Green's function for boundary value problems

Let $\Omega \subseteq \mathbb {R}^{3}$ be bounded and open with smooth boundary. Let $u \in C^1(\overline{\Omega }) \cap C^2(\Omega )$ solve

Derive an appropriate mean-value property for the solution.

At the point $x = \xi$ of a homogeneous string ($\rho = 1$), subject to the action of a strong tension $T_{0}$ and fastened at the ends $x = 0$ and $x = 1$, a weight $Q = 1$ is suspended (Fig. 15.22).

Fig. 15.22

Prove that the equation of the sag of the string has the form

A heavy homogeneous string is fastened at the point $x = 1$ and is subject at the point $x = \xi$ to the action of a horizontal force $Q = 1$ (Fig. 15.23).

Fig. 15.23

Prove that the equilibrium equation of the string has the form

A homogeneous string of length $l = 1$ is fastened by the end $x = 0$ to a fixed axis and rotates about this axis with constant angular velocity. To the point $x = \xi$ of such a string a force $Q = 1$ is applied (Fig. 15.24).

Fig. 15.24

Prove that the equation of relative equilibrium of the string has the form

Reduce to an integral equation the problem of finding the solutions of the equation

under the boundary conditions: $y(0)$ bounded, $y(1) = 0$.

Reduce to an integral equation the problem of finding the solutions of the equation

under the boundary conditions: $y(0) = 0$, $y(1) = 0$.

Construct the integral equation whose fundamental functions are the Legendre functions of order $m$:

boundary condition: $y$ bounded at $x = -1$ and $x = +1$.

Solve problem 5943 for the case $m = 0$ of the Legendre polynomials.

Construct the integral equation whose fundamental functions are the Hermite polynomials:

$y = 0$ at $x = \pm \infty$.

Construct the integral equation whose fundamental functions are the Laguerre polynomials:

$y$ bounded at $x = 0$ and $x = +\infty$.

Having found the corresponding Green's function, replace the equation

by an integral one.

Having found the corresponding Green's function, replace the equation

by an integral equation.

Having found the corresponding Green's function $G(\xi ,x)$, replace the equation

by an integral one.

Having verified that the fundamental functions of the equation of problem 5950 (problem 5950: $y''+k^{2}y = f(x)$, $y(0) = 0$, $y'(1) = -hy(1)$, with Green's function $G(\xi ,x)$) equal

where $k_{n}$ --- the root of the equation

solve the equation

Verify that for the equation

in the case of a disk of radius $R$ with boundary condition $u = 0$, Green's function equals

where $r_{1}$ and $r_{2}$ --- the distances from the point $(x,y)$ to the points

Verify the symmetry of the function $G(x,y,\zeta ,\eta )$.

Verify that for the equation

in the case of a sphere of radius $R$ with boundary condition $u = 0$, Green's function equals

where $r_{1}$ and $r_{2}$ --- the distances from the point $(x,y,z)$ to the points $(\xi ,\eta ,\zeta )$ and

Verify that the characteristic numbers and fundamental functions of the equation

where $G$ --- Green's function of problem 5952 (problem 5952: $G(x,y,\zeta ,\eta ) = -\frac{1}{2\pi }\ln \frac{r_{1}}{r_{2}}+\frac{1}{2\pi }\ln \frac{\sqrt{\zeta^{2}+\eta^{2}}}{R}$ for a disk of radius $R$ with boundary condition $u=0$), and $\Omega$ --- a disk of radius $R$, can be found by finding the characteristic numbers and fundamental functions of the equation

where

$\rho$ and $l$ --- the distances from the points $(\zeta ,\eta )$, $(x,y)$ to the center of the disk.

Find the characteristic numbers and fundamental functions of the equation of problem 5954 (problem 5954: $\varphi (x,y) = \lambda \iint_{\Omega } G(x,y,\zeta ,\eta )\varphi (\zeta ,\eta )\, d\zeta \, d\eta$, where $G$ --- Green's function of problem 5952 for a disk of radius $R$).

Find the characteristic numbers and fundamental functions of the equation

where $G(x,y,z,\xi ,\eta ,\zeta )$ --- Green's function of problem 5953 (problem 5953: $G(x,y,z,\xi ,\eta ,\zeta ) = \frac{1}{4\pi }\left(\frac{1}{r_{1}}-\frac{R}{r_{2}\sqrt{\xi^{2}+\eta^{2}+\zeta^{2}}}\right)$ for a sphere of radius $R$ with boundary condition $u=0$), and $\Omega$ --- a sphere of radius $R$.

Find the solution of the equation

where $a = \sqrt{x^{2}+y^{2}+z^{2}}$, satisfying on the boundary of a sphere of radius $R$ centered at the origin the condition $V = 0$; the function $V$ must be given inside the sphere.

The fundamental functions of the equation

where $r$ --- the distance between the points $(x,y,z)$ and $(\xi ,\eta ,\zeta )$, are called the universal Korn functions.

Find the universal Korn functions, assuming that the region $\Omega$ is a sphere of radius $R$.

Find the Green's function of the Dirichlet problem for the Laplace operator in the unit disk $\left\{ z:\left|z\right| < 1\right\}$.

Find the Green's function of the Dirichlet problem for the Laplace operator in the upper half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$.

Find the Green's function of the Dirichlet problem for the Laplace operator in the unit half-disk $\left\{ z:\left|z\right| < 1, \mathrm{Im}\left(z\right) > 0\right\}$.

Suppose $d \geq 3,\left(X_{t}, \mathbb {P}[x]\right)$ is a $d$-dimensional Brownian motion, and

Show that if $f$ is bounded and measurable with compact support, then $U f$ is continuous and $\left|U f(x)\right| \rightarrow 0$ as $\left|x\right| \rightarrow \infty$. Show that if $f \in C^{2}$ with compact support, then $U f$ is $C^{2}$. Show that $\frac{1}{2} \Delta U f=-f$.