3.1: Linear Equations of Second Order

The general nonlinear partial differential equation of second order is

$$F(x,u,Du,D^2u)=0,$$
where $n$, $u:\ \Omega\subset\mathbb{R}\mapsto\mathbb{R}^1$, $Du\equiv\nabla u$ and $u$ stands for all second derivatives. The function $F$ is given and sufficiently regular with respect to its $2n+1+n^2$ arguments.

In this section we consider the case
$$\begin{equation} \label{linsecond} \sum_{i,k=1}^na^{ik}(x)u_{x_ix_k}+f(x,u,\nabla u)=0. \end{equation}$$
The equation is linear if
$$f=\sum_{i=1}^nb^i(x)u_{x_i}+c(x)u+d(x).$$
Concerning the classification the main part
$$\sum_{i,k=1}^n a^{ik}(x)u_{x_ix_k}$$
plays the essential role. Suppose $u\in C^2$, then we can assume, without restriction of generality, that $a^{ik}=a^{ki}$, since
$$\sum_{i,k=1}^n a^{ik}u_{x_ix_k}=\sum_{i,k=1}^n (a^{ik})^\star u_{x_ix_k},$$
where
$$(a^{ik})^\star=\frac{1}{2}(a^{ik}+a^{ki}).$$
Consider a hypersurface $\mathcal{S}$ in $n$ defined implicitly by $\chi(x)=0$, $\nabla\chi\not=0$, see Figure 3.1.1

Figure 3.1.1: Initial Manifold $\mathcal{S}$

Assume $u$ and $\nabla u$ are given on $\mathcal{S}$.

Problem: Can we calculate all other derivatives of $u$ on $\mathcal{S}$ by using differential equation ($\ref{linsecond}$\) and the given data?

We will find an answer if we map $\mathcal{S}$ onto a hyperplane $\mathcal{S}_0$ by a mapping
$$\begin{eqnarray*} \lambda_n&=&\chi(x_1,\ldots,x_n)\\ \lambda_i&=&\lambda_i(x_1,\ldots,x_n),\ i=1,\ldots,n-1, \end{eqnarray*}$$
for functions $\lambda_i$ such that
$$\det\frac{\partial(\lambda_1,\ldots,\lambda_n)}{\partial(x_1,\ldots,x_n)}\not=0$$
in $n$. It is assumed that $i$ and $i$ are sufficiently regular. Such a mapping $\lambda=\lambda(x)$ exists, see an exercise.

The above transform maps $\mathcal{S}$ onto a subset of the hyperplane defined by $\lambda_n=0$, see Figure 3.1.2.

Figure 3.1.2: Transformed flat manifold $\mathcal{S}$

We will write the differential equation in these new coordinates. Here we use Einstein's convention, that is, we add terms with repeating indices. Since
$$u(x)=u(x(\lambda))=:v(\lambda)=v(\lambda(x)),$$
where $x=(x_1,\ldots,x_n)$ and $\lambda=(\lambda_1,\ldots,\lambda_n)$, we get
$$\begin{eqnarray} \label{known} u_{x_j}&=&v_{\lambda_i}\frac{\partial\lambda_i}{\partial x_j},\\ u_{x_jx_k}&=&v_{\lambda_i\lambda_l}\frac{\partial\lambda_i}{\partial x_j}\frac{\partial\lambda_l}{\partial x_k}+v_{\lambda_i}\frac{\partial^2\lambda_i}{\partial x_j\partial x_k}.\nonumber \end{eqnarray}$$
Then the differential equation ($\ref{linsecond}$) in the new coordinates is given by
$$a^{jk}(x)\frac{\partial\lambda_i}{\partial x_j}\frac{\partial\lambda_l}{\partial x_k}v_{\lambda_i\lambda_l}+\mbox{terms known on}\ \mathcal{S}_0=0.$$
Since $v_{\lambda_k}(\lambda_1,\ldots,\lambda_{n-1},0)$, $n$, are known, see ($\ref{known}$), it follows that $v_{\lambda_k\lambda_l}$, $l=1,\ldots,n-1$, are known on $\mathcal{S}_0$. Thus we know all second derivatives $v_{\lambda_i\lambda_j}$ on $\mathcal{S}_0$ with the only exception of $v_{\lambda_n\lambda_n}$.

We recall that, provided $v$ is sufficiently regular,
$$v_{\lambda_k\lambda_l}(\lambda_1,\ldots,\lambda_{n-1},0)$$
is the limit of
$$\frac{v_{\lambda_k}(\lambda_1,\ldots,\lambda_l+h,\lambda_{l+1},\ldots,\lambda_{n-1},0)- v_{\lambda_k}(\lambda_1,\ldots,\lambda_l,\lambda_{l+1},\ldots,\lambda_{n-1},0)}{h}$$
as $h\to0$.

