Auscher, Rosen and Rule solve the Dirichlet, Dirichlet-regularity and Neumann problems with L2(w) boundary data for divergence-form second-order elliptic systems div(A grad u)=0 on the upper half-space whose coefficient matrix A is only comparable in size to a Muckenhoupt A2 weight w(x) rather than bounded above and below by constants. Coefficients may be complex and the system may have m components, so no interior pointwise regularity is available and none is used. The second-order system is recast as a first-order Cauchy-Riemann-type evolution equation d_t f + DBf = 0 for the w-normalized conormal gradient f = (w^-1 d_nu u, grad_x u), and everything rests on one weighted quadratic (square-function) estimate for the bisectorial operator DB on L2(w), whose Tb/Carleson-measure proof introduces a corona decomposition sorting dyadic cubes into families on which the average of log w barely moves. From that estimate follow a bounded holomorphic functional calculus, semigroup representations of solutions, non-tangential maximal estimates, Fatou-type convergence of Whitney averages, a well-posedness criterion as invertibility on a spectral subspace, Dirichlet-regularity duality, perturbation stability, and well-posedness for hermitian or block-triangular t-independent coefficients; the block-diagonal case recovers the Kato square root problem for degenerate elliptic operators.
Treats the Dirichlet, Dirichlet-regularity and Neumann problems for second-order divergence-form elliptic systems div(A grad u) = 0 on the upper half-space, in the degenerate case: instead of being b…