,y) = \Delta_x g_N(x,y) = \delta(x,y)$. $\endgroup$
    – Igor Khavkine
    Commented Jun 14, 2017 at 18:11
  • $\begingroup$ Hmm you are right about the fact that $g_N$ is self adjoint. But what if we have a non-symmetric smoothing right inverse. I still think the operator will be nice as the difference of the green functions should always be 'nice' as somehow you are removing the singularity. $\endgroup$
    – Ali
    Commented Jun 14, 2017 at 18:57

0

Reset to default

You must log in to answer this question.