Then the differential equation can be written as
$$\sum_{j,k=1}^na^{jk}(x)\frac{\partial\lambda_n}{\partial x_j}\frac{\partial\lambda_n}{\partial x_k}v_{\lambda_n\lambda_n}=\mbox{terms known on}\ \mathcal{S}_0.$$
It follows that we can calculate $v_{\lambda_n\lambda_n}$ if
$$\begin{equation} \label{nonchar} \sum_{i,j=1}^na^{ij}(x)\chi_{x_i}\chi_{x_j}\not=0 \end{equation}$$

on $\mathcal{S}$. This is a condition for the given equation and for the given surface $\mathcal{S}$.

Definition. The differential equation
$$\sum_{i,j=1}^na^{ij}(x)\chi_{x_i}\chi_{x_j}=0$$
is called it characteristic differential equation associated to the given differential equation ($\ref{linsecond}$).

If $\chi$, $\nabla \chi\not=0$, is a solution of the characteristic differential equation, then the surface defined by $\chi=0$ is called characteristic surface.

Remark. The condition ($\ref{nonchar}$) is satisfied for each $\chi$ with $\nabla\chi\not=0$ if the quadratic matrix $(a^{ij}(x))$ is positive or negative definite for each $x\in\Omega$, which is equivalent to the property that all eigenvalues are different from zero and have the same sign. This follows since there is a $\lambda(x)>0$ such that, in the case that the matrix $(a^{ij})$ is poitive definite,
$$\sum_{i,j=1}^na^{ij}(x)\zeta_i\zeta_j\ge\lambda(x)|\zeta|^2$$
for all $\zeta\in\mathbb{R}$. Here and in the following we assume that the matrix $(a^{ij})$ is real and symmetric.

The characterization of differential equation ($\ref{linsecond}$) follows from the signs of the eigenvalues of $(a^{ij}(x))$.

Definition. The differential equation ($\ref{linsecond}$) is said to be of type $(\alpha,\beta,\gamma)$ at $x\in\Omega$ if $\alpha$ eigenvalues of $(a^{ij})(x)$ are positive, $\beta$ eigenvalues are negative and $\gamma$ eigenvalues are zero ( $\alpha+\beta+\gamma=n$).
In particular, the equation is called

elliptic if it is of type $(n,0,0)$ or of type $(0,n,0)$, that is, all eigenvalues are different from zero and have the same sign,\
parabolic if it is of type $(n-1,0,1)$ or of type $(0,n-1,1)$, that is, one eigenvalue is zero and all the others are different from zero and have the same sign,
hyperbolic if it is of type $(n-1,1,0)$ or of type $(1,n-1,0)$, that is, all eigenvalues are different from zero and one eigenvalue has another sign than all the others.

Remarks:

1. According to this definition there are other types aside from elliptic, parabolic or hyperbolic equations.

2. The classification depends in general on $x\in\Omega$. An example is the Tricomi equation, which appears in the theory of transsonic flows,
$$yu_{xx}+u_{yy}=0.$$
This equation is elliptic if $y>0$, parabolic if $y=0$ and hyperbolic for $y<0$.

Examples:

Example 3.1.1:

The Laplace equation in $\mathbb{R}^3$ is $\triangle u=0$, where
$$\triangle u:=u_{xx}+u_{yy}+u_{zz}.$$
This equation is elliptic since for every manifold $\mathcal{S}$ given by $\{(x,y,z):\ \chi(x,y,z)=0\}$, where $\chi$ is an arbitrary sufficiently regular function such that $\nabla \chi\not=0$, all derivatives of $u$ are known on $\mathcal{S}$, provided $u$ and $\nabla u$ are known on $\mathcal{S}$.

Example 3.1.2:

The wave equation $u_{tt}=u_{xx}+u_{yy}+u_{zz}$, where $u=u(x,y,z,t)$, is hyperbolic. Such a type describes oscillations of mechanical structures, for example.

Example 3.1.3:

The heat equation $u_t=u_{xx}+u_{yy}+u_{zz}$, where $u=u(x,y,z,t)$, is
parabolic. It describes, for example, the propagation of heat in a domain.

Example 3.1.4:

Consider the case that the (real) coefficients $a^{ij}$ in equation ($\ref{linsecond}$) are {\it constant}. We recall that the matrix $A=(a^{ij})$ is symmetric, that is, $A^T=A$. In this case, the transform to principle axis leads to a normal form from which the classification of the equation is obviously. Let $U$ be the associated orthogonal matrix, then
$$U^TAU=\left(\begin{array}{llcl} \lambda_1 & 0&\cdots & 0\\ 0 & \lambda_2&\cdots&0\\ ... & ... & ... & ...\\ 0&0&\cdots&\lambda_n \end{array}\right).$$
Here is $U=(z_1,\ldots,z_n)$, where $z_l$, $l=1,\ldots,n$, is an orthonormal system of eigenvectors to the eigenvalues $\lambda_l$.

Set $y=U^Tx$ and $v(y)=u(Uy)$, then
$$\begin{equation} \label{hauptachs} \sum_{i,j=1}^na^{ij}u_{x_ix_j}=\sum_{i=1}^n\lambda_iv_{y_iy_j}. \end{equation}$$

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.