Let $\Omega\subset\mathbb{R}^2$ be a bounded open set with Lipschitz boundary. Let $v\in H^{l+1}(\Omega)$. I have to prove that
There exists a unique $p\in\mathbb{P}_l$ such that $$ \int_\Omega D^\alpha v=-\int_\Omega D^\alpha p $$ for every $|\alpha|\le l$, where $\mathbb{P}_l$ is the space of real polynomials defined on $\Omega$ and of degree less or equal to $l$.
Do you have any ideas?
P.S. Merry Christmas :)
Let $$ p(x) = \sum_{|\beta|\leq l} c_\beta x^\beta $$ for some coefficients $c_\alpha$, which you have to find based on the system of equations $$\int_\Omega D^\alpha p=\sum_{|\beta|\leq l,|\beta|\geq|\alpha|}c_\beta D^\alpha x^{\beta} = -\int_\Omega D^\alpha v,\qquad \forall \ |\alpha|\leq l.$$ The uniqueness comes from the fact this is a tri-diagonal system. In other words, you can solve the coefficients $c_\beta$ for the highest order $x^{\beta}$ first for $|\beta|=l$, and then the coefficients $c_\beta$ for $|\beta|=l-1$, etc.