Minkowski inequality - Revision history

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en>Spinoza1989: /* Minkowski's integral inequality */

2014-02-10T08:57:33Z

Minkowski's integral inequality

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	~~In [[physics]], the '''screened Poisson equation'''~~ is ~~a [[partial differential equation]], which arises in (for example) [[Hideki Yukawa\|Yukawa~~'s~~]] theory of [[meson]]s and [[electric field screening]] in [[Plasma (physics)\|plasma]]s.~~

	~~==Statement of~~ the ~~equation==~~

	~~:<math>~~
	~~\left[ \Delta - \lambda^2 \right] u(\mathbf{r}) = - f(\mathbf{r})~~
	~~</math>~~

	~~Where <math>\Delta</math> is the [[Laplace operator]], ''λ'' is~~ a ~~constant, ''f'' is an arbitrary function~~ of ~~position (known as the "source function") and ''u'' is the function to be determined~~.

	~~In the homogenous case (f=0), the screened Poisson equation is the same as the time-independent [[Klein–Gordon equation]]~~. In the inhomogeneous case, the screened Poisson equation is very similar to the [[Helmholtz_equation#Inhomogeneous_Helmholtz_equation\|inhomogeneous Helmholtz equation]], the only difference being the sign within the brackets.

	~~==Solutions==~~

	~~===Three dimensions===~~

	~~Without loss of generality, we will take ''λ~~'~~' to be non-negative~~. ~~When ''λ''~~ is ~~[[0 (number)\|zero]], the equation reduces to [[Poisson's equation]]~~. Therefore, when ''λ'' is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension <math>n=3</math>, is a superposition of 1/''r'' functions weighted by the source function ''f'':

	:~~<math>~~
	~~u(\mathbf{r})_{(\text{Poisson})} = \iiint \mathrm{d}^3r' \frac{f(\mathbf{r}')}{4\pi \|\mathbf{r} - \mathbf{r}'\|}.~~
	~~</math>~~

	~~On the other hand, when ''λ'' is extremely large, ''u'' approaches the value ''f~~/~~λ²'', which goes to zero as ''λ'' goes to infinity. As we shall see, the solution for intermediate values of ''λ'' behaves as a superposition of '''screened''' (or damped) 1~~/~~''r'' functions, with ''λ'' behaving as the strength of the screening.~~

	~~The screened Poisson equation can be solved for general ''f'' using the method of~~ [~~[Green's function]]s. The Green's function ''G'' is defined by~~

	:~~<math>~~
	~~\left[ \Delta - \lambda^2 \right] G(\mathbf{r}) = - \delta^3(\mathbf{r}).~~
	~~</math>~~

	~~Assuming ''u'' and its derivatives vanish at large ''r'', we may perform a [[continuous Fourier transform]] in spatial coordinates:~~

	~~:<math>~~
	~~G(\mathbf{k}) = \iiint \mathrm{d}^3r \; G(\mathbf{r}) e^{-i \mathbf{k} \cdot \mathbf{r}}~~
	~~</math>~~

	~~where the integral is taken over all space. It is then straightforward to show that~~

	~~:<math>~~
	~~\left[ k^2 + \lambda^2 \right] G(\mathbf{k}) = 1.~~
	</~~math>~~

	~~The Green's function in ''r'' is therefore given by the inverse Fourier transform,~~

	~~:<math>~~
	~~G(\mathbf{r}) = \frac{1}{(2\pi)^3} \; \iiint \mathrm{d}^3\!k \; \frac{e^{i \mathbf{k} \cdot \mathbf{r}}}{k^2 + \lambda^2}.~~
	</~~math>~~

	~~This integral may be evaluated using [[Spherical coordinate system\|spherical coordinates]] in ''k''-space~~. ~~The integration over the angular coordinates is straightforward, and the integral reduces to one over the radial [[wavenumber]] <math> k_r </math>:~~

