$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

**Problem 1.**

Consider Dirichlet problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty < x < \infty, y>0, \\ &u|_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

Consider Neumann problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty< x< \infty, y>0, \\ &u_y|_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition must satisfy $f$?

**Problem 2.**

Consider Dirichlet problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty<x<\infty, 0<y<1, \\ &u|_{y=0}=f(x),\quad u|_{y=1}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

Consider Dirichlet-Neumann problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty<x<\infty, 0<y<1, \\ &u|_{y=0}=f(x), \quad u_y|_{y=1}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

Consider Neumann problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty< x<\infty, 0< y< 1, \\ &u_y|_{y=0}=f(x), \quad u_y|_{y=1}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition must satisfy $f,g$?

**Problem 3.**

Consider Robin problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty< x<\infty, y>0, \\ &(u_y+\alpha u)|_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition (if any) must satisfy $f$?

*Hint.* Consider separately cases

- $\alpha\in \mathbb{C}\setminus [0,\infty)$ and
- $\alpha\in [0,\infty)$.

**Problem 4.**

Consider problem \begin{align} &\Delta^2u=0,\qquad -\infty< x<\infty, y>0, \\ &u|_{y=0}=f(x),\quad u_y|_{y=0}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

Consider problem \begin{align} &\Delta^2u=0,\qquad -\infty< x<\infty, y>0, \\ &u|_{y=0}=f(x),\quad \Delta u|_{y=0}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

Consider problem \begin{align} &\Delta^2u=0,\qquad -\infty< x<\infty, y>0, \\ &\Delta u|_{y=0}=f(x),\quad \Delta u_{y}|_{y=0}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition must satisfy $f,g$?