Extrapolation of well posedness for higher order elliptic systems with rough coefficients

In this paper we study boundary value problems for higher order elliptic differential operators in divergence form. We establish well posedness for problems with boundary data in Besov spaces $\dot B^{p,p}_s$, $p\leq 1$, given well posedness for appropriate values of $s$ and $p>1$. We work with smoothness parameter $s$ between $0$ and $1$; this allows us to consider inhomogeneous differential equations. Combined with results of Maz'ya, I. Mitrea, M. Mitrea, and Shaposhnikova, this allows us to establish new well posedness results for higher order operators whose coefficients are in or close to the space $VMO$, for the biharmonic operator, and for fourth-order operators close to the biharmonic operator.