Caccioppoli inequality

A Caccioppoli inequality is of the form

\displaystyle r^p\int_{B_{\frac{r}{2}}(x_0)}|\nabla u|^pdx\leq C\int_{B_r(x_0)}\bigg|u(x)-(u)_{B_{r}(x_0)}\bigg|^pdx\quad\quad (1)

where C is a constant independent of r and u, 1<p<\infty, and \displaystyle (u)_{B_{r}(x_0)}=\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}udx.

Caccioppoli inequality can be considered as the reverse Poincare inequality

\displaystyle \int_{B_r(x_0)}\bigg|u(x)-(u)_{B_{r}(x_0)}\bigg|^pdx\leq Cr^p\int_{B_{r}(x_0)}|\nabla u|^pdx

holds for u\in W^{1,p}(B_r(x_0)).

Of course (1) does not hold for any u\in W^{1,p}(B_r(x_0)), but if u satisfies some second order elliptic equation or certain variational problems, u can satisfies (1).

For example, if u is the weak solution of the divergence form

\displaystyle \text{div}(A\nabla u)=0 or (a^{ij}\partial_ju)_i=0 in B_r(x_0)

here a^{ij} are just bounded measurable and strictly elliptic. \nu I\leq A\leq \nu^{-1}I.

Choose a cut off function \eta\in C^\infty_c(B_r(x_0)) and \eta=1 in B_{\frac{r}{2}}(x_0) and \displaystyle |\nabla\eta|\leq \frac{4}{r}. Let v=\eta^2(u-\overline{u})\in H^1_0(B_r(x_0)), here we use \overline{u}=(u)_{B_{r}(x_0)} for abbreviation. Then

\displaystyle \int_{B_r(x_0)}a^{ij}\partial_ju\partial_iv=0

\displaystyle \int_{B_r(x_0)} a^{ij}\partial_ju\eta^2\partial_iu=-\int_{B_r(x_0)} 2a^{ij}\partial_ju\eta\nabla\eta(u-\overline{u})

\displaystyle \nu\int_{B_r(x_0)}\eta^2|\nabla u|^2dx\leq C\int_{B_r(x_0)} |\partial_ju\eta\nabla\eta(u-\overline{u})|

\displaystyle \int_{B_r(x_0)}\eta^2|\nabla u|^2dx\leq C(\nu)\int_{B_r(x_0)}|\nabla\eta(u-\overline{u})|^2dx\leq \frac{C}{r^2}\int_{B_r(x_0)} |u-\overline{u}|^2dx

Noticing that \eta=1 on B_{\frac{r}{2}(x_0)}, we can get (1)

\textbf{Remark:} If u satisfies (a^{ij}\partial_ju)_i+b^i\partial_iu=0, we can also get the same conclusion.

MA6000A: Theory of Partial Differential Equations. Roger Moser

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Comments

  • Hafid Younsi  On October 28, 2013 at 6:03 pm

    Dear Professor ;

    I am interested in your Website and in the Caccioppoli inequality and in the Navier-Stokes equations.

    I write to you for asked if we can apply the Caccioppoli inequality for the unsteady Stokes system on the weak solutions of the unsteady 3D Navier-Stokes system.

    Or if there existes anothre type Caccioppoli inequality or reverse Poincare Inequality for the unsteady 3D Navier-Stokes equations.

    Best regards,

    Hafid

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