2.4: Ecuaciones no lineales en\(\mathbb{R}^n\)
- Page ID
- 118139
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Aquí consideramos la ecuación diferencial no lineal
\ begin {ecuación}
\ label {nonlineal2}
F (x, z, p) =0,
\ end {ecuación}
donde
$$
x =( x_1,\ ldots, x_n),\ z=u (x):\\ Omega\ subconjunto\ mathbb {R} ^n\ mapsto\ mathbb {R} ^1,\ p=\ nabla u.
$$
El siguiente sistema de ecuaciones diferenciales\(2n+1\) ordinarias se denomina sistema característico.
\ begin {eqnarray*}
x' (t) &=&\ nAbla_PF\\
z' (t) &=&p\ cdot\ Nabla_PF\\
p' (t) &=&-\ Nabla_xf-f_zp.
\ end {eqnarray*}
Dejar
$$
x_0 (s) =( x_ {01} (s),\ ldots, x_ {0n} (s)),\ s =( s_1,\ ldots, s_ {n-1}),
$$
ser un dado regular (n-1) -dimensional\(C^2\) -hipersurface in\(\mathbb{R}^n\), i. e., suponemos
$$
\ mbox {rank}\ frac {\ parcial x_0 (s)} {\ parcial s} =n-1.
$$
Aquí\(s\in D\) hay un parámetro de un dominio de parámetro\((n-1)\) -dimensional\(D\).
Por ejemplo,\(x=x_0(s)\) define en el caso tridimensional una superficie regular en\(\mathbb{R}^3\).
Supongamos que
$$
z_0 (s):\ D\ mapsto\ mathbb {R} ^1,\ p_0 (s) =( p_ {01} (s),\ ldots, p_ {0n} (s))
$$
se les dan funciones suficientemente regulares.
El\((2n+1)\) -vector
$$
(x_0 (s), z_0 (s), p_0 (s))
$$
se llama colector de tira inicial y la condición
$$
\ frac {\ parcial z_0} {\ parcial s_l} =\ suma_ {i=1} ^ {n-1} p_ {0i} (s)\ frac {\ parcial x_ {0i}} {\ s_l parcial},
$$
\(l=1,\ldots,n-1\), condición de tira.
Se dice que el colector de tira inicial no es característico si
$$
\ det\ left (\ begin {array} {llcl} F_ {p_1} &F_ {p_2} &\ cdots & F_ {p_n}\
\ frac {\ parcial x_ {01}} {\ partial s_1} &\ frac {\ parcial x_ {02}} {\ parcial s_1} &\ cpuntos &\ frac {\ parcial x_ { 0n}} {\ s_1 parcial}\\
... &... &... &... \
\ frac {\ parcial x_ {01}} {\ parcial s_ {n-1}} &\ frac {\ parcial x_ {02}} {\ parcial s_ {n-1}} &\ cdots &\ frac {\ parcial x_ {0n}} {\ parcial s_ {n-1}}\ end {array}\ derecha)\ no=0,
$$
donde el argumento de\(F_{p_j}\) es el colector de tira inicial.
Problema de valor inicial de Cauchy. Buscar una solución\(z=u(x)\) de la ecuación diferencial (\ ref {nonlineal2}) tal que el colector inicial sea un subconjunto de\(\{(x,u(x),\nabla u(x)):\ x\in \Omega\}\).
Como en el caso bidimensional tenemos bajo suposiciones adicionales de regularidad
Teorema 2.3. Supongamos que el colector de tira inicial no es característico y satisface la ecuación diferencial (\ ref {nonlinear2}), es decir,
\(F(x_0(s),z_0(s),p_0(s))=0\). Luego hay un vecindario del colector inicial\((x_0(s),z_0(s))\) tal que existe una solución única del problema de valor inicial de Cauchy.
Croquis de prueba. Que
$$
x=x (s, t),\ z=z (s, t),\ p=p (s, t)
$$
sea la solución del sistema característico y que
$$
s=s (x),\ t=t (x)
$$
sea la inversa del\(x=x(s,t)\) cual existe en un barrio de \(t=0\). Entonces, resulta que
$$
z=u (x) := z (s_1 (x_1,\ ldots, x_n),\ ldots, s_ {n-1} (x_1,\ ldots, x_n), t (x_1,\ ldots, x_n))
$$
es la solución del problema.