\documentstyle{amsart} \begin{document} {\noindent\small {\em Electronic Journal of Differential Equations}, Vol.\ 1997(1997), No.\ 01, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \thanks{\copyright 1997 Southwest Texas State University and University of North Texas.} \vspace{1.5cm} \title[\hfilneg EJDE--1997/01\hfil Sub-elliptic boundary value problems] {Sub-elliptic boundary value problems for quasilinear elliptic operators} \author[D.K. Palagachev \& P.R. Popivanov\hfil EJDE--1997/01\hfilneg] {Dian K. Palagachev \& Peter R. Popivanov} \address{Dian K. Palagachev\\ Department of Mathematics, Technological University of Sofia, 8~``Kl. Okhridski''~blvd., 1756 Sofia, Bulgaria} \email{dian@@bgcict.acad.bg} \address{Peter R. Popivanov\\ Institute of Mathematics, Bulgarian Academy of Sciences, ``G. Bonchev'' str., bl. 8, 1113 Sofia, Bulgaria} \email{popivano@@bgearn.acad.bg} \date{} \thanks{Submitted October 28, 1996. Published January 8, 1997.} \thanks{Partially supported by the Bulgarian Ministry of Education, Science and Technologies \newline\indent under Grant MM--410.} \subjclass{35J65, 35R25} \keywords{Quasilinear elliptic operator, degenerate oblique derivative problem, \newline\indent sub-elliptic estimates} \begin{abstract} Classical solvability and uniqueness in the H\"older space $C^{2+\alpha}(\overline{\Omega})$ is proved for the oblique derivative problem $$ \begin{cases} a^{ij}(x)D_{ij}u + b(x,\,u,\,Du)=0 \quad \text{in\ }\Omega,\\ \partial u/\partial \ell =\varphi(x)\quad \text{on\ }\partial \Omega \end{cases} $$ in the case when the vector field $\ell(x)=(\ell^1(x),\ldots,\ell^n(x))$ is tangential to the boundary $\partial \Omega$ at the points of some non-empty set $S\subset\partial \Omega,$ and the nonlinear term $b(x,\,u,\,Du)$ grows quadratically with respect to the gradient $Du$. \end{abstract} \maketitle \newtheorem{thm}{Theorem}[subsection] %% Definition of Theorem \newtheorem{lem}[thm]{Lemma} %% Definition of Lemma \newtheorem{crlr}[thm]{Corollary} %% Definition of Corollary \newtheorem{prp}[thm]{Proposition} %% Definition of Proposition \newtheorem{defin}{Definition}[section]%% Definition of Definition \theoremstyle{remark} \newtheorem{rem}{Remark}[section] %% Remark \makeatletter \def\theequation{\thesection.\@arabic\c@equation} \def\thethm{\thesection.\@arabic\c@thm} \def\thelem{\thesection.\@arabic\c@thm} \def\thecrlr{\thesection.\@arabic\c@thm} \def\theprp{\thesection.\@arabic\c@thm} \def\therem{\thesection.\@arabic\c@thm} \makeatother \def\ol{\overline} \def\ds{\displaystyle} \newcommand{\R}{{\Bbb R}} \setcounter{section}{-1} \section{Introduction} \setcounter{equation}{0} The paper is devoted to the study of so-called {\it oblique derivative problem\/} firstly posed by H.~Poincar\'e (\cite{Poi}): {\it given a domain $\Omega$, find a solution in $\Omega$ of an elliptic differential equation that satisfies boundary condition in terms of directional derivative with respect to a vector field $\ell$ defined on the boundary $\partial \Omega$.\/} More precisely, we shall be concerned with the problem \begin{equation}\label{0.1} \begin{cases} a^{ij}(x)D_{ij}u + b(x,\,u,\,Du)=0 \quad \text{in\ }\Omega,\\ \partial u/\partial \ell\equiv \ell^{\,i}(x) D_i u =\varphi(x)\quad \text{on\ }\partial \Omega \end{cases} \end{equation} in the degenerate (tangential) case, i.e. a situation when the vector field $\ell(x)=(\ell^{\,1}(x),\ldots,\ell^{\,n}(x))$ prescribing the boundary operator becomes tangential to $\partial \Omega$ at the points of some non-empty set $S.$ This way, the well-known Shapiro--Lopatinskii complementary condition is violated on the set $S$ and the classical theory (cf. \cite{GT}) cannot be applied to the problem \eqref{0.1}. The linear tangential problem ($b(x,\,z,\,p)=b^i(x)p_i + c(x)z$) has been very well studied in the last three decades. The pioneering works of Bicadze~\cite{B} and H\"or\-man\-der~\cite{H} indicated how the solvability and uniqueness properties depend on the way in which the normal component of $\ell(x)$ changes its sign across $S.$ More precisely, suppose $S$ to be a submanifold of $\partial \Omega$ of co-dimension one, and let $\ell(x)=\tau(x)+\gamma(x)\nu(x).$ Here $\nu(x)$ is the unit outward normal to $\partial \Omega$ and $\tau(x)$ is a tangential field to $\partial \Omega$ such that $|\ell(x)|=1.