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I am trying to estimate, in either direction, the following integral:

$\int_{B} a^{i,j} X_j D_ig dx$ . We are integrating over a unit ball, $B$, and $[a^{i,j}]$ is positive.

I have tried to use ellipticity, but it didn't work. I was wondering if some one can show me: a. How to use ellipticity to estimate the integral, or b. Is there any other inequality that can be useful to estimating the integral. Thanks in advance.

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Can you write down the form at least of the estimate that you are hoping to prove? –  Glen Wheeler Feb 11 '11 at 17:55
    
Thank you so much guys! Very helpful. –  user7015 Feb 13 '11 at 11:33
    
@Praire: please do not use answers to comment. –  Qiaochu Yuan Feb 13 '11 at 11:33
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1 Answer

Estimating above is just a matter of Cauchy-Schwarz (if you also assume $a^{ij}$ is symmetric). Since $a^{ij}$ is positive, you can define the matrix $[b] = [a]^{1/2}$, and rewrite

$$ \int_B a^{ij}X_jD_ig dx = \int_B \sum_k b^{ik}b^{kj}X_jD_ig dx $$

apply Cauchy-Schwarz to the terms $V^k = b^{ik}D_ig$ and $W^k = b^{kj}X_j$ you get

$$ \left\lvert\int_B a^{ij}X_jD_ig dx\right\rvert \leq \left(\int_B V^kV^k dx\right)^{1/2}\left(\int_B W^kW^k dx\right)^{1/2} = \left(\int_B a^{ij}X_jX_i dx\right)^{1/2} \left(\int_B a^{ij}D_ig D_jg dx\right)^{1/2} $$

In general: Cauchy-Schwarz can be used for any positive semi-definite symmetric bilinear form.

One direction of the ellipticity definition can then let you control the RHS by a constant factor times

$$ \left(\int_B |X|^2 dx\right)^{1/2} \left(\int_B |Dg|^2 dx\right)^{1/2} $$


Another thing you can do, if you know the behaviour of $u$ on the boundary of $B$, is to integrate by parts:

$$ \int_{B} a^{ij}X_j D_i g dx = \int_{\partial B} a^{ij}X_j n_i g d\sigma - \int_{B} g D_i( a^{ij}X_j ) dx $$

which can be useful if you know something about the derivatives of the vector $a^{ij}X_j$. $n_i$ in the first term on the right hand side is just the unit out-ward normal on $\partial B$.


In general you cannot estimate from below. Ellipticity does not rule out the possibility that $X_j$ and $D_i g$ are pointwise orthogonal relative to the bilinear form $a^{ij}$.


If you say more precisely what kinds of estimates you are looking for, or what possible additional properties about $X_j$ and $D_i g$ that you have, I may be able to give more precise answers.

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@Theo: many, many thanks. I have no idea how I managed to type the first one right while screwing up the next two times. –  Willie Wong Feb 11 '11 at 15:04
    
No problem, it's actually a pet peeve of mine, like Leibniz. +1 by the way for this very nice answer. With the little data you had to begin with, you couldn't have been more helpful. –  t.b. Feb 11 '11 at 15:23
    
+1 for somehow answering such a vague and ill-defined question. –  Glen Wheeler Feb 11 '11 at 17:54
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