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Mathematics LibreTexts

7.E: Elliptic Equations of Second Order (Exercises)

Q7.1

Let \(\gamma(x,y)\) be a fundamental solution to \(\triangle\), \(y\in \Omega\). Show that
$$
-\int_\Omega\gamma(x,y)\ \triangle\Phi(x)\ dx=\Phi(y)\quad\hbox{for all}\ \
\Phi\in C_0^2(\Omega)\ .
$$

Hint: See the proof of the representation formula.

Q7.2

Show that \(|x|^{-1}\sin(k|x|)\) is  a solution of the Helmholtz equation
$$
\triangle u+k^2u=0 \ \mbox{in}\ \mathbb{R}^n\setminus\{0\}.
$$

Q7.3

Assume \(u\in C^2(\overline{\Omega})\), \(\Omega\) bounded and sufficiently regular, is a solution of
\begin{eqnarray*}
\triangle u &=& u^3\ \mbox{in}\ \Omega\\
u&=&0\ \mbox{on}\ \partial\Omega.
\end{eqnarray*}
Show that \(u=0\) in \(\Omega\).

Q7.4

Let \(\Omega_\alpha=\{x\in {\mathbb R}^2:\ x_1>0, 0<x_2<x_1\tan\alpha\}\), \(0<\alpha\le\pi\). Show that
$$
u(x)=r^{{\pi\over\alpha}k}\sin\left({\pi\over\alpha}k\theta\right)
$$
is a harmonic function in \(\Omega_\alpha\) satisfying \(u=0\) on \(\partial\Omega_\alpha\), provided \(k\) is an integer. Here  \((r,\theta)\) are polar coordinates with the center at \((0,0)\).

Q7.5

Let \(u\in C^2(\overline{\Omega})\) be a solution of \(\triangle u=0\) on the quadrangle \(\Omega=(0,1)\times (0,1)\) satisfying the boundary conditions \(u(0,y)=u(1,y)=0\) for all \(y\in[0,1]\) and \(u_y(x,0)=u_y(x,1)=0\) for all \(x\in[0,1]\). Prove that \(u\equiv 0\) in \(\overline{\Omega}\).

Q7.6

Let \(u\in C^2({\mathbb R}^n)\) be a solution of \(\triangle u=0\) in \({\mathbb R}^n\) satisfying \(u\in L^2({\mathbb R}^n)\), i. e., \(\int_{{\mathbb R}^n}\ u^2(x)\ dx<\infty.\) Show that \(u\equiv 0\) in \({\mathbb R}^n\).

Hint: Prove
$$
\int_{B_R(0)}\ |\nabla u|^2\ dx\le {const.\over R^2}
\int_{B_{2R}(0)}\ |u|^2\ dx,
$$
where \(c\) is a constant independent of \(R\).
To show this inequality, multiply the differential equation by \(\zeta:=\eta^2 u\), where
\(\eta\in C^1\) is a cut-off function with properties: \(\eta\equiv1\) in \(B_R(0)\), \(\eta\equiv0\) in the exterior of \(B_{2R}(0)\),
\(0\le\eta\le1\), \(|\nabla\eta|\le C/R\). Integrate the product, apply integration by parts and use
the formula \(2ab\le\epsilon a^2 + {1\over \epsilon}
b^2\), \(\epsilon>0\).

Q7.7

Show that a bounded harmonic function defined on \(\mathbb{R}^n\) must be a constant (a theorem of Liouville).

Q7.8

Assume \(u\in C^2(B_1(0))\cap C(\overline{B_1(0)}\setminus\{(1,0)\})\) is a solution of
\begin{eqnarray*}
\triangle u&=&0\ \ \mbox{in}\ B_1(0)\\
u&=&0\ \ \mbox{on}\ \partial B_1(0)\setminus\{(1,0)\}.
\end{eqnarray*}
Show that there are at least two solutions.

Hint: Consider
$$
u(x,y)=\frac{1-(x^2+y^2)}{(1-x)^2+y^2}.
$$

Q7.9

Assume \(\Omega\subset\mathbb{R}^n\) is bounded and \(u,\ v\in C^2(\Omega)\cap C(\overline{\Omega})\) satisfy \(\triangle u=\triangle v\) and \(\max_{\partial\Omega}|u-v|\le\epsilon\) for given \(\epsilon>0\). Show that \(\max_{\overline{\Omega}}|u-v|\le\epsilon\).

Q7.10

Set \(\Omega={\mathbb R}^n\setminus\overline{B_1(0)}\) and let \(u\in C^2(\overline {\Omega})\) be a harmonic function in \(\Omega\) satisfying  \(\lim_{|x|\to\infty}u(x)=0\). Prove that
$$
\max_{\overline{\Omega}}|u|=\max_{\partial\Omega}|u|\ .
$$

Hint: Apply the maximum principle to \(\Omega\cap B_R(0)\), \(R\) large.