$ There are three possible behaviors of $\ell(x)$ near the set $S=\{x\in\partial \Omega\colon \gamma(x)=0\}:$ \begin{itemize} \item[\it a)] $\ell(x)$ {\it is of neutral type:\/} $\gamma(x)\geq0$ or $\gamma(x)\leq0$ on $\partial\Omega;$ \item[\it b)] $\ell(x)$ {\it is of emergent type:\/} the sign of $\gamma(x)$ changes from $-$ to $+$ in the positive direction on $\tau$-integral curves through the points of $S;$ \item[\it c)] $\ell(x)$ {\it is of submergent type:\/} the sign of $\gamma(x)$ changes from $+$ to $-$ along the $\tau$-integral curves through $S.$ \end{itemize} H\"ormander's results were refined by Egorov and Kondrat'ev~\cite{EK} who pro\-ved that the linear problem \eqref{0.1} is of Fredholm type in the neutral case {\it a).} Moreover, they showed that either the values of $u$ should be prescribed on $S$ in order to get uniqueness in the case {\it b),\/} or to accept jump discontinuity on $S$ in order to have existence in the case {\it c).} What is the universal property of the linear problem \eqref{0.1}, however, no matter the type of $\ell(x),$ is that a loss of regularity of the solution occurs in contrast to the regular ($S=\emptyset$) oblique derivative problem. Later, precise studies were carried out in order to indicate the exact regularity that a solution of the linear problem \eqref{0.1} gains on the data both in Sobolev and H\"older spaces. We refer the reader to \cite{E}, \cite{M}, \cite{MPh}, \cite{Gu}, \cite{Sm}, \cite{W1}--\cite{W4}, and most recently to \cite{GuS1} and \cite{GuS2}. The investigations on the quasilinear problem \eqref{0.1} (especially, in the weak nonlinear case $b(x,\,z,\,p)= b^i(x,\,z)p_i+c(x,\,z)$) were initiated by the papers \cite{PK1} and \cite{PK2}. In our previous study \cite{PPa}, classical solvability results were obtained for \eqref{0.1} both in the cases of neutral and emergent $C^\infty$-vector field $\ell(x)$ supposing $C^\infty$ structure of the elliptic operator. Moreover, we assumed in \cite{PPa} that $\ell(x)$ has a contact of order $k<\infty$ with $\partial \Omega,$ and $|b|,$ $|b_x|=O(|p|^2),$ $|b_z|=o(|p|^2),$ $|b_p|=o(|p|)$ as $|p|\to\infty,$ uniformly on $x$ and $z.$ The general aim of the present article is to improve the results of \cite{PPa} weakening the growth assumptions on $b(x,\,z,\,p)$ with respect to $p.$ Let us note that although our results here hold true both for neutral and emergent fields $\ell(x),$ for the sake of simplicity we have restricted ourselves to consider the case of emergent field only. (Detailed exposition of the study on degenerate problem with a neutral vector field $\ell$ can be found in \cite{PPa2}.) That is why, according to the above mentioned result of Egorov and Kondrat'ev, we consider the problem \eqref{0.1} supplied with the extra condition \begin{equation}\label{0.2} u=\psi(x)\quad \text{on the set of tangency\ } S. \end{equation} Concerning the problem \eqref{0.1}, \eqref{0.2}, we prove its solvability and uniqueness in the H\"older space $C^{2+\alpha}(\ol\Omega)$ assuming $a^{ij}\in C^{\alpha}(\ol\Omega),$ $b(x,\,z,\,p)\in C^{\alpha}(\ol\Omega\times\R\times \R^n),$ $\ell^{\,i}\in C^{2+\alpha}(\partial \Omega)$ and $|b(x,\,z,\,p)|\leq \mu(|u|)(1+|p|^2)$ with a non-decreasing function $\mu$ (no growth assumptions on the derivatives of $b$ are required!). Further on, suitable conditions due to P.~Guan and E.~Sawyer~\cite{GuS2} and concerning behavior of $\ell(x)$ on $\partial\Omega$ are imposed. It is worth noting that our growth condition on $b(x,\,z,\,p)$ includes these in \cite{PPa}, as well as the natural structural conditions in the treatment of regular oblique derivative problems for nonlinear elliptic equations (see~\cite{LT}). The main tool in proving our results is the Leray--Schauder fixed point theorem, that reduces solvability of \eqref{0.1}, \eqref{0.2} to the establishment of an a~priori $C^{1+\beta}(\ol\Omega)$-estimate for the solutions of related problems. The bound for $\|u\|_{C^0(\ol\Omega)}$ is a simple consequence of the maximum principle. In order to estimate the $C^{\beta}(\ol\Omega)$-norm of the gradient $Du,$ we use an approach due to F.~Tomi~\cite{To} (see \cite{AC} also) that imbeds the problem \eqref{0.1}, \eqref{0.2} into a family of similar problems depending on a parameter $\rho\in[0,\,1]$ and having solutions $u(\rho;\,x).$ Then the norm $\|Du\|_{C^\beta(\ol\Omega)}=\|D_xu(1;\,x)\|_{C^\beta(\ol\Omega)}$ can be estimated in terms of $\|D_xu(0;\,x)\|_{C^\beta(\ol\Omega)}$ after iterations on $\rho,$ assuming the difference $u(\rho_1;\,x)- u(\rho_2;\,x)$ to be under control for small $\rho_1-\rho_2.