# 狄利克雷原理

$\Delta u + f = 0\,$

$\partial\Omega$$u=g \,$

$E[v] = \int_\Omega \left(\frac{1}{2}|\nabla v|^2 - vf\right)\,\mathrm{d}x$

## 证明

$E[v] = \int_\Omega \left(\frac{1}{2}|\nabla v|^2 - vf\right)\,\mathrm{d}x$

$E[u+\varepsilon w] = \int_\Omega \left(\frac{1}{2}|\nabla u + \varepsilon \nabla w|^2 - uf - \varepsilon w f\right)\,\mathrm{d}x \geqslant \int_\Omega \left(\frac{1}{2}|\nabla u|^2 - uf\right)\,\mathrm{d}x$

$\int_\Omega \left(\varepsilon \nabla u \cdot \nabla w + \frac{1}{2} \varepsilon^2 | \nabla w|^2 - \varepsilon w f\right)\,\mathrm{d}x \geqslant 0$

$\int_\Omega \left(\nabla u \cdot \nabla w - w f\right)\,\mathrm{d}x = 0$

\begin{align} 0 &= \int_{\partial\Omega} w \left( \nabla u \cdot \mathbf{n} \right)\,\mathrm{d}\sigma = \int_{\Omega} \operatorname{div} \left( w \cdot \nabla u \right)\,\mathrm{d}x \\ &= \int_{\Omega} \left( w \Delta u + \nabla u \cdot \nabla w \right)\,\mathrm{d}x = \int_{\Omega} w \left( \Delta u + f \right)\,\mathrm{d}x \end{align}

## 参考来源

1. ^ Mark.A.Prinsky. Partial Differential Equations and Boundary Value Problems With Applications. Waveland Pr Inc. 2003. ISBN 978-1577662754.