	~~:<math>~~
	~~G(\mathbf{r}) = \frac{1}{2\pi^2 r} \; \int_0^{+\infty} \mathrm{d}k_r \; \frac{k_r \, \sin k_r r }{k_r^2 + \lambda^2}~~.
	</~~math>~~

	~~This may be evaluated using [[line integral\|contour integration]]. The result is:~~

	~~:<math>~~
	~~G(\mathbf{r}) = \frac{e^{- \lambda r}}{4\pi r}.~~
	</~~math>~~

	~~The solution to the full problem is then given by~~

	~~:<math>~~
	~~u(\mathbf{r})~~ = ~~\int \mathrm{d}^3r' G(\mathbf{r} - \mathbf{r}') f(\mathbf{r}')~~
	~~= \int \mathrm{d}^3r' \frac{e^{- \lambda \|\mathbf{r} - \mathbf{r}'\|}}{4\pi \|\mathbf{r} - \mathbf{r}'\|} f(\mathbf{r}').~~
	~~</math>~~

	As stated above, this is a superposition of screened 1/''r'' functions, weighted by the source function ''f'' and with ''λ'' acting as the strength of the screening. The screened 1/''r'' function is often encountered in physics as a screened Coulomb potential, also called a "[[Yukawa potential]]".

	~~===Two dimensions===~~
	~~In two dimensions:~~
	~~In the case of a magnetized plasma, the [[screened Poisson equation]] is quasi-2D:~~
	: <~~math~~> ~~\left( \Delta_\perp -\frac{1}{\rho^2} \right)u(\mathbf{r}_\perp) = -f(\mathbf{r}_\perp)~~ <~~/math>~~
	~~with <math>\Delta_\perp=\nabla\cdot\nabla_\perp</math> and <math>\nabla_\perp=\nabla-\frac{\mathbf{B}}{B}\cdot \nabla</math>, with <math>\mathbf{B}</math> the magnetic field and <math>\rho</math~~> is ~~the (ion) [[Larmor radius]].~~
	~~The two~~-~~dimensional [[Fourier Transform]]~~ of ~~the associated [[Green's function]] is:~~
	~~: <math> G(\mathbf{k_\perp}) = \iint d^2 r~G(\mathbf{r}_\perp)e^{~~-~~i\mathbf{k}_\perp\cdot\mathbf{r}_\perp}. </math>~~
	~~The 2D [~~[~~screened Poisson equation]] yields~~:
	~~: <math> \left( k_\perp^2 +\frac{1}{\rho^2} \right)G(\mathbf{k}_\perp) = 1 <~~/~~math>.~~
	~~The [[Green's function]] is therefore given by the [[inverse Fourier transform]]:~~
	~~:<math>~~
	~~G(\mathbf{r}_\perp) = \frac{1}{4\pi^2} \; \iint \mathrm{d}^2\!k \; \frac{e^{i \mathbf{k}_\perp \cdot \mathbf{r}_\perp}}{k_\perp^2 + 1~~ / ~~\rho^2}~~.
	~~</math>~~
	~~This integral can be calculated using [[polar coordinates]] in [[k-space]]:~~
	~~: <math> \mathbf{k}_\perp = (k_r\cos(\theta),k_r\sin(\theta)) </math>~~
	~~The integration over~~ the ~~angular coordinate gives a [[Bessel function]~~], ~~and the integral reduces to one over the radial [[wavenumber]] <math> k_r </math>:~~
	~~:<math>~~
	~~G(\mathbf{r}_\perp) = \frac{1}{2\pi} \; \int_{0}^{+\infty} \mathrm{d}k_r \; \frac{k_r \, J_0(k_r r_\perp)}{k_r^2 + 1 / \rho^2} = \frac{1}{2\pi} K_0(r_\perp \, / \, \rho).~~
	~~</math>~~

	~~==See also==~~
	* [[Yukawa interaction]]

	~~{{DEFAULTSORT:Screened Poisson Equation}}~~
	~~[[Category:Partial differential equations]]~~
	~~[[Category:Plasma physics]]~~
	~~[[Category:Electrostatics]]~~

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