$ To realize this strategy, we use the refined sub-elliptic estimates in Sobolev and H\"older spaces proved very recently by Guan and Sawyer~\cite{GuS2}. At the end, uniqueness for the solutions of \eqref{0.1}, \eqref{0.2} follows by the maximum principle. \subsubsection*{Acknowledgements} The authors are indebted to Professor Pengfei Guan for supplying them with the text of manuscript~\cite{GuS2} before its publication. \section{Statement of the problem and main results} \setcounter{equation}{0} \setcounter{thm}{0} Let $\Omega\subset\R^n,$ $n\geq 2,$ be a bounded domain. On the boundary $\partial \Omega$ a unit vector field $\ell(x)=(\ell^{\,1}(x),\ldots,\ell^{\,n}(x))$ is defined, which can be decomposed into $$ \ell(x)=\tau(x)+\gamma(x)\nu(x)\qquad x\in\partial \Omega, $$ where $\nu(x)$ is the unit outward normal to $\partial \Omega$ and $\tau(x)$ is the tangential projection of $\ell(x)$ on $\partial \Omega.$ Let $$ S=\{x\in\partial \Omega\colon\ \gamma(x)=0\} $$ be the set of tangency between $\ell(x)$ and $\partial \Omega.$ Throughout the paper we consider the case $S\not\equiv\emptyset.$ In order to describe our technique, we shall consider the case of emergent field $\ell(x)$ only. In other words, we suppose that $\gamma(x)$ changes its sign from $-$ to $+$ in the positive direction on the $\tau$-integral curves passing through the points of $S.$ Moreover, to avoid unessential complications, we assume that $S$ is a closed submanifold of $\partial \Omega,$ $\text{codim}_{\partial \Omega}S=1,$ $\partial \Omega=\partial \Omega_+\cup \partial \Omega_-\cup S$ where $\partial \Omega_{\pm}=\{x\in\partial \Omega\colon \gamma(x){}^{>}_{<}0\},$ and let the field $\ell(x)$ be strictly transversal to $S$ at each point $x\in S$ (indeed, $\ell\equiv\tau$ there). We aimed to study the classical solvability of the degenerate oblique derivative problem: \begin{equation}\label{1.1} \begin{cases} a^{ij}(x)D_{ij}u + b(x,\,u,\,Du)=0 \quad \text{in\ }\Omega,\\ \partial u/\partial \ell\equiv \ell^{\,i}(x) D_i u =\varphi(x)\quad \text{on\ }\partial \Omega,\qquad u=\psi(x)\quad \text{on\ } S. \end{cases} \end{equation} Hereafter, the standard summation convention is adopted and $Du$ denotes the gradient $(D_1u,\ldots,D_nu)$ of $u(x)$ with $D_i\equiv \partial/\partial x_i.$ Further on, the symbol $C^{k+\alpha}(\ol\Omega),$ $k\geq0$ integer, stands for the H\"older functional space equipped with the norm $\|\cdot\|_{C^{k+\alpha}(\ol\Omega)}$ (see~\cite{GT}). The letter $C$ will denote a constant, independent of $u,$ that may vary from a line into another. In order to state our result, we give a list of assumptions. {\it Uniform ellipticity:\/} there exists a positive constant $\lambda$ such that \begin{equation}\label{1.2} a^{ij}(x)\xi^i\xi^j\geq \lambda|\xi|^2\quad \forall x\in\ol\Omega,\ \forall \xi\in \R^n,\quad \ a^{ij}=a^{ji}; \end{equation} {\it Regularity conditions:\/} for some $\alpha\in (0,\,1)$ \begin{equation}\label{1.3} \begin{cases} a^{ij}\in C^\alpha(\ol\Omega),\ b(x,\,z,\,p)\in C^\alpha(\ol\Omega\times\R\times\R^n),\\ b(x,\,z,\,p)\ \text{is continuously differentiable with respect to\ } z\ \text{and\ } p,\\[2pt] \ell^{\,i}(x)\in C^{2+\alpha}(\partial \Omega),\quad \partial \Omega\in C^{3+\alpha},\quad S\in C^{2+\alpha}; \end{cases} \end{equation} {\it Monotonicity condition:\/} there exists a positive constant $b_0$ such that \begin{equation}\label{1.4} b_z(x,\,z,\,p)\leq -b_0<0\quad \forall (x,\,z,\,p)\in \ol\Omega\times\R\times \R^n\quad (b_z=\partial b/\partial z); \end{equation} {\it Quadratic growth with respect to the gradient:\/} there exists a positive and non-decreasing function $\mu(t)$ such that \begin{equation}\label{1.5} |b(x,\,z,\,p)|\leq \mu(|z|)\Big(1+|p|^2\Big)\quad \forall (x,\,z,\,p)\in \ol\Omega\times\R\times\R^n. \end{equation} Denote by $\omega(t,\,x)$ the parameterization of the $\tau$-integral curve passing through the point $x\in\partial \Omega,$ i.e. $\frac{d}{dt}\omega(t,\,x)=\tau(\omega(t,\,x)),$ $\omega(0,\,x)=x.$ The next notions were introduced by Guan and Sawyer in \cite{GuS2}. \def\thedefin{1} \begin{defin}\em The vector field $\ell(x)$ satisfies condition ${\cal A}^{\mp}_p$ on $S$ if for each $y\in S$ there exist constants $r>0,$ $R^-<0n\ \text{and\ } \theta\in[0,\,1),\quad \theta\neq \alpha. \end{cases} \end{equation} We are in a position now to state the main result of the paper. \begin{thm}\label{thm1.1} Suppose assumptions $\eqref{1.2}-\eqref{1.6}$ to be fulfilled. Then the degenerate oblique derivative problem $\eqref{1.1}$ admits a unique classical $C^{2+\alpha}(\ol\Omega)$ solution for each $\varphi\in C^{2+\alpha-\theta}(\partial \Omega)$ and $\psi\in C^{2+\alpha}(S).$ \end{thm} \addtocounter{thm}{1} \begin{rem}\label{rem1.2} 1. The requirements in \eqref{1.3} on $b(x,\,z,\,p)$ to be differentiable with respect to $z$ and $p$ may be replaced by its Lipschitz continuity in $z$ and $p.