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The simplest case of the problem I'm thinking about involves an elliptic differential operator, $Lu = -u'' + qu$, on the interval $(0,1)$, with homogeneous Dirichlet boundary conditions. I want to show that the bilinear form on $H_0^1 \subset H_1$ defined by

$a(u,v) = \int_0^1 u'v' + quv~dx$

is bounded for the $H^1$-norm, i.e., $|a(u,v)| \leq M\|u\|_1\|v\|_1$ for some constant $M>0$.

My question: can I assume that the linear coefficient $q$ is $L^1$ or even $L^2$ and still guarantee boundedness?

I was thinking that this is possible, but the only books that I have lying around discussing this consider only the case when $q$ is smooth or $L^\infty$. I've played around with the Cauchy-Schwartz inequality for the term $\int quv$ but am not getting anywhere.

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I think this is just a standard Sobolev inequality... – Helge Aug 13 '10 at 18:25
up vote 1 down vote accepted

Of course, as Helge says, the $H^1(0,1)$ norm controls the $L^\infty(0,1)$ norm, so you just write $$ \left| \int quv \right| \le \|q\|_{L^1} \|u\| _{L^\infty} \|v\| _{L^\infty} \le C \|q\| _{L^1} \|u\| _{H^1} \|v\| _{H^1}.$$

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And to control the $L^2$ case, use this estimate once and C-S on $\langle q, u \rangle$. – Helge Aug 13 '10 at 21:30
Right, thanks! I was hoping it would be that straightforward. Was just suffering from temporary myopia. – Jerry Gagelman Aug 13 '10 at 22:36

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