| Abstract: |
| We investigate the relationship between Sobolev-type embeddings and the solvability of the divergence equation $\mathrm{div}\,u = f$ in the setting of Banach function spaces. Given two spaces $X, Y \subset L^{1}(Q_{0})$ where $Q_0=[0,1]^n$, we denote by $\mathbf{Y}$ the space of vector fields whose components belong to $Y$, and by $X`$, $Y`$ their associate spaces. We show that the existence of solutions $u \in \mathbf{Y}$ to $\mathrm{div}\,u = f$ for every $f \in X$ is equivalent to a dual Sobolev embedding of the form
\[
W^{1}_{0}Y` \hookrightarrow X`.
\]
This establishes a general duality principle linking divergence-type equations and Sobolev embeddings beyond the classical Lebesgue framework. We also present examples highlighting borderline phenomena, where standard regularity and gradient representation fail.
This work is based on a joint project with Gianluigi Manzo. |
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