Energia di Dirichlet

In matematica, l'energia di Dirichlet, il cui nome si deve a Peter Gustav Lejeune Dirichlet, è un funzionale quadratico definito sullo spazio di Sobolev $H^{1}$ che è strettamente legato all'equazione di Laplace.

Dato un insieme aperto $\Omega \subseteq \mathbb {R} ^{n}$ e una funzione $u:\Omega \to \mathbb {R}$ , l'energia di Dirichlet è il numero reale:

E[u]={\frac {1}{2}}\int _{\Omega }\|\nabla u(x)\|^{2}\,\mathrm {d} V

dove $\nabla u:\Omega \to \mathbb {R} ^{n}$ è il campo vettoriale gradiente di $u$ . Dal momento che l'integrale è una quantità non-negativa, l'energia di Dirichlet è essa stessa non-negativa, ovvero $E[u]\geq 0$ per ogni scelta di $u$ .

Risolvere l'equazione di Laplace:

-\Delta u(x)=0\qquad \forall x\in \Omega

con appropriate condizioni al contorno, è equivalente alla soluzione del problema variazionale di trovare la funzione $u$ che soddisfa le condizioni al contorno e minimizza l'energia di Dirichlet.

Bibliografia

(EN) Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, 1998, ISBN 978-0-8218-0772-9.

Voci correlate

Collegamenti esterni

(EN) Eric W. Weisstein, Energia di Dirichlet, su MathWorld, Wolfram Research.
(EN) Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička - Dirichlet Energy Integral and Laplace Equation (Springer)

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