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# 3.E: Classification (Exercises)

### Q3.1

Let $$\chi$$: $${\mathbb{R}^n}\to {\mathbb{R}^1}$$ in $$C^1$$, $$\nabla\chi\not=0$$. Show that for given $$x_0\in {\mathbb{R}^n}$$ there is in a neighborhood of $$x_0$$ a local diffeomorphism $$\lambda=\Phi(x)$$, $$\Phi:\ (x_1,\ldots,x_n)\mapsto(\lambda_1,\ldots,\lambda_n)$$, such that $$\lambda_n=\chi(x)$$.

### Q3.2

Show that the differential equation

$$a(x,y)u_{xx}+2b(x,y)u_{xy}+c(x,y)u_{yy}+\mbox{lower order terms}=0$$

is  elliptic  if $$ac-b^2>0$$, parabolic if $$ac-b^2=0$$ and hyperbolic if $$ac-b^2<0$$.

### Q3.3

Show that in the hyperbolic case there exists a solution of $$\phi_x+\mu_1\phi_y=0$$, see equation (3.9), such that $$\nabla\phi\not=0$$.

Hint: Consider an appropriate Cauchy initial value problem.

### Q3.4

Show equation (3.4).

### Q3.5

Find the type of

$$Lu:=2u_{xx}+2u_{xy}+2u_{yy}=0$$

and transform this equation into an equation with vanishing mixed derivatives by using the orthogonal mapping (transform to principal axis) $$x=Uy,\ U$$ orthogonal.

### Q3.6

Determine the type of the following equation at $$(x,y)=(1,1/2)$$.

$$Lu:=xu_{xx}+2yu_{xy}+2xyu_{yy}=0.$$

### Q3.7

Find all $$C^2$$-solutions of

$$u_{xx}-4u_{xy}+u_{yy}=0.$$

Hint: Transform to principal axis and stretching of axis lead to the wave equation.

### Q3.8

Oscillations of a beam are described by

\begin{eqnarray*}
w_x-{1\over E}\sigma_t&=& 0\\
\sigma_x-\rho w_t&=&0,
\end{eqnarray*}

where $$\sigma$$ stresses, $$w$$ deflection of the beam and $$E,\ \rho$$ are positive constants.

1. Determine the type of the system.
2. Transform the system into two uncoupled equations, that is, $$w,\ \sigma$$ occur only in one equation, respectively.
3. Find non-zero solutions.

### Q3.9

Find nontrivial solutions ($$\nabla \chi\not=0$$) of the characteristic equation to

$$x^2u_{xx}-u_{yy}=f(x,y,u,\nabla u),$$

where $$f$$ is given.

### Q3.10

Determine the type of

$$u_{xx}-xu_{yx}+u_{yy}+3u_x=2x,$$

where $$u=u(x,y)$$.

### Q3.11

Transform equation

$$u_{xx}+(1-y^2)u_{xy}=0,$$

$$u=u(x,y)$$, into its normal form.

### Q3.12

Transform  the Tricomi-equation

$$yu_{xx}+u_{yy}=0,$$

$$u=u(x,y)$$, where $$y<0$$, into its normal form.

### Q3.13

Transform  equation

$$x^2u_{xx}-y^2u_{yy}=0,$$

$$u=u(x,y)$$, into its normal form.

### Q3.14

Show that

$$\lambda=\dfrac{1}{\left(1+|p|^2\right)^{3/2}},\ \ \Lambda=\dfrac{1}{\left(1+|p|^2\right)^{1/2}}.$$

are the minimum and maximum of eigenvalues of the matrix $$(a^{ij})$$, where

$$a^{ij}=\left(1+|p|^2\right)^{-1/2}\left(\delta_{ij}-\dfrac{p_ip_j}{1+|p|^2}\right).$$

### ﻿Q3.15

Show that Maxwell equations are a hyperbolic system.

### Q3.16

Consider Maxwell equations and prove that $$\text{div}\ E=0$$ and $$\text{div}\ H=0$$ for all $$t$$ if these equations are satisfied for a fixed time $$t_0$$.

Hint. $$\text{div}\ \text{rot} \ A=0$$ for each $$C^2$$-vector field $$A=(A_1,A_2,A_3)$$.

