Ngô Quốc Anh

August 22, 2010

Liouville’s theorem and related problems

Filed under: Giải tích 7 (MA4247), PDEs — Tags: — Ngô Quốc Anh @ 6:35

The following theorem is well-known

Theorem (Liouville). Let \Omega be a simply connected domain in \mathbb R^2. Then all real solutions of

\displaystyle \Delta u +2Ke^u=0

in \Omega where K a constant, are of the form

\displaystyle u=\log\frac{|f'|^2}{\left(1+\frac{K}{4}|f|^2\right)^2}

where f is a locally univalent meromorphic function in \Omega.

In geometry, our PDE

\displaystyle \Delta u +2Ke^u=0

says that under the case \Omega=\mathbb R^2, it holds

e^u|dz|^2=f^*g_K

where g_K denotes the standard metric on \mathbb S^2 with constant curvature K. Thus we have

Corollary. All solutions of the PDE in \mathbb R^2 with K>0 and

\displaystyle \int_{\mathbb R^2} e^u<\infty

are of the form

\displaystyle u(x)=\log\frac{16\lambda^2}{\left(4+\lambda^2K|x-x_0|^2\right)^2},\quad \lambda>0, \quad x_0 \in \mathbb R^2.

(more…)

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