Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty }$ coefficients satisfying the small Carleson measure norm condition

Abstract

We consider divergence form elliptic equations $Lu:=\nabla \cdot (A\nabla u)=0$ in the half space ${\mathbb{R}}_{+}^{n+1}:=\{(x,t)\in {\mathbb{R}}^n\times (0,\infty )\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In addition, we suppose that $A$ satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy $A(x,t)-A(x,0)$ satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in $L^p$, Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for $L$, with data in $L^p({\mathbb{R}}^n)$, $BMO({\mathbb{R}}^n)$, and $C^{\alpha }({\mathbb{R}}^n)$, and solvability of the Neumann and Regularity problems, with data in the spaces $L^p({\mathbb{R}}^n)/H^p({\mathbb{R}}^n)$ and $L_1^p({\mathbb{R}}^n)/H^{1,p}({\mathbb{R}}^n)$ respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials in for the $t$-independent operator $L_0:=-\nabla \cdot (A(\cdot ,0)\nabla )$.

Submitted, 71 pages

Cited by in corpus (2)

• The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^{\infty }$ coefficients
• Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients