Ngô Quốc Anh

April 25, 2011

The Monge-Ampère equations: An invariant

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 3:16

Howdy, today, we study a simple invariant concerning the Monge-Ampère equations. Let us consider the following simple case

\displaystyle\det (u_{x_ix_j})=u^p

in \mathbb R^n, where 0<p<n. For a beginner, we refer to an entry from wikipedia [here].

We now look for a function of the form

w(x)=m^\alpha u(mx)

where \alpha is a constant to be determined. A quick calculation shows that

w_{x_i}(x)=m^{\alpha+1} u_{x_i}(mx)

and

w_{x_ix_j}(x)=m^{\alpha+2} u_{x_ix_j}(mx).

Thus

\displaystyle \det (w_{x_ix_j}) = m^{(\alpha+2)n}\det (u_{x_ix_j}(mx))=m^{(\alpha+2)n} u(mx)^p=m^{(\alpha+2)n-p\alpha} w(x)^p.

Therefore, \alpha is chosen so that (\alpha+2)n-p\alpha=0. Hence

\displaystyle\alpha=\frac{2n}{p-n}.

In conclusion,

\displaystyle m^\frac{2n}{p-n} u(mx)

is also a solution. This property can be found in a paper by Kutev [here].

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