REGULARITY ESTIMATES FOR THE OBLIQUE DERIVATIVE PROBLEM ON NON-SMOOTH DOMAINS （I）

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REGULARITY ESTIMATES FOR THE OBLIQUE DERIVATIVE PROBLEM ON NON-SMOOTH DOMAINS (I)

The authors consider the existence and regularity of the oblique derivative problem: $$ \left\{ \aligned Pu & = f \quad \roman{in}\ \Omega, \ \overset{\rightarrow }\to{\ell }u & = g \quad \ \roman{on}\ \partial\Omega, \endaligned \right. $$ where $P$ is a second order elliptic differential operator on $R^n$, $\Omega $ is a bounded domain in $R^n$ and $\overset{\rightarrow }\to{\ell }$ 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 solution $u$ has an elliptic gain from $f$ in Holder spaces ( Theorem 1.1). The authors obtain $L^p$ regualrity of solution in Theorem 1.3, which generalizes the results in 7] to the limited smooth case. Some of the application nonlinear problems are also discussed.