We establish solvability methods for strongly elliptic second order systems in divergence form with lower order (drift) terms on a domain above a Lipschitz graph, satisfying L p -boundary data for p near 2. The main novel aspect of our result is that the coefficients of the operator do not have to be constant or have very high regularity, instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A particular example of a system where this result can be applied is the Lamé operator for isotropic inhomogeneous materials.The systems case poses substantial new challenges not present in the scalar case. In particular, there is no maximum principle for general elliptic systems and De Giorgi -Nash -Moser theory is also not available. Despite this we are able to establish estimates for the square and nontangential maximal functions for the solution of the elliptic system and use these estimates to establish L p solvability for p near 2.

We establish L p , 2 ≤ p ≤ ∞ solvability of the Dirichlet boundary value problem for a parabolic equation ut − div(A∇u) − B · ∇u = 0 on time-varying domains with coefficient matrices A = [a ij ] and B = [b i ] that satisfy a small Carleson condition. The results are sharp in the following sense. For a given value of 1 < p < ∞ there exists operators that satisfy Carleson condition but fail to have L p solvability of the Dirichlet problem. Thus the assumption of smallness is sharp. Our results complements results of [18,31,32] where solvability of parabolic L p (for some large p) Dirichlet boundary value problem for coefficients that satisfy large Carleson condition was established. We also give a new (substantially shorter) proof of these results.In this metric, we consider the distance function δ of a point (X, t) to the boundary ∂Ω δ(X, t) = inf (Y,τ )∈∂Ω d[(X, t), (Y, τ )].

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We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has C 1,1 boundary. We also obtain pointwise bounds for the Green functions and its derivatives.

We consider a coupled system consisting of a degenerate porous medium type of Keller-Segel system and Stokes system modeling the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global existence of weak solutions and Hölder continuous solutions in dimension three, under the assumption that the power of degeneracy is above a certain number depending on given parameter values. To show Hölder continuity of weak solutions, we consider a single degenerate porous medium equation with lower order terms, and via a unified method of proof, we obtain Hölder regularity, which is of independent interest.