Regularity estimates for the oblique derivative problem on non-smooth domains. I (Q1899637)

The authors consider the existence and regularity of the oblique derivative problem \(Pu= f\) in \(\Omega\), \(\vec lu= g\) on \(\partial\Omega\), where \(P\) is a second-order elliptic differential operator on \(\mathbb{R}^n\), \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) and \(\vec l\) is a unit vector field on the boundary of \(\Omega\) (which may be tangential to the boundary). All above are assumed with limited smoothness. The authors show that the solution \(u\) has an elliptic gain from \(f\) in Hölder spaces and they obtain \(L^p\) regularity of \(u\) generalizing some results from \textit{P. F. Guan} and \textit{E. Sawyer} [Ann. Math., II. Ser. 137, 1-70 (1993; Zbl 0792.35039)] to the limited smooth case. Some applied nonlinear problems are also discussed.