Let $\Omega \subset \mathbb R^n$ a $C^{2,\alpha}$ domain, $f \in C^{0,\alpha}(\overline{\Omega})$, $g \in C^{1,\alpha}(\overline{\Omega})$, $h \in C^{1,\alpha}(\overline{\Omega};\mathbb{R}^n)$ such that $$ \int_{\Omega} f = \int_{\partial \Omega}g $$ and consider the following elliptic PDE with oblique boundary condition $$ \left\{ \begin{array}[c]{cl} \text{div} (\nabla u + hu) = f & \text{in}\ \Omega \\ \left \langle \nabla u + hu ; \nu \right \rangle = g & \text{on}\ \partial \Omega. \end{array}\right.$$

The standard estimate for classical solutions of oblique problems is $$ ||u||_{C^{2,\alpha}} \leq C (||u||_{C^{0}} + ||f||_{C^{0,\alpha}} + ||g||_{C^{1,\alpha}}), $$ with $C$ depending only on $h$.

My question is the following : is it possible to improve this estimate to get $$ ||u||_{C^{2,\alpha}} \leq C (||f||_{C^{0,\alpha}} + ||g||_{C^{1,\alpha}}), $$ in an appropriate function space (say zero mean functions), for general data $h$ ?


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