7.1: Solución Fundamental
- Page ID
- 118051
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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 soluciones particulares de la ecuación de Laplace en\(\mathbb{R}^n\) del tipo
$$u (x) =f (|x-y|),\]
donde\(y\in\mathbb{R}^n\) es fijo y\(f\) es una función la cual determinaremos tal que\(u\) define una solución si la ecuación de Laplace.
Establezca\(r=|x-y|\), luego
\ begin {eqnarray*}
u_ {x_i} &=&f' (r)\ frac {x_i-y_i} {r}\\
u_ {x_ix_i} &=&f "(r)\ frac {(x_i-y_i) ^2} {r^2} +f' (r)\ left (\ frac {1} {r} -\ frac {(x_i-y_i) ^2} {r^3}\ derecha)\\
\ triángulo u&=&f" (r) +\ frac {n-1} {r} f' (r).
\ end {eqnarray*}
Por lo tanto, una solución de\(\triangle u=0\) es dada por
$$f (r) =\ left\ {\ begin {array} {r@ {\ qua d:\quad} l}
c_1\ ln r+c_2&n=2\\
c_1r^ {2-n} +c_2&n\ ge3
\ end {array}\ derecha.\]
con constantes\(c_1\),\(c_2\).
Definición. Set\(r=|x-y|\). La función
$$
s (r) :=\ left\ {\ begin {array} {r@ {\ qua d:\quad} l}
-\ frac {1} {2\ pi}\ ln r&n=2\\
\ frac {r^ {2-n}} {(n-2)\ omega_n} &n\ ge3
\ end {array}\ derecha.
\]
se llama función de singularidad asociada a la ecuación de Laplace. Aquí está\(\omega_n\) el área de la esfera unitaria n-dimensional que viene dada por\(\omega_n=2\pi^{n/2}/\Gamma(n/2)\), donde
$$\ Gamma (t) :=\ int_0^\ infty\ e^ {-\ rho}\ rho^ {t-1}\ d\ rho,\\ t>0,\]
es la función Gamma.
Definición. Una función
$$\ gamma (x, y) =s (r) +\ phi (x, y)\]
se llama solución fundamental asociada a la ecuación de Laplace si\(\phi\in C^2(\Omega)\) y\(\triangle_x\phi=0\) para cada fijo\(y\in\Omega\).
OBSERVACIÓN. La solución fundamental\(\gamma\) satisface para cada fijo\(y\in\Omega\) la relación
$$-\ int_\ Omega\\ gamma (x, y)\ triangle_x\ Phi (x)\ dx=\ Phi (y)\\\ mbox {para todos}\\ Phi\ en C_0^2 (\ Omega),\]
ver un ejercicio. Esta fórmula se deriva de consideraciones similares a la siguiente sección.
En el lenguaje de distribución, esta relación puede escribirse por definición como
$$-\ triangle_x\ gamma (x, y) =\ delta (x-y),\]
donde\(\delta\) está la distribución Dirac, que se llama\(\delta\) -function.