$ 2. The quadratic growth assumption \eqref{1.5} includes for example the natural conditions in studying regular oblique derivative problems for fully nonlinear elliptic operators (cf.~\cite{LT}), as well as the structure conditions on $b(x,\,z,\,p)$ imposed in \cite{PPa}. 3. Conditions ${\cal A}^{\mp}_p$ and ${\cal T}_\theta$ correspond to the requirement of ``finite type'' vector field $\ell$ in the $C^\infty$ case (cf.~\cite{Gu}, \cite{PPa}, \cite{GuS1}). In fact, supposing $\partial\Omega\in C^\infty,$ $\ell\in C^\infty,$ we say that the field $\ell(x)$ is of finite type if there exists an integer $k,$ such that $$ \sum_{i=1}^k\left|\frac{\partial^i}{\partial t^i}\gamma(\omega(t,\,x))\Big|_{t=0} \right|>0\quad \text{for all\ } x\in\partial \Omega. $$ (Indeed, the number $k$ is exactly the order of contact between $\ell(x)$ and $\partial \Omega.$) Now, if $\ell$ is of type $k,$ then \cite[Lemma~C.1]{Tr} implies condition ${\cal T}_\theta$ with $\theta=1/(k+1).$ Moreover, it follows from \cite{GuS1} that the ${\cal A}^{\mp}_p$ condition is satisfied for all $p$ in the range $(1,\,\infty).$ 4. Careful analysis on the condition ${\cal T}_\theta$ shows that, if it is satisfied by a field $\ell(x)$ which becomes tangential to $\partial\Omega$ then the exponent $\theta$ is necessary strictly less than one. \end{rem} \section{Some preliminaries} \setcounter{equation}{0} \setcounter{thm}{0} For the sake of completeness we will sketch in this section some of the results proved by Guan and Sawyer in~\cite{GuS2}. Define the linear uniformly elliptic operator $$ {\cal L}\equiv a^{ij}(x)D_{ij}+ b^i(x)D_i + c(x) $$ with $C^\alpha(\ol\Omega)$ coefficients ($0<\alpha<1$) and assume $\ell(x)$ to be an emergent type vector field as in the preceding section, with \eqref{1.2} and \eqref{1.3} being fulfilled. Let us consider the linear tangential oblique derivative problem \begin{equation}\label{2.1} \begin{cases} {\cal L}u\equiv a^{ij}(x)D_{ij}u + b^i(x)D_iu +c(x)u=f(x) \quad \text{in\ }\Omega,\\ \partial u/\partial \ell =g(x)\quad \text{on\ }\partial \Omega, \qquad u=h(x)\quad \text{on\ } S. \end{cases} \end{equation} The following result is a special case of \cite[Theorem~10]{GuS2} that concerns the properties of the problem \eqref{2.1} in H\"older spaces. \begin{lem}\label{lem2.1} Let the field $\ell$ satisfy condition ${\cal T}_\theta$ for some $\theta\geq0,$ and $c(x)\leq0.$ Then for each $(f,\,g,\,h)\in C^\alpha(\ol\Omega)\times C^{2+\alpha-\theta}(\partial \Omega)\times C^{2+\alpha}(S)$ there exists a unique solution $u\in C^{2+\alpha}(\ol\Omega)$ of the problem $\eqref{2.1}.$ Moreover, if $u\in C^{2+\alpha'}(\ol\Omega)$ $(0<\alpha'<\alpha)$ satisfies $\eqref{2.1}$ with $f,$ $g$ and $h$ as above, then $u\in C^{2+\alpha}(\ol\Omega)$ and there is a constant $C$ (independent of $u$) such that \begin{equation}\label{2.2} \|u\|_{C^{2+\alpha}(\ol\Omega)} \leq C\Big(\! \|f\|_{C^{\alpha}(\ol\Omega)}\!+\! \|g\|_{C^{2+\alpha-\theta}(\partial \Omega)}\!+\! \|h\|_{C^{2+\alpha}(S)}\!+\! \|u\|_{C^{0}(\ol\Omega)}\!\Big)\!. \end{equation} \end{lem} To summarize the corresponding results in the Sobolev functional scale, denote by $H^s_p(\Omega)$ and $B^{s,\,p}(\Omega)$ the Sobolev and Besov $L^p$-spaces, respectively (\cite{Ad}). Theorem~12 and Remark~3 of~\cite{GuS2} yield the following \begin{lem}\label{lem2.2} Let the field $\ell(x)$ satisfy condition ${\cal T}_\theta$ on $\partial \Omega$ $(\theta\geq0),$ condition ${\cal A}^{\mp}_p$ on $S$ $(p>1),$ and $c(x)\leq0.$ For each $(f,\,g,\,h)\in L^p(\Omega)\times B^{2-\theta-1/p,\,p}(\partial \Omega)\times B^{2-\theta/p-1/p,\,p}(S)$ there exists a unique solution $u\in H^2_p(\Omega)$ of the problem $\eqref{2.1},$ and there is a constant $C$ such that \begin{align}\label{2.3} \|u\|_{H^2_p(\Omega)}\leq C\Big(\|f\|_{L^p(\Omega)} & + \|g\|_{B^{2-\theta-1/p,\,p}(\partial \Omega)}\\[2pt] \nonumber & + \|h\|_{B^{2-\theta/p-1/p,\,p}(S)}+ \|u\|_{L^{p}(\Omega)}\Big)\!. \end{align} \end{lem} The remaining part of this section is devoted to comparison principles for linear and quasilinear elliptic operators. \begin{lem}\label{lem2.3} Suppose conditions $\eqref{1.2}$ and $c(x)\leq 0$ to be fulfilled and let $u\in C^2(\Omega)\cap C^1(\ol\Omega)$ satisfy $$ \begin{cases} {\cal L}u\equiv a^{ij}(x)D_{ij}u + b^i(x)D_iu +c(x)u\geq 0 \quad \text{in\ }\Omega,\\ \partial u/\partial \ell =0\quad \text{on\ }\partial \Omega, \qquad u\leq 0\quad \text{on\ } S. \end{cases} $$ Then $u\leq 0$ on $\ol\Omega.