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i have this exercice: Let $f\in L^2(\mathbb{R}^n)$. 1- Prouve that the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admeit a unique solution $u \in H^1(\mathbb{R}^n)$?

2- Prouve the exustance of constant $C \geq 0$ such that $||u||_{H^1} \leq C ||f||_{L^2}$?

3- Prouve the existante of a constant $M \geq 0$ such that for all $u \in H^2(\mathbb{R}^n)$ we have $||u||_{H^2} \leq M (||u||_{L^2}+||\Delta u||_{L^2}$.

For question 1, i suppose existence of two solutions $u_1$ and $u_2$, and i set $w=u_1-u_2$ solution of $\Delta w = w =0$ and my prolem is what we can this imply that $w=0$? and how we prouve the existence? For question 2 and 3, i don't have idea. Thank's for the help.

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  • $\begingroup$ Try Lax-Milgram theorem $\endgroup$ – daw May 5 '14 at 12:06
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Since you tagged this (distribution-theory) I'm going to assume that you're familiar with tempered distributions and the Fourier transform. Then all three questions can be solved easily. Uniqueness follows from $$\Delta u - u = \partial_k f \iff -\|\xi\|^2 \mathcal{F}u - \mathcal{F}u = (-i\xi_k)\mathcal{F}f \iff \mathcal{F}u = \frac{i\xi_k}{\|\xi\|^2+1}\mathcal{F}f$$ This also proves existence of a solution in $\mathcal{S}'$. The characterizations $H^s(\mathbb{R}^n) = \{u\in\mathcal{S}' \mid (1+\|\xi\|^2)^{s/2} \mathcal{F}u \in L^2\}$ and $\|u\|_{H^s} = \|(1+\|\xi\|^2)^{1/2}\mathcal{F}u\|_{L^2}$ (*) together with $(1+\|\xi\|^2)^{1/2}\mathcal{F}u = \frac{i\xi_k}{(1+\|\xi\|^2)^{1/2}} \mathcal{F}f$ show that $u$ is in fact in $H^1$ and gives the inequality $$\|u\|_{H^1} = \|(1+\|\xi\|^2)^{1/2} \mathcal{F}u\|_{L^2} \leq \| \frac{i\xi_k}{(1+\|\xi\|^2)^{1/2}} \|_{L^\infty} \|f\|_{L^2} \leq \|f\|_{L^2}$$ as well as $$\|u\|_{H^2} = \|(1+\|\xi\|^2)\mathcal{F}u\|_{L^2} = \|\mathcal{F}u - \mathcal{F}(\Delta u)\|_{L^2} \leq \|u\|_{L^2} + \|\Delta u\|_{L^2}$$


(*) Your definition of the Sobolev-Norm might differ, but the norm I'v given is equivalent to yours. So one can choose $C=M=1$ if you choose my norm, but might have to use other constants if you don't.

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