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 the equation $$ Lu=f\quad\text{ in $D$},$$ $$ u=g\quad\text{ on $\partial D$}. $$

Are there necessary and sufficient conditions for the solvability of this problem?

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?

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 the equation $$ Lu=f\quad\text{ in $D$},$$ $$ u=g\quad\text{ on $\partial D$}. $$

Are there solvability conditions for this problem?

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 the equation $$ Lu=f\quad\text{ in $D$},$$ $$ u=g\quad\text{ on $\partial D$}. $$

Are there necessary and sufficient conditions for the solvability of this problem?