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Mathematics LibreTexts

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

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    Here we consider the nonlinear differential equation
    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.
    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 \(\mathbb{R}^3\).

    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
    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.