# 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}