**Laplasova jednačina'** je eliptička parcijalna diferencijalna jednačina drugoga reda oblika:

{\displaystyle \qquad \nabla ^{2}\varphi =0}
Rešenja Laplasove jednačine su harmoničke funkcije. Laplasova jednačina je značajna u matematici, elektromagnetizmu, astronomiji i dinamici fluida.

Definicija
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U tri demenzije Laplasiva jednačina može da se prikaže u različitim koordinatnim sistemima.
U kartezijevom koordinatnom sistemu je oblika:

{\displaystyle \Delta f={\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}=0.}
U cilindričnom koordinatnom sistemu je:

{\displaystyle \Delta f={1 \over r}{\partial \over \partial r}\left(r{\partial f \over \partial r}\right)+{1 \over r^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}=0}
U sfernom koordinatnom sistemu je:

{\displaystyle \Delta f={1 \over \rho ^{2}}{\partial \over \partial \rho }\!\left(\rho ^{2}{\partial f \over \partial \rho }\right)\!+\!{1 \over \rho ^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over \rho ^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}=0.}
U zakrivljenom koordinatnom sistemu je:

{\displaystyle \Delta f={\partial \over \partial \xi ^{i}}\!\left({\partial f \over \partial \xi ^{k}}g^{ki}\right)\!+\!{\partial f \over \partial \xi ^{j}}g^{jm}\Gamma \_{mn}^{n}=0,}
ilir

{\displaystyle \Delta f={1 \over {\sqrt {|g|}}}{\partial \over \partial \xi ^{i}}\!\left({\sqrt {|g|}}g^{ij}{\partial f \over \partial \xi ^{j}}\right)=0,\quad (g=\mathrm {det} \{g\_{ij}\}).}

Dvodimenzionalni sistem
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U polarnom koordinatnom dvodimenzionalnom sistemu je oblika:

{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial u}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}u}{\partial \phi ^{2}}}=0}
U dvodimenzionalnom kartezijevom sistemu je:

{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}

Grinova funkcija
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Laplasova jednačina se često rešava uz pomoć Grinove funkcije i Grinova teorema:

{\displaystyle \int \_{V}(\phi \nabla ^{2}\psi -\psi \nabla ^{2}\phi )dV=\int \_{S}(\phi \nabla \psi -\psi \nabla \phi )\cdot d{\hat {\sigma }}.}
Definicija Grinove funkcije je:

{\displaystyle \nabla ^{2}G(x,x')=\delta (x-x').}
Uvrstimo u Grinov teorem {\displaystyle \psi =G} pa dobijamo:

{\displaystyle {\begin{aligned}&{}\quad \int \_{V}\left[\phi (x')\delta (x-x')-G(x,x')\nabla ^{2}\phi (x')\right]\ d^{3}x'\\[6pt]&=\int \_{S}\left[\phi (x')\nabla 'G(x,x')-G(x,x')\nabla '\phi (x')\right]\cdot d{\hat {\sigma }}'.\end{aligned}}}
Sada možemo da rešimo Laplasovu jednačinu {\displaystyle \nabla ^{2}\phi (x)=0} u slučaju Nojmanovih ili Dirihleovih rubnih uslova. Uzimajući u obzir:

{\displaystyle \int \limits \_{V}{\phi (x')\delta (x-x')\ d^{3}x'}=\phi (x)}
pa se jednačina svodi na:

{\displaystyle \phi (x)=\int \_{V}G(x,x')\rho (x')\ d^{3}x'+\int \_{S}\left[\phi (x')\nabla 'G(x,x')-G(x,x')\nabla '\phi (x')\right]\cdot d{\hat {\sigma }}'.}
Kada nema rubnih uslova Grinova funkcija je:

{\displaystyle G(x,x')={\dfrac {1}{|x-x'|}}.}

Literatura
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* Sommerfeld A, Partial Differential Equations in Physics, New York: Academic Press (1949)
* Korn GA and Korn TM. (1961) *Mathematical Handbook for Scientists and Engineers*, McGraw-Hill.
* Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York:. Šablon:Page1
* Laplasova jednačina