### Q3.17

Assume a characteristic surface $$\mathcal{S}(t)$$ in $$\mathbb{R}^3$$ is defined by $$\chi(x,y,z,t)=const.$$ such that $$\chi_t=0$$ and $$\chi_z\not=0$$. Show that $$\mathcal{S}(t)$$ has a nonparametric representation $$z=u(x,y,t)$$ with $$u_t=0$$, that is $$\mathcal{S}(t)$$ is independent of $$t$$.

### Q3.18

Prove formula (3.22) for the normal on a surface.

### Q3.19

Prove formula (3.23) for the speed of the surface $$\mathcal{S}(t)$$.

### Q3.20

Write the Navier-Stokes system as a system of type (3.4.1).

### Q3.21

Show that the following system (linear elasticity, stationary case of (3.4.1.1) in the two-dimensional case) is elliptic
$$\mu\triangle u+(\lambda+\mu)\mbox{\ grad(div}\ u)+f=0,$$
where $$u=(u_1,u_2)$$. The vector $$f=(f_1,f_2)$$ is given and
$$\lambda,\ \mu$$ are positive constants.

### Q3.22

Discuss the type of the following system in stationary gas dynamics (isentrop flow) in $$\mathbb{R}^2$$.
\begin{eqnarray*}
\rho u u_x+\rho v u_y+ a^2\rho_x&=&0\\
\rho u v_x+\rho v v_y+ a^2\rho_y&=&0\\
\rho (u_x+v_y)+u\rho_x+ v\rho_y&=&0.
\end{eqnarray*}
Here are $$(u,v)$$ velocity vector, $$\rho$$ density and
$$a=\sqrt{p'(\rho)}$$ the sound velocity.

### Q3.23

Show  formula 7. (directional derivative).

Hint: Induction with respect to $$m$$.

### Q3.24

Let $$y=y(x)$$ be the solution of:
\begin{eqnarray*}
y'(x)&=&f(x,y(x))\\
y(x_0)&=&y_0,
\end{eqnarray*}
where $$f$$ is  real analytic in a neighborhood of $$(x_0,y_0)\in \mathbb{R}^2$$.
Find the polynomial $$P$$ of degree 2 such that
$$y(x)=P(x-x_0)+O(|x-x_0|^3)$$
as $$x\to x_0$$.

### Q3.25

Let $$u$$ be the solution of
\begin{eqnarray*}
\triangle u&=&1\\
u(x,0)&=&u_y(x,0)=0.
\end{eqnarray*}
Find the polynomial $$P$$ of degree 2 such that
$$u(x,y)=P(x,y)+O((x^2+y^2)^{3/2})$$
as $$(x,y)\to(0,0)$$.

### Q3.26

Solve the Cauchy initial value problem
\begin{eqnarray*}
V_t&=&{Mr\over r-s-NV}(1+N(n-1)V_s)\\
V(s,0)&=&0.
\end{eqnarray*}

Hint: Multiply the differential equation with $$(r-s-NV)$$.

### Q3.27

Write $$\triangle^2 u=-u$$ as a system of first order.

Hint: $$\triangle^2 u\equiv\triangle(\triangle u)$$.

### Q3.28

Write the minimal surface equation

$${\partial\over\partial x}\left({u_x\over\sqrt{1+u_x^2+u_y^2}}\right)+{\partial\over\partial y}\left({u_y\over\sqrt{1+u_x^2+u_y^2}}\right)=0$$

as a system of first order.

Hint: $$v_1:= u_x/\sqrt{1+u_x^2+u_y^2},\ v_2:=u_y/\sqrt{1+u_x^2+u_y^2}.$$

### Q3.29

Let $$f:\ \mathbb{R}^1\times\mathbb{R}^m\to\mathbb{R}^m$$ be real analytic in $$(x_0,y_0)$$. Show that a real analytic solution in a neighborhood of $$x_0$$ of the problem

\begin{eqnarray*}
y'(x)&=&f(x,y)\\
y(x_0)&=&y_0
\end{eqnarray*}

exists and is equal to the unique $$C^1[x_0-\epsilon, x_0+\epsilon]$$-solution from the Picard-Lindel\"of theorem, $$\epsilon>0$$ sufficiently small.

### Q3.30

Show (see the proof of Proposition A7)

$$\dfrac{\mu\rho(r-x_1-\ldots-x_n)}{\rho r-(\rho+mM)(x_1+\ldots+x_n)} <<\dfrac{\mu\rho r}{\rho r-(\rho+mM)(x_1+\ldots+x_n)}.$$

Hint: Leibniz's rule.

### Contributors

• Integrated by Justin Marshall.