Q7.11

Let \(\Omega_\alpha=\{x\in \mathbb{R}^2:\ x_1>0,\ 0<x_2<x_1\tan\alpha\}\), \(0<\alpha\le\pi\), \(\Omega_{\alpha,R}=\Omega_\alpha\cap B_R(0)\), and assume \(f\) is given and bounded on \(\overline{\Omega_{\alpha,R}}\).

Show that for each solution \(u\in C^1(\overline{\Omega_{\alpha,R}})\cap C^2(\Omega_{\alpha,R})\) of \(\triangle u=f\) in
\(\Omega_{\alpha,R}\) satisfying \(u=0\) on \(\partial\Omega_{\alpha,R}\cap B_R(0)\), holds:

For given \(\epsilon>0\) there is a constant \(C(\epsilon)\) such that
$$
|u(x)|\le C(\epsilon)\ |x|^{{\pi\over\alpha}-\epsilon}\qquad\hbox{in}\
\Omega_{\alpha,R}.
$$

Hint: (a) Comparison principle (a consequence from the maximum principle): Assume \(\Omega\) is bounded, \(u,v\in C^2(\overline{\Omega})\cap C(\overline{\Omega})\) satisfying \(-\triangle u\le -\triangle v\) in \(\Omega\) and \(u\le v\) on \(\partial\Omega\). Then \(u\le v\) in \(\Omega\).

(b) An appropriate comparison function is
$$
v=Ar^{{\pi\over\alpha}-\epsilon}\sin(B(\theta+\eta))\ ,
$$
\(A,\ B,\ \eta\) appropriate constants, \(B,\ \eta\) positive.

Q7.12

Let  \(\Omega\) be the quadrangle \((-1,1)\times(-1,1)\) and \(u\in C^2(\Omega)\cap C(\overline{\Omega})\) a solution of the boundary value problem \(-\triangle u=1\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). Find a lower and an upper bound for \(u(0,0)\).

Hint: Consider the comparison function \(v=A(x^2+y^2)\), \(A=const.\)

Q7.13

Let \(u\in C^2(B_a(0))\cap C(\overline{B_a(0)})\) satisfying \(u\ge0,\ \triangle u =0\) in \(B_a(0)\). Prove (Harnack's inequality):
$$
{a^{n-2}(a-|\zeta|)\over (a+|\zeta|)^{n-1}}u(0)\le u(\zeta)\le
{a^{n-2}(a+|\zeta|)\over (a-|\zeta|)^{n-1}}u(0)\ .
$$

Hint: Use the formula (see Theorem 7.2)
$$
u(y)={a^2-|y|^2\over a\omega_n}\int_{|x|=a}\ {u(x)\over |x-y|^n}\ dS_x
$$
for \(y=\zeta\) and \(y=0\).

Q7.14

Let \(\phi(\theta)\) be a \(2\pi\)-periodic \(C^4\)-function with the Fourier series
$$
\phi(\theta)=\sum_{n=0}^\infty \left(a_n\ cos(n\theta)+b_n\sin(n\theta)
\right)\ .
$$
Show that
$$
u=\sum_{n=0}^\infty \left(a_n\ cos(n\theta)+b_n\sin(n\theta)
\right) r^n
$$
solves the Dirichlet problem in \(B_1(0)\).

Q7.15

Assume \(u\in C^2(\Omega)\) satisfies \(\triangle u=0\) in \(\Omega\). Let \(B_a(\zeta)\) be a ball such that its closure is in \(\Omega\).
Show that
$$
|D^\alpha u(\zeta)|\le M\left(\frac{|\alpha|\gamma_n}{a}\right)^{|\alpha|},
$$
where \(M=\sup_{x\in B_a(\zeta)}|u(x)|\) and \(\gamma_n=2n\omega_{n-1}/((n-1)\omega_n)\).

Hint: Use the formula of Theorem 7.2, successively to the k th derivatives in balls with radius \(a(|\alpha|-k)/m\), \(k=o,1,\ldots,m-1\).

Q7.16

Use the result of the previous exercise to show that \(u\in C^2(\Omega)\) satisfying \(\triangle u=0\) in \(\Omega\) is real analytic in \(\Omega\).

Hint: Use Stirling's formula
$$
n!=n^ne^{-n}\left(\sqrt{2\pi n}+O\left(\frac{1}{\sqrt{n}}\right)\right)
$$
as \(n\to\infty\), to show that \(u\) is in the class \(C_{K,r}(\zeta)\), where \(K=cM\) and \(r=a/(e\gamma_n)\). The constant \(c\) is the constant in the estimate \(n^n\le ce^nn!\) which follows from Stirling's formula. See Section 3.5 for the definition of a real analytic function.