$ \end{lem} \begin{pf} We argue by contradiction. If $u(x)$ assumes positive values on $\ol\Omega$ then there exists $x_0\in\ol\Omega$ such that $u(x_0)=\max_{\Omega}u>0$ and the strong interior maximum principle asserts $x_0\in\partial \Omega.$ Further, $u\leq0$ on $S$ and it remains $x_0\in \partial \Omega\setminus S$ which is impossible since $\partial u/\partial \ell =0$ on $\partial \Omega\setminus S$ while the boundary maximum principle yields $|\partial u/\partial\ell|>0$ at the point $x_0$ ($\ell$ is strictly transversal to $\partial \Omega$ on $\partial \Omega\setminus S$). \end{pf} \begin{crlr}\label{crlr2.4} Let $\eqref{1.2}$ hold true and suppose the function $b(x,\,z,\,p)$ to be non-increasing in $z$ for each $(x,\,p)\in \Omega\times\R^n$ and differentiable with respect to $p$ in $\Omega\times\R\times\R^n.$ Let $u,\ v\in C^2(\Omega)\cap C^1(\ol\Omega)$ satisfy $$ \begin{cases} a^{ij}(x)D_{ij}u + b(x,\,u,\,Du)\geq a^{ij}(x)D_{ij}v + b(x,\,v,\,Dv)\quad \text{in\ }\Omega,\\ \partial u/\partial \ell = \partial v/\partial \ell =0\quad \text{on\ }\partial \Omega, \qquad u\leq v\quad \text{on\ } S. \end{cases} $$ Then $u\leq v$ on $\ol\Omega.$ \end{crlr} \begin{pf} Defining $w=u-v,$ we have $$ {\cal L}w\equiv a^{ij}(x)D_{ij}w +b^i(x)D_iw\geq 0\quad\text{on\ } \{x\in\Omega\colon w(x)>0\}, $$ where $$ b^i(x)=\int_0^1 b_{p_i}(x,\,v(x),\,sDw(x)+Dv(x))\;ds. $$ Furthermore, $$ \partial w/\partial \ell=0\quad\text{on\ } \partial \Omega\qquad\text{and} \qquad w\leq 0\quad\text{on\ }S. $$ Thus, the assertion of Corollary~\ref{crlr2.4} follows by Lemma~\ref{lem2.3}. \end{pf} \section{A priori estimates} \setcounter{equation}{0} \setcounter{thm}{0} Theorem~\ref{thm1.1} will be proved with the aid of the Leray--Schauder fixed point theorem that reduces the classical solvability of \eqref{1.1} to the establishment of an a~priori estimate in the Banach space $C^{1+\beta}(\ol\Omega)$ ($\beta\in(0,\,1)$ is a suitable number) for all solutions to a family of problems related to \eqref{1.1}. This section deals with deriving of these estimates. To making our exposition more clear, we shall start with the homogeneous case, i.e. we take $\varphi\equiv0,$ $\psi\equiv0$ and consider the problem \begin{equation}\label{3.1} \begin{cases} a^{ij}(x)D_{ij}u + b(x,\,u,\,Du)=0 \quad \text{in\ }\Omega,\\ \partial u/\partial \ell=0\quad \text{on\ }\partial \Omega,\qquad u=0\quad \text{on\ } S \end{cases} \end{equation} instead of \eqref{1.1}. \subsection{A priori estimate for $\|u\|_{C^0(\ol\Omega)}$} \begin{lem}\label{lem3.1} Suppose the conditions $\eqref{1.2},$ $\eqref{1.3}$ and $\eqref{1.4}$ to be fulfilled. Then $$ \|u\|_{C^0(\ol\Omega)}\equiv \max_{\ol\Omega}|u(x)|\leq \frac{1}{b_0} \max_{\ol\Omega}|b(x,\,0,\,0)| $$ for each solution $u\in C^2(\Omega)\cap C^1(\ol\Omega)$ of the problem $\eqref{3.1}.$ \end{lem} \begin{pf} Choosing the positive constant $M$ such that $$ M\geq \frac{1}{b_0} \max_{\ol\Omega}|b(x,\,0,\,0)|, $$ one has \begin{align*} a^{ij}(x)D_{ij}u+ b(x,\,u,\,Du)& \geq -Mb_0 +\ds \max_{\ol\Omega}|b(x,\,0,\,0)|\\[4pt] &\ds\geq M\int_0^1 b_z(x,\,sM,\,0)\;ds +b(x,\,0,\,0)=b(x,\,M,\,0)\\[4pt] &=a^{ij}(x)D_{ij}(M) + b(x,\,M,\,DM)\quad\text{in\ }\Omega \end{align*} as consequence of \eqref{1.4}. Moreover, $$ \partial u/\partial \ell =0=\partial M/\partial \ell\quad\text{on\ }\Omega, \qquad u=0n$} In view of the Morrey lemma ($H_q^2(\Omega)\subset C^{2-n/q}(\ol\Omega),$ $q>n$), the a~priori bound for the $C^{\beta}$-H\"older norm of the gradient $Du$ with $\beta=1-n/q$ (and therefore, the solvability of \eqref{3.1}) is equivalent to an estimate of the $H^{2}_{q}(\Omega)$-norm of $u.$ On the other hand, Lemma~\ref{lem2.2} (and especially \eqref{2.3}) reduces that bound to a uniform with respect to $u$ estimate of $\|b(x,\,u,\,Du)\|_{L^{q}(\Omega)},$ which becomes equivalent to an a~priori estimate of $\|Du\|_{L^{2q}(\Omega)}$ through the quadratic growth assumption \eqref{1.5} and Lemma~\ref{lem3.1}. We shall employ a technique inspired by Amann--Crandall's approach (cf.~\cite{AC}) in proving an $L^\infty(\Omega)$ gradient estimate for semilinear elliptic equations. \addtocounter{thm}{1} \begin{lem}\label{lem3.2} Let conditions $\eqref{1.2},$ $\eqref{1.3},$ $\eqref{1.5}$ and $\eqref{1.6}$ be satisfied. Then there exists a constant $C$ depending on known quantities only and on $\|u\|_{C^0(\ol\Omega)},$ such that \begin{equation}\label{3.2} \|Du\|_{L^{2q}(\Omega)}\leq C \end{equation} for each solution $u\in C^{2+\alpha}(\ol\Omega)$ of the problem $\eqref{3.1}.