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Let $D=D_1(0)\subset\mathbb{R}^2$ and $\lambda\in\mathbb{R}$, $\lambda>0$. Consider the Helmholtz operator $L=(\Delta +\lambda I).$

Let $f\in Ker_0(L)$, that is $f$ solves $$ Lf=0\quad\text{ in $D$}, $$ $$ f=0\quad\text{ on $\partial D$}.$$

Consider, for a given boundary datum $g$, the problem $$ Lu=f\quad\text{ in $D$},$$ $$ u=g\quad\text{ on $\partial D$}. $$

Are there necessary and sufficient conditions for the solvability of this problem? If so, which would be the correct formulation in terms of function spaces?

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  • $\begingroup$ Try integrate $\|f\|_{L^2}=\int fLu$ by parts. $\endgroup$
    – Fan Zheng
    Commented Dec 3, 2015 at 18:56
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    $\begingroup$ You might find this related. $\endgroup$
    – BigM
    Commented Dec 3, 2015 at 21:50
  • $\begingroup$ This might be useful. Where can I find the references cited in this PDF? $\endgroup$
    – gin111
    Commented Dec 4, 2015 at 11:09

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One necessary condition follows from integration by parts:

$$ \int_D f^2=\int_D fLu=\int_{\partial D} (f\partial_nu-u\partial_nf)+\int_D uLf=\int_{\partial D} g\partial_nf. $$

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