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Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$. ...

Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE $$ \begin{cases} -\Delta_p u=f\;\text{in $D$}...

I want to solve the equation: $$ \begin{cases} \nabla \times (A \mathbf v)=f, \quad x\in \Omega \\ \operatorname{div}(\mathbf v)=0, \end{cases} $$ where $\Omega \subset\mathbb{R}^n$, is an open set, $...