Ngô Quốc Anh

September 30, 2013

The Lichnerowicz equation under some variable changes

Filed under: Linh Tinh, Nghiên Cứu Khoa Học, PDEs — Tags: — Ngô Quốc Anh @ 4:54

Let us consider the so-called Lichnerowicz equation

-\Delta_g u + hu = fu^{2^\star-1}+au^{-2^\star-1} \quad u>0

on (M,g), a Riemannian manifold of dimension n \geq 3. Here h, f, and a are smooth function with a \geq 0.

  • We first use the the following variable change

    \displaystyle v=\log u \quad u=e^ v.

    Clearly,

    \displaystyle\Delta v = \frac{\Delta u}{u} - \frac{|\nabla u|^2}{u^2}

    and

    \displaystyle |\nabla v|^2 = \frac{|\nabla u|^2}{u^2}.

    Therefore, we can write

    \displaystyle -\Delta v =-\frac{\Delta u}{u} +|\nabla v|^2.

    Using this rule, we can rewrite the equation as follows

    \displaystyle \boxed{-\Delta v = -h+fu^{2^\star-2}+au^{-2^\star-2}+|\nabla v|^2=-h+fe^{(2^\star-2)v}+ae^{-(2^\star+2)v}+|\nabla v|^2. }

Clearly, under this variable change, we have killed u after h in the original equation.

  • We now use the change

    \displaystyle v=e^u \quad u=\log v.

    Clearly,

    \displaystyle\Delta v = v\Delta u + v|\nabla u|^2

    and

    \displaystyle |\nabla v|^2 = v^2|\nabla u|^2.

    Therefore, we can write

    \displaystyle -\Delta v =-\frac{|\nabla v|^2}{v}-v\Delta u.

    Again, our PDE becomes

    \displaystyle \boxed{-\Delta v = huv-fvu^{2^\star-1}-avu^{-2^\star-1}-\frac{|\nabla v|^2}{v}. }

This seems that this variable change is not useful and is useful only when we deal with PDEs with nonlinearities of power type.

  • Finally, we use the change

    \displaystyle v=u^\alpha \quad u=v^{1/\alpha}.

    Clearly,

    \displaystyle\Delta v = \alpha(\alpha-1)u^{\alpha-2}|\nabla u|^2+\alpha u^{\alpha-1}\Delta u=\alpha(\alpha-1)v\frac{|\nabla u|^2}{u^2}+\alpha v\frac{\Delta u}{u}

    and

    \displaystyle |\nabla v|^2 = \alpha^2u^{2(\alpha-1)}|\nabla u|^2=\alpha^2v^2\frac{|\nabla u|^2}{u^2}.

    Therefore, we can write

    \displaystyle -\Delta v =-\frac{\alpha-1}{\alpha}\frac{|\nabla v|^2}{v}-\alpha v\frac{\Delta u}{u}.

    Again, our PDE becomes

    \displaystyle \boxed{-\Delta v =-\frac{\alpha-1}{\alpha}\frac{|\nabla v|^2}{v}-\alpha hv+\alpha fvu^{2^\star-2}+\alpha avu^{-2^\star-2}. }

Depending on each problem, we may choose a suitable variable change.

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