Boundary value problems in Lipschitz domains for equations with drifts

2017/01/28 by Sakellaris, Georgios
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1701.08332

Abstract

In this work we establish solvability and uniqueness for the D₂ Dirichlet problem and the R₂ Regularity problem for second order elliptic operators L=-\rm div(A∇⋅)+b∇⋅ in bounded Lipschitz domains, where b is bounded, as well as their adjoint operators L^t=-\rm div(A^t∇⋅)-\rm div(b ⋅). The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to D₂ for L and R₂ for L^t leads us to impose a specific size condition on \rm divb in order to obtain solvability. On the other hand, we show that R₂ for L and D₂ for L^t are uniquely solvable, assuming only that A is Lipschitz continuous (and not necessarily symmetric) and b is bounded.

Related

• Boundary value problems in Lipschitz domains for equations with lower order coefficients
• The Boundary value problems for second order elliptic operators satisfying a Carleson condition
• Boundary Value Problems for Second‐Order Elliptic Operators Satisfying a Carleson Condition
• Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains
• The regularity problem for elliptic operators with boundary data in Hardy-Sobolev space HS¹