Elliptic and parabolic equations with rough boundary data in Sobolev spaces with degenerate weights

Abstract: We investigate the inhomogeneous boundary value problem for elliptic and parabolic equations in divergence form in the half space ${x_d > 0}$, where the coefficients are measurable, singular or degenerate, and depend only on $x_d$. The boundary data are considered in Besov spaces of distributions with negative orders of differentiability in the range $(-1,0]$. The solution spaces are weighted Sobolev spaces with power weights that decay rapidly near the boundary, and are outside the Muckenhoupt $A_p$ class. Sobolev spaces with such weights contain functions that are very singular near the boundary and do not possess a trace on the boundary. Consequently, solutions may not exist for arbitrarily prescribed boundary data and right-hand sides of the equations. We establish a natural structural condition on the right-hand sides of the equations under which the boundary value problem is well-posed.