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im struggling how to get a weak formulation for the following problem. After this i want to apply the Lax-Milgram Theorem to show that there exists a unique solution to this problem.

$$(*)\begin{cases} -\Delta u +c \partial_{x}^{-1}u=f, & \text{in } \Omega=(0,1)^2 \\ u=0, & \text{on } \partial\Omega \end{cases}$$ where $$\partial_{x}^{-1}u(x,y)=\int_{0}^{x} u(z,y) dz$$ and $$f \in L^2(\Omega) \text{ and } c\in \mathbb{R} \text{ is a constant.}$$

I know, the common way is to multiply by a testfunction from a suitable vectorspace V and use integration by parts. My main problem is how to handle the $ \partial_{x}^{-1} $ operator on the step int by parts and get the bilinear form. Thx in advance!

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  • $\begingroup$ My idea is to apply $\partial_x$ to both sides of the equation, and reformulate with $g =\partial_x f$ known. Then you might be able to make a bilinear form out of the LHS $\endgroup$ – George Dewhirst Apr 1 at 16:57
  • $\begingroup$ @GeorgeDewhirst yea i got your idea, reformulate the right-hand-side is no problem. i have some trouble with the left-hand-side. The application of the $\partial_{x}$ operator on the term $c \partial_{x}^{-1}u$ reduce it to $cu$, since c is a const. But idk if there is a simplification on the term $-\Delta u$ which gets to $ -\partial_{x}\Delta u$. On that step i'm more confused how to apply int by parts and get the bilinear form afterwards. $\endgroup$ – mathastic27182 Apr 2 at 11:40
  • $\begingroup$ Yes i agree with that. I discussed this with my phd friends and we think the bilinear form would be $B(u,v) = <\nabla u, \nabla v> + c(int u)(int v)$ $\endgroup$ – George Dewhirst Apr 2 at 11:48
  • $\begingroup$ This is without reformulation $\endgroup$ – George Dewhirst Apr 2 at 11:49

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