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Mathematics LibreTexts

2.4: Nonlinear Equations in \(\mathbb{R}^n\)

Here we consider the nonlinear differential equation
\begin{equation}
\label{nonlinear2}
F(x,z,p)=0,
\end{equation}
where
$$
x=(x_1,\ldots,x_n),\ z=u(x):\ \Omega\subset\mathbb{R}^n\mapsto\mathbb{R}^1,\ p=\nabla u.
$$
The following system of \(2n+1\) ordinary differential equations is called characteristic system.
\begin{eqnarray*}
x'(t)&=&\nabla_pF\\
z'(t)&=&p\cdot\nabla_pF\\
p'(t)&=&-\nabla_xF-F_zp.
\end{eqnarray*}
Let
$$
x_0(s)=(x_{01}(s),\ldots,x_{0n}(s)),\ s=(s_1,\ldots,s_{n-1}),
$$
be a given regular (n-1)-dimensional \(C^2\)-hypersurface in \(\mathbb{R}^n\), i. e., we assume
$$
\mbox{rank}\frac{\partial x_0(s)}{\partial s}=n-1.
$$
Here  \(s\in D\) is a parameter from an \((n-1)\)-dimensional parameter domain \(D\).

For example, \(x=x_0(s)\) defines in the three dimensional case a regular surface in \(\mathbb{R}^3\).

Assume
$$
z_0(s):\ D\mapsto\mathbb{R}^1,\ p_0(s)=(p_{01}(s),\ldots,p_{0n}(s))
$$
are given sufficiently regular functions.

The \((2n+1)\)-vector
$$
(x_0(s),z_0(s),p_0(s))
$$
is called initial strip manifold and the condition
$$
\frac{\partial z_0}{\partial s_l}=\sum_{i=1}^{n-1}p_{0i}(s)\frac{\partial x_{0i}}{\partial s_l},
$$
\(l=1,\ldots,n-1\), strip condition.

The initial strip manifold is said to be non-characteristic if
$$
\det\left(\begin{array}{llcl}F_{p_1}&F_{p_2}&\cdots & F_{p_n}\\
\frac{\partial x_{01}}{\partial s_1}&\frac{\partial x_{02}}{\partial s_1}&\cdots & \frac{\partial x_{0n}}{\partial s_1}\\
... & ... & ... & ...\\
\frac{\partial x_{01}}{\partial s_{n-1}}&\frac{\partial x_{02}}{\partial s_{n-1}}&\cdots & \frac{\partial x_{0n}}{\partial s_{n-1}}\end{array}\right)\not=0,
$$
where the argument of \(F_{p_j}\) is the initial strip manifold.

Initial value problem of Cauchy. Seek a solution \(z=u(x)\) of the differential equation (\ref{nonlinear2}) such that the initial manifold is a subset of \(\{(x,u(x),\nabla u(x)):\ x\in \Omega\}\).

As in the two dimensional case we have under additional regularity assumptions

Theorem 2.3. Suppose the initial strip manifold is not characteristic and satisfies  differential equation (\ref{nonlinear2}), that is,
\(F(x_0(s),z_0(s),p_0(s))=0\). Then there is a  neighborhood of the initial manifold \((x_0(s),z_0(s))\) such that there exists a unique solution of the Cauchy initial value problem.

Sketch of proof. Let
$$
x=x(s,t),\ z=z(s,t),\ p=p(s,t)
$$
be the solution of the characteristic system and let
$$
s=s(x),\ t=t(x)
$$
be the inverse of \(x=x(s,t)\) which exists in a neighborhood of \(t=0\). Then, it turns out that
$$
z=u(x):= z(s_1(x_1,\ldots,x_n),\ldots,s_{n-1}(x_1,\ldots,x_n),t(x_1,\ldots,x_n))
$$
is the solution of the problem.

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