$ \end{lem} \begin{pf} The function $u\in C^{2+\alpha}(\ol\Omega)$ solves the equation $$ a^{ij}(x)D_{ij}u + B(x)|Du|^2 -u(x) = F(x)\quad\text{in\ } \Omega, $$ where \begin{equation}\label{3.3} \begin{cases} B(x)=\ds\frac{b(x,\,u(x),\,Du(x))}{1+|Du|^2}\in C^\alpha(\ol\Omega),\\[12pt] F(x)=-u(x)-\ds\frac{b(x,\,u(x),\,Du(x))}{1+|Du|^2}\in C^\alpha(\ol\Omega). \end{cases} \end{equation} For the fixed solution $u(x)$ we imbed \eqref{3.1} into the one-parameter family of tangential oblique derivative problems \begin{equation}\label{3.4} \begin{cases} a^{ij}(x)D_{ij}u(\rho;\,x) + B(x)|Du(\rho;\,x)|^2 -u(\rho;\,x) = \rho F(x)\quad\text{in\ }\Omega,\\ \partial u(\rho;\,x)/\partial \ell=0\quad \text{on\ } \partial \Omega,\qquad u(\rho;\,x)=0\quad\text{on\ } S \end{cases} \end{equation} with solutions $u(\rho;\,x)\in H^2_q(\Omega)$ $(\rho\in [0,\,1])$ if they do exist. Let us point out that $q>n$ and Sobolev's imbedding theorem ensure that the values of $u(\rho;\,x)$ and its derivatives on $\partial \Omega$ are well defined. Indeed, $u(0;\,x)=0$ and $u(1;\,x)\equiv u(x)$ is the fixed solution of \eqref{3.1}. Our aim is to estimate $\|D_xu(\rho_2;\,x)\|_{L^{2q}(\Omega)}$ in terms of $\|D_xu(\rho_1;\,x)\|_{L^{2q}(\Omega)}$ when $\rho_2-\rho_1>0$ is small enough. After that, having in addition the unique solvability of \eqref{3.4} in $H^2_q(\Omega)$ for each value $\rho\in[0,\,1],$ it will be easy to derive the desired estimate \eqref{3.2} by iteration of the $L^{2q}(\Omega)$-norms of $Du(\rho;\,x)$ for $\rho<1.$ {\it Step 1.\/} To realize our program, we shall estimate at first the difference between two solutions of \eqref{3.4} in terms of the difference between the corresponding values of the parameter $\rho.$ Let $u(\rho_1;\,x),$ $u(\rho_2;\,x)\in H^2_q(\Omega)$ solve \eqref{3.4} with $\rho_1\leq \rho_2.$ Then \begin{equation}\label{3.5} \|u(\rho_1;\,x)-u(\rho_2;\,x)\|_{C^0(\ol\Omega)}\leq (\rho_2-\rho_1)\left[\mu\left(\|u\|_{C^0(\ol\Omega)}\right)+ \|u\|_{C^0(\ol\Omega)}\right]. \end{equation} To prove this, put $w(x)=u(\rho_1;\,x)-u(\rho_2;\,x)$ and observe that $w\in H^2_q(\Omega)$ solves the linearized problem \begin{equation}\label{3.6} \begin{cases} a^{ij}(x)D_{ij}w + B^i(x)D_iw -w=(\rho_1-\rho_2)F(x) \quad\text{in\ }\Omega,\\ \partial w/\partial \ell=0\quad \text{on\ } \partial \Omega,\qquad w=0\quad\text{on\ } S \end{cases} \end{equation} with $$ B^i(x)=2B(x) \int_0^1 \Big(sD_iw+D_i u(\rho_2;\,x)\Big)\;ds \in C^{\min(\alpha,\,1-n/q)}(\ol\Omega). $$ Now, the result of Lemma~\ref{lem3.1} can be applied to \eqref{3.6} whence $$ \|w\|_{C^0(\ol\Omega)}\leq (\rho_2-\rho_1)\max_{\ol\Omega}|F(x)| \leq (\rho_2-\rho_1)\left[\mu\left(\|u\|_{C^0(\ol\Omega)}\right) +\|u\|_{C^0(\ol\Omega)}\right] $$ by means of \eqref{1.5}. The only difference we have to point out is that the Aleksandrov--Pucci maximum principle (\cite[Theorem~9.6]{GT}) is to be used ($w\in H^2_q(\Omega)\subset C^{2-n/q}(\ol\Omega)$, $q>n$) instead of the strong interior maximum principle. The estimate \eqref{3.5} is proved. \addtocounter{thm}{1} \begin{rem}\label{rem3.3} Putting $\rho_1=\rho_2$ in \eqref{3.5} we obtain uniqueness of solutions to \eqref{3.4} for each value of $\rho\in[0,\,1].$ \end{rem} {\it Step 2.\/} Let $\rho_1<\rho_2$ be two arbitrary numbers and suppose there exist solutions $u(\rho_1;\,x)$ and $u(\rho_2;\,x)\in H^2_q(\Omega)$ of \eqref{3.4}. The difference $w(x)=u(\rho_1;\,x)-u(\rho_2;\,x)\in H^2_q(\Omega)$ solves $$ \begin{cases} a^{ij}(x)D_{ij}w =(\rho_1\!-\!\rho_2)F(x)\!-\! B(x)\Big(\!|Du(\rho_1;\,x)|^2\!-\!|Du(\rho_2;\,x)|^2\!\Big)\!+w\quad \text{a.e.\,}\Omega,\\[4pt] \partial w/\partial \ell=0\quad \text{on\ } \partial \Omega,\qquad w=0\quad\text{on\ } S, \end{cases} $$ and therefore Lemma~\ref{lem2.2} yields \begin{align*} \|w\|_{H^2_q(\Omega)}\leq C\Bigg(&\|w\|_{L^q(\Omega)}\\[4pt] +&\left\|(\rho_1-\rho_2)F(x)- B(x)\Big(|Du(\rho_1;\,x)|^2-|Du(\rho_2;\,x)|^2\Big) \right\|_{L^q(\Omega)}\Bigg). \end{align*} The conditions \eqref{1.5}, \eqref{3.3} and \eqref{3.5} lead to \begin{equation}\label{3.7} \|w\|_{H^2_q(\Omega)}\leq C\left( 1+ \|Dw\|^2_{L^{2q}(\Omega)}+\|Du(\rho_1;\,\cdot)\|^2_{L^{2q} (\Omega)}\right) \end{equation} with a new constant $C$ that depends on $\|u\|_{C^0(\ol\Omega)}$ in addition, but it is independent of $\rho_1-\rho_2.$ We utilize Gagliardo--Nirenberg's interpolation inequality (see~\cite{Ga}, \cite{N}) and the bound \eqref{3.5} in order to obtain \begin{align*} \|Dw\|^2_{L^{2q}(\Omega)} & \leq C\|D^2w\|_{L^q(\Omega)} \|w\|_{L^\infty(\Omega)}\\[4pt] & \leq C(\rho_2-\rho_1)\left[\mu\left( \|u\|_{C^0(\ol\Omega)}\right)+ \|u\|_{C^0(\ol\Omega)}\right]\|D^2w\|_{L^q(\Omega)}. \end{align*} Making use of \eqref{3.7} one has $$ \|Dw\|^2_{L^{2q}(\Omega)}\leq C\left( 1+ (\rho_2-\rho_1)\|Dw\|^2_{L^{2q}(\Omega)} +\|Du(\rho_1;\,\cdot)\|^2_{L^{2q}(\Omega) } \right) $$ with a constant $C$ independent of $\rho_1-\rho_2.