Non-symmetric elliptic operators on bounded Lipschitz domains in the plane

Rule, David J.
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Texas State University-San Marcos, Department of Mathematics
We consider divergence form elliptic operators L = div A∇ in ℝ2 with a coefficient matrix A = A(x) of bounded measurable functions independent of the t-direction. The aim of this note is to demonstrate how the proof of the main theorem in [4] can be modified to bounded Lipschitz domains. The original theorem states that the Lp Neumann and regularity problems are solvable for 1 < p < p0 for some p0 in domains of the form {(x, t) : φ(x) < t}, where φ is a Lipschitz function. The exponent p0 depends only on the ellipticity constants and the Lipschitz constant of φ. The principle modification of the argument for the original result is to prove the boundedness of the layer potentials on domains of the form {X = (x, t) : φ(e ⋅ X) < e⊥ ⋅ X}, for a fixed unit vector e = (e1, e2) and e⊥ = (e2, e1. This is proved in [4] only in the case e = (1, 0). A simple localisation argument then completes the proof.
T(b) Theorem, Layer potentials, Lp Neumann problem, Lp regularity problem, Non-symmetric elliptic equations
Rule, D. J. (2007). Non-symmetric elliptic operators on bounded Lipschitz domains in the plane. <i>Electronic Journal of Differential Equations, 2007</i>(144), pp. 1-8.