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Let $\Omega \subset R^n$ be bounded and open , $u\in C^2(\Omega)\cap C(\bar \Omega)$ be a solution of $-\triangle u=f$ in $\Omega$ , $u=0$ on $\partial \Omega$. Prove that there exists a constant $C$, depending only on $n$ and $diam(\Omega )$ such that $||u||_{L^\infty} \le C ||f||_{L^\infty}$ . I have got some idea like comparing $u$ with parabolas and using maximum principle or so . Some how i am not able do anything on this problem . Any kind of help or solution is appreciated. Thanks a lot.

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1 Answer 1

up vote 3 down vote accepted

Your basic idea is correct.

Let $M = \|f\|_\infty$, and let $x_0\in\Omega$. The functions

$$ \tilde{u}^\pm(x) = u(x) \pm \frac{M}{2n} (x - x_0)^2 $$

are seen to have the property that

$$ -\triangle \tilde{u}^+ = -\triangle u - M = f - M \leq 0 $$

is subharmonic and

$$ -\triangle \tilde{u}^- = - \triangle u + M = f + M \geq 0 $$

is superharmonic.

So by the maximal/minimal principles for sub/super harmonic functions you have that

$$ u(x) \leq \tilde{u}^+(x) \leq \sup_{\partial\Omega} \tilde{u}^+ \leq \frac{M}{2n} \mathrm{diam}(\Omega)^2 $$

and

$$ u(x) \geq \tilde{u}^-(x) \geq \inf_{\partial\Omega}\tilde{u}^- \geq - \frac{M}{2n} \mathrm{diam}(\Omega)^2 $$

And so you get

$$ \|u\|_\infty \leq \frac{\mathrm{diam}(\Omega)^2}{2n} \|f\|_\infty $$

and the constant indeed depends only on the dimension and the diameter of the set $\Omega$.

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Thanks. I would be glad if you could tell me the construction of the first equation .Why did you particularly choose that function ? –  Theorem Jun 19 '12 at 15:32
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@Theorem: I choose those two functions so that they are sub and super harmonic respectively :-p. The basic idea is the same as the following fact: "any $C^2$ function can be made convex by adding to it a sufficiently convex function". Similarly, any function with bounded Laplacian can be made to be subharmonic by adding to it a sufficiently subharmonic function. Adding/subtracting or (as you observed) comparing against subharmonic functions (often paraboloids) is a very convenient tool when one wants to use the maximum principle. –  Willie Wong Jun 19 '12 at 15:40
    
Sir nicely explained . Thank you. If you have some reference text where i can learn more about it i would be interested . –  Theorem Jun 19 '12 at 15:43
1  
Umm... that's a hard question. It is sort of a general toolkit you just pick up. I'd say try Gilbarg and Trudinger Chapter 3. Similar constructions are also used for the "barrier method" in constructing solutions to elliptic PDEs, as well as for constructions using viscosity solutions. –  Willie Wong Jun 19 '12 at 15:57
    
Sir, i have a doubt here , can you wrote the inequality relating $sup$ with $M, diam(\Omega)$ i am not able to relate it. –  Theorem Jul 23 '12 at 17:53

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