Pokhozhaev's identity

Here is a general form due to H. Berestycki and P.-L. Lions.{{ cite journal

|author=Berestycki, H. and Lions, P.-L.

|title=Nonlinear scalar field equations, I. Existence of a ground state

|journal=Arch. Rational Mech. Anal.

|volume=82

|issue=4

|year=1983

|pages=313–345

|doi=10.1007/BF00250555

|bibcode=1983ArRMA..82..313B

|s2cid=123081616

}}

Let $g(s)$ be continuous and real-valued, with $g(0)=0$.

Denote $G(s)=\backslash int\_0^s\; g(t)\backslash ,dt$.

Let

:$u\backslash in\; L^\backslash infty\_\{\backslash mathrm\{loc\}\}(\backslash R^n),$

\qquad

\nabla u\in L^2(\R^n),

\qquad

G(u)\in L^1(\R^n),

\qquad

n\in\N,

be a solution to the equation

:$-\backslash nabla^2\; u=g(u)$,

in the sense of distributions.

Then $u$ satisfies the relation

:$\backslash frac\{n-2\}\{2\}\backslash int\_\{\backslash R^n\}|\backslash nabla\; u(x)|^2\backslash ,dx=n\backslash int\_\{\backslash R^n\}G(u(x))\backslash ,dx.$

There is a form of the virial identity for the stationary nonlinear Dirac equation in three spatial dimensions (and also the Maxwell-Dirac equations){{ cite journal

|author=Esteban, M. and Séré, E.

|title=Stationary states of the nonlinear Dirac equation: A variational approach

|journal=Commun. Math. Phys.

|volume=171

|issue=2

|year=1995

|pages=323–350

|doi=10.1007/BF02099273

|bibcode=1995CMaPh.171..323E

|s2cid=120901245

|url=http://projecteuclid.org/euclid.cmp/1104273565

}} and in arbitrary spatial dimension.{{ cite book

|author=Boussaid, N. and Comech, A.

|title=Nonlinear Dirac equation. Spectral stability of solitary waves

|publisher=American Mathematical Society

|series=Mathematical Surveys and Monographs

|volume=244

|year=2019

|doi=10.1090/surv/244

|isbn=978-1-4704-4395-5

|s2cid=216380644

}}

Let $n\backslash in\backslash N,\backslash ,N\backslash in\backslash N$

and let $\backslash alpha^i,\backslash ,1\backslash le\; i\backslash le\; n$ and $\backslash beta$ be the self-adjoint Dirac matrices of size $N\backslash times\; N$:

:$$

\alpha^i\alpha^j+\alpha^j\alpha^i=2\delta_{ij}I_N,

\quad

\beta^2=I_N,

\quad

\alpha^i\beta+\beta\alpha^i=0,

\quad

1\le i,j\le n.

Let $D\_0=-\backslash mathrm\{i\}\backslash alpha\backslash cdot\backslash nabla=-\backslash mathrm\{i\}\backslash sum\_\{i=1\}^n\backslash alpha^i\backslash frac\{\backslash partial\}\{\backslash partial\; x^i\}$ be the massless Dirac operator.

Let $g(s)$ be continuous and real-valued, with $g(0)=0$.

Denote $G(s)=\backslash int\_0^s\; g(t)\backslash ,dt$.

Let $\backslash phi\backslash in\; L^\backslash infty\_\{\backslash mathrm\{loc\}\}(\backslash R^n,\backslash C^N)$ be a spinor-valued solution that satisfies the stationary form of the nonlinear Dirac equation,

:$$

\omega\phi=D_0\phi+g(\phi^\ast\beta\phi)\beta\phi,

in the sense of distributions,

with some $\backslash omega\backslash in\backslash R$.

Assume that

:$$

\phi\in H^1(\R^n,\C^N),\qquad

G(\phi^\ast\beta\phi)\in L^1(\R^n).

Then $\backslash phi$ satisfies the relation

:$$

\omega\int_{\R^n}\phi(x)^\ast\phi(x)\,dx

=\frac{n-1}{n}\int_{\R^n}\phi(x)^\ast D_0\phi(x)\,dx

+\int_{\R^n}G(\phi(x)^\ast\beta\phi(x))\,dx.

• Virial theorem
• Derrick's theorem

Category:Mathematical_identities

Category:Theorems in mathematical physics

Category:Physics theorems