The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients

TL;DR

This paper proves well-posedness for the Dirichlet problem of higher order elliptic systems with rough coefficients in Lipschitz domains, using boundary data in Besov spaces, under optimal oscillation conditions.

Contribution

It establishes the well-posedness of the Dirichlet problem for complex-valued elliptic systems with minimal regularity assumptions on coefficients and boundary geometry.

Findings

Well-posedness is achieved under optimal oscillation conditions.

The results apply to systems with rough, measurable coefficients.

Boundary data in Besov spaces is effectively handled.

Abstract

We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in Besov spaces. The main hypothesis under which our principal result is established is in the nature of best possible and requires that, at small scales, the mean oscillations of the unit normal to $\partial\Omega$ and of the coefficients of the differential operator ${\mathcal L}(x,D_x)$ are not too large.