$ Now, if $\rho_2-\rho_1\leq\varepsilon$ where $C\varepsilon<1/2,$ we have \begin{equation}\label{3.8} \|Du(\rho_2;\,\cdot)\|^2_{L^{2q}(\Omega)}\leq C_1 +C_2 \|Du(\rho_1;\,\cdot)\|^2_{L^{2q}(\Omega)} \end{equation} whenever $\rho_2-\rho_1\leq\varepsilon$. In particular, taking $\rho_1=0$ and $\rho_2=\varepsilon$ above, the uniqueness result (Remark~\ref{rem3.3}) implies \begin{equation}\label{3.9} \|Du(\varepsilon;\,\cdot)\|^2_{L^{2q}(\Omega)}\leq C_1 \end{equation} whenever there exists a solution $u(\varepsilon;\,x)\in H^2_q(\Omega)$ of \eqref{3.4} with $\rho=\varepsilon.$ {\it Step 3.\/} The Leray--Schauder fixed point theorem (\cite[Theorem~11.3]{GT}) will be used to prove solvability of the problem \eqref{3.4} for $\rho=\varepsilon.$ For this goal, define the compact nonlinear operator $$ {\cal F}\colon H^1_{2q}(\Omega) \longrightarrow H^2_q(\Omega) \underset{\text{compactly}}{\hookrightarrow} H^1_{2q}(\Omega) $$ as follows: for each $v\in H^1_{2q}(\Omega)$ the image ${\cal F}v\in H^2_q(\Omega)$ is the unique solution of the {\it linear\/} oblique derivative problem: $$ \begin{cases} a^{ij}(x)D_{ij}({\cal F}v) =\varepsilon F(x)- B(x)|Dv|^2+v\quad \text{a.e.\ }\Omega,\\ \partial ({\cal F}v)/\partial \ell=0\quad \text{on\ } \partial \Omega,\qquad {\cal F}v=0\quad\text{on\ } S. \end{cases} $$ Indeed, this problem is uniquely solvable in $H^2_q(\Omega)$ in view of \eqref{3.3} and Lem\-ma~\ref{lem2.2}. Clearly, each fixed point of $\cal F$ will be a solution to \eqref{3.4} with $\rho=\varepsilon.$ The estimate \eqref{2.3} shows that $\cal F$ is a continuous mapping from $H^1_{2q}(\Omega)$ into itself. Moreover, it follows by \eqref{3.9} an a~priori estimate (uniformly with respect to $\sigma$ and $v$) for each solution of the equation $v=\sigma{\cal F}v,$ $\sigma\in[0,\,1],$ that is equivalent to the problem $$ \begin{cases} a^{ij}(x)D_{ij}v =\ds\sigma\left(\varepsilon F(x)- B(x)|Dv|^2+v\right)\quad \text{a.e.\ }\Omega,\\ \partial v/\partial \ell=0\quad \text{on\ } \partial \Omega,\qquad v=0\quad\text{on\ } S. \end{cases} $$ Hence, Leray--Schauder's theorem asserts existence of a fixed point of $\cal F$ that proves solvability in $H^2_q(\Omega)$ of the problem \eqref{3.4} with $\rho=\varepsilon.$ To complete the proof of Lemma~\ref{lem3.2}, put $\rho_1=k\varepsilon$ and $\rho_2=(k+1)\varepsilon$ $(k=1,\,2,\ldots)$ in \eqref{3.8}. Applying finitely many times the above procedure we get the desired estimate \eqref{3.2} for $u(x)\equiv u(1;\,x).$ \end{pf} \begin{crlr}\label{crlr3.3} Let conditions $\eqref{1.2}-\eqref{1.6}$ be fulfilled. Then there is the bound $$ \|u\|_{H^2_q(\Omega)}\leq C $$ for each solution $u\in H^2_q(\Omega)$ of the problem $\eqref{3.1}.$ \end{crlr} \begin{pf} It follows by the estimate \eqref{2.3} and Lemmas~\ref{lem3.1} and \ref{lem3.2}. \end{pf} \begin{crlr}\label{crlr3.4} Assume conditions $\eqref{1.2}-\eqref{1.6}$ to be satisfied. Then there exists a constant $C$ such that \begin{equation}\label{3.10} \|u\|_{C^{2-n/q}(\ol\Omega)}\leq C \end{equation} for each solution $u\in C^{2+\alpha}(\ol\Omega)$ of the problem $\eqref{1.1}$ with $\varphi\in C^{2+\alpha-\theta}(\partial \Omega)$ and $\psi\in C^{2+\alpha}(S).$ \end{crlr} \begin{pf} Taking into account the imbedding $H^2_q(\Omega)\subset C^{2-n/q}(\ol\Omega)$ for $q>n$, the estimate \eqref{3.10} is an immediate consequence of Corollary~\ref{crlr3.3} if $u\in C^{2+\alpha}(\ol\Omega)$ solves the problem \eqref{3.1}. To handle with the non-homogeneous problem \eqref{1.1} we solve at first the linear problem $$ \begin{cases} a^{ij}(x)D_{ij}\delta =0\quad \text{in\ }\Omega,\\ \partial \delta/\partial \ell=\varphi\quad \text{on\ } \partial \Omega,\qquad \delta=\psi\quad\text{on\ } S. \end{cases} $$ Indeed, there exists a unique solution $\delta\in C^{2+\alpha}(\ol\Omega)$ of that problem by virtue of Lemma~\ref{lem2.1}. Thus, if $u(x)$ solves \eqref{1.1} then the function $v=u-\delta$ is a solution of the homogeneous problem $$ \begin{cases} a^{ij}(x)D_{ij}v + b'(x,\,v,\,Dv)=0\quad \text{in\ }\Omega,\\ \partial v/\partial \ell=0\quad \text{on\ } \partial \Omega,\qquad v=0\quad\text{on\ } S, \end{cases} $$ where $b'(x,\,z,\,p)=b(x,\,z+\delta(x),\,p+D\delta(x))$ and conditions of the type \eqref{1.4} and \eqref{1.5} are fulfilled by $b'(x,\,z,\,p).