Q7.17

Assume \(\Omega\) is connected and \(u\in C^2(\Omega)\) is a solution of \(\triangle u=0\) in \(\Omega\). Prove that \(u\equiv0\) in \(\Omega\) if \(D^\alpha u(\zeta)=0\) for all \(\alpha\), for a point \(\zeta\in\Omega\). In particular, \(u\equiv0\) in \(\Omega\) if \(u\equiv0\) in an open subset of \(\Omega\).

Q7.18

Let \(\Omega=\{(x_1,x_2,x_3)\in\mathbb{R}^3:\ x_3>0\}\), which is a half-space of \(\mathbb{R}^3\). Show that
$$
G(x,y)=\frac{1}{4\pi|x-y|}-\frac{1}{4\pi|x-\overline{y}|},
$$
where \(\overline{y}=(y_1,y_2,-y_3)\), is the Green function to \(\Omega\).

Q7.19

Let \(\Omega=\{(x_1,x_2,x_3)\in\mathbb{R}^3:\ x_1^2+x_2^2+x_3^2<R^2,\ x_3>0\}\), which is half of a ball in \(\mathbb{R}^3\). Show that
\begin{eqnarray*}
G(x,y)&=&\frac{1}{4\pi|x-y|}-\frac{R}{4\pi|y||x-y^\star|}\\
&&\quad -\frac{1}{4\pi|x-\overline{y}|}+\frac{R}{4\pi|y||x-\overline{y}^\star|},
\end{eqnarray*}
where \(\overline{y}=(y_1,y_2,-y_3)\), \(y^\star=R^2y/(|y|^2)\) and \(\overline{y}^\star=R^2\overline{y}/(|y|^2)\),
is the Green function to \(\Omega\).

Q7.20

Let \(\Omega=\{(x_1,x_2,x_3)\in\mathbb{R}^3:\ x_2>0,\ x_3>0\}\), which is a wedge in \(\mathbb{R}^3\). Show that
\begin{eqnarray*}
G(x,y)&=&\frac{1}{4\pi|x-y|}-\frac{1}{4\pi|x-\overline{y}|}\\
&&\quad -\frac{1}{4\pi|x-y'|}+\frac{1}{4\pi|x-\overline{y}'|},
\end{eqnarray*}
where \(\overline{y}=(y_1,y_2,-y_3)\), \(y'=(y_1,-y_2,y_3)\) and \(\overline{y}'=(y_1,-y_2,-y_3)\),
is the Green function to \(\Omega\).

Q7.21

Find Green's function for the exterior of a disk, i. e., of the domain \(\Omega=\{x\in\mathbb{R}^2:\ |x|>R\}\).

Q7.22

Find Green's function for the angle domain \(\Omega=\{z\in\mathbb{C}:\ 0<\arg z<\alpha\pi\}\), \(0<\alpha<\pi\).

Q7.23

Find Green's function for the slit domain \(\Omega=\{z\in\mathbb{C}:\ 0<\arg z<2\pi\}\).

Q7.24

Let for a sufficiently regular domain \(\Omega\in\mathbb{R}^n\), a ball or a quadrangle for example,
$$
F(x)=\int_\Omega\ K(x,y)\ dy,
$$
where \(K(x,y)\) is continuous in \(\overline{\Omega}\times\overline{\Omega}\) where \(x\not=y\), and which satisfies
$$
|K(x,y)|\le\frac{c}{|x-y|^\alpha}
$$
with a constants \(c\) and \(\alpha\), \(\alpha<n\).

Show that \(F(x)\) is continuous on \(\overline{\Omega}\).

Q7.25

Prove (i) of the lemma of Section 7.5.

Hint: Consider the case \(n\ge3\). Fix a function \(\eta\in C^1(\mathbb{R}^1)\) satisfying \(0\le\eta\le1\), \(0\le\eta'\le2\), \(\eta(t)=0\) for \(t\le1\), \(\eta(t)=1\) for \(t\ge2\) and consider for \(\epsilon>0\) the regularized integral
$$
V_\epsilon(x):=\int_\Omega\ f(y)\eta_\epsilon\frac{dy}{|x-y|^{n-2}},
$$
where \(\eta_\epsilon=\eta(|x-y|/\epsilon)\). Show that \(V_\epsilon\) converges uniformly to \(V\) on compact subsets of \(\mathbb{R}^n\) as \(\epsilon\to0\), and that \(\partial V_\epsilon(x)/\partial x_i\) converges uniformly on compact subsets of \(\mathbb{R}^n\) to
$$
\int_\Omega\ f(y)\frac{\partial}{\partial x_i}\left(\frac{1}{|x-y|^{n-2}}\right)\ dy
$$
as \(\epsilon\to0\).

Q7.26

Consider the inhomogeneous Dirichlet problem \(-\triangle u=f\) in \(\Omega\), \(u=\phi\) on \(\partial\Omega\). Transform this problem into a Dirichlet problem for the Laplace equation.

Hint: Set \(u=w+v\), where \(w(x):=\int_\Omega\ s(|x-y|)f(y)\ dy\).

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