Abstract / Description of output
In this paper, we continue the study of a class of second order elliptic operators of the form L=div(A∇⋅) in a domain above a Lipschitz graph in R n, where the coefficients of the matrix A satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of operators, it is known (since 2001) that the L q Dirichlet problem is solvable for some 1<q<∞. Moreover, further studies completely resolved the range of L q solvability of the Dirichlet, Regularity, Neumann problems in Lipschitz domains, when the Carleson measure norm of the oscillation is sufficiently small. We show that there exists p reg>1 such that for all 1<p<p reg the L p Regularity problem for the operator L=div(A∇⋅) is solvable. Furthermore [Formula presented] where q ⁎>1 is the number such that the L q Dirichlet problem for the adjoint operator L ⁎ is solvable for all q>q ⁎. Additionally when n=2, there exists p neum>1 such that for all 1<p<p neum the L p Neumann problem for the operator L=div(A∇⋅) is solvable. Furthermore [Formula presented] where q ⁎>1 is the number such that the L q Dirichlet problem for the operator L 1=div(A 1∇⋅) with matrix A 1=A/detA is solvable for all q>q ⁎.
Keywords / Materials (for Non-textual outputs)
- Boundary value problem
- Carleson measure
- Elliptic second order PDE
- Regularity problem