7.1: Fundamental Solution
- Page ID
- 2162
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Here we consider particular solutions of the Laplace equation in \(\mathbb{R}^n\) of the type
$$u(x)=f(|x-y|),\]
where \(y\in\mathbb{R}^n\) is fixed and \(f\) is a function which we will determine such that \(u\) defines a solution if the Laplace equation.
Set \(r=|x-y|\), then
\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}\right)\\
\triangle u&=&f''(r)+\frac{n-1}{r}f'(r).
\end{eqnarray*}
Thus a solution of \(\triangle u=0\) is given by
$$f(r)=\left\{\begin{array}{r@{\quad:\quad}l}
c_1\ln r+c_2&n=2\\
c_1r^{2-n}+c_2&n\ge3
\end{array}\right.\]
with constants \(c_1\), \(c_2\).
Definition. Set \(r=|x-y|\). The function
$$
s(r):=\left\{\begin{array}{r@{\quad:\quad}l}
-\frac{1}{2\pi}\ln r&n=2\\
\frac{r^{2-n}}{(n-2)\omega_n}&n\ge3
\end{array}\right.
\]
is called singularity function associated to the Laplace equation. Here is \(\omega_n\) the area of the n-dimensional unit sphere which is given by\(\omega_n=2\pi^{n/2}/\Gamma(n/2)\), where
$$\Gamma(t):=\int_0^\infty\ e^{-\rho}\rho^{t-1}\ d\rho,\ \ t>0,\]
is the Gamma function.
Definition. A function
$$\gamma(x,y)=s(r)+\phi(x,y)\]
is called fundamental solution associated to the Laplace equation if \(\phi\in C^2(\Omega)\) and \(\triangle_x\phi=0\) for each fixed \(y\in\Omega\).
Remark. The fundamental solution \(\gamma\) satisfies for each fixed \(y\in\Omega\) the relation
$$-\int_\Omega\ \gamma(x,y)\triangle_x\Phi(x)\ dx=\Phi(y)\ \ \mbox{for all}\ \Phi\in C_0^2(\Omega),\]
see an exercise. This formula follows from considerations similar to the next section.
In the language of distribution, this relation can be written by definition as
$$-\triangle_x\gamma(x,y)=\delta(x-y),\]
where \(\delta\) is the Dirac distribution, which is called \(\delta\)-function.
Contributors and Attributions
Integrated by Justin Marshall.