$ Since the bound \eqref{3.10} is satisfied by the function $v(x),$ it will be true for $u(x)$ also, with a new constant $C$ depending on $\|\delta\|_{C^{2+\alpha}(\ol\Omega)}$ in addition. \end{pf} \section{Proof of Theorem 1.1} \setcounter{equation}{0} The uniqueness assertion of Theorem~\ref{thm1.1} follows immediately by \eqref{1.4} and Corollary~\ref{crlr2.4}. To prove existence, Leray--Schauder's fixed point theorem will be used again. Let us set $\beta=1-n/q,$ and for $v\in C^{1+\beta}(\ol\Omega)$ consider the linear tangential oblique derivative problem: $$ \begin{cases} a^{ij}(x)D_{ij}u + b(x,\,v,\,Dv)=0 \quad \text{in\ }\Omega,\\ \partial u/\partial \ell=\varphi\quad \text{on\ }\partial \Omega,\qquad u=\psi\quad \text{on\ } S. \end{cases} $$ Since $b(x,\,v,\,Dv)\in C^{\alpha\beta}(\ol\Omega)$ (cf. \eqref{1.3}), it follows by Lemma~\ref{lem2.1} that there exists a unique solution $u\in C^{2+\alpha\beta}(\ol\Omega)$ of the above problem. This way, a nonlinear operator $$ {\cal F}\colon C^{1+\beta}(\ol\Omega)\longrightarrow C^{2+\alpha\beta}(\ol\Omega) $$ is defined by the formula ${\cal F}v=u.$ The mapping $\cal F$ is a continuous (in view of \eqref{2.2}) and compact ($C^{2+\alpha\beta}(\ol\Omega) \hookrightarrow C^{1+\beta}(\ol\Omega)$ compactly) mapping acting from $C^{1+\beta}(\ol\Omega)$ into itself. Moreover, the bound \eqref{3.10} provides an a~priori estimate with a constant $C,$ independent of $u$ and $\sigma\in[0,\,1],$ for each solution to the equation $u=\sigma {\cal F}u$ that is equivalent to the problem $$ \begin{cases} a^{ij}(x)D_{ij}u + \sigma b(x,\,u,\,Du)=0 \quad \text{in\ }\Omega,\\ \partial u/\partial \ell=\sigma\varphi\quad \text{on\ }\partial \Omega,\qquad u=\sigma\psi\quad \text{on\ } S. \end{cases} $$ Therefore, the Leray--Schauder theorem ensures existence of a fixed point $u={\cal F}u\in C^{2+\alpha\beta}(\ol\Omega)$ that is a solution of \eqref{1.1}. Finally, the assertion $u\in C^{2+\alpha}(\ol\Omega)$ follows easily by Lemma~\ref{lem2.1} and by using standard bootstrapping arguments. \begin{thebibliography}{GuS22} \bibitem[Ad]{Ad} {R.~Adams,} {\em Sobolev Spaces,} Academic Press, New York, 1975. \bibitem[A-C]{AC} {H.~Amann and M.~Crandall,} {\em On some existence theorems for semi-linear elliptic equations}, {Indiana Univ. Math. J.} {\bf 27} (1978), 779--790. \bibitem[B]{B} {A. V. Bicadze,} {\em A homogeneous problem for the slant derivative for harmonic functions in three-dimensional regions}, {Sov. Math. Dokl.} {\bf 4} (1963), 156--159. \bibitem[E]{E} {Y. Egorov,} {\em On subelliptic pseudodifferential operators}, {Sov. Math. Dokl.} {\bf 10} (1969), 1056--1059. \bibitem[E-K]{EK} {Y. Egorov and V. Kondrat'ev,} {\em The oblique derivative problem}, {Math. USSR Sbornik} {\bf 7} (1969), 139--169. \bibitem[Ga]{Ga} {E. Gagliardo,} {\em Teoremi di esistenza e di unicit\`a per problemi al contorno relativi ad equazioni paraboliche lineari e quasi lineari in $n$ variabili}, {Ricerche Mat.} {\bf 5} (1956), 239--257. \bibitem[G-T]{GT} {D.~Gilbarg and N.~S.~Trudinger,} {\em Elliptic Partial Differential Equations of Second Order,} 2nd ed., Springer--Verlag, Berlin, 1983. \bibitem[Gu]{Gu} {P. Guan,} {\em H\"older regularity of subelliptic pseudodifferential operators}, {Duke Math. J.} {\bf 60} (1990), 563--598. \bibitem[Gu-S1]{GuS1} {P. Guan and E. Sawyer,} {\em Regularity estimates for the oblique derivative problem}, {Ann. of Math.} {\bf 137} (1993), 1--70. \bibitem[Gu-S2]{GuS2} {P. Guan and E. Sawyer,} {\em Regularity estimates for the oblique derivative problem on non-smooth domains~I}, {Chinese Ann. of Math. Ser.~B} {\bf 16} (1995), No.~3, 299--324; {\em II}, ibid. {\bf 17} (1996), No.~1, 1--34. \bibitem[H]{H} {L. H\"ormander,} {\em Pseudodifferential operators and non-elliptic boundary value problems}, {Ann. of Math.} {\bf 83} (1966), 129--209. \bibitem[L-T]{LT} {G. Lieberman and N. S. Trudinger,} {\em Nonlinear oblique boundary value problems for nonlinear elliptic equations}, {Trans. Amer. Math. Soc.} {\bf 295} (1986), 509--546. \bibitem[M]{M} {V. Maz'ya,} {\em On a degenerating problem with directional derivative}, {Math. USSR Sbornik} {\bf 16} (1972), 429--469. \bibitem[M-Ph]{MPh} {V. Maz'ya and B. P. Paneyakh,} {\em Coercive estimates and regularity of the solution of degenerate elliptic pseudodifferential equations}, {Funct. Anal. Appl.} {\bf 4} (1970), 299--311. \bibitem[N]{N} {L.~Nirenberg,} {\em On elliptic partial differential equations}, {Ann. Scuola Norm. Super. Pisa, Sci. Fis. Mat., Ser. III} {\bf 123} (1959), 115--162. \bibitem[Pa-P]{PPa2} {D. K. Palagachev and P. Popivanov,} {\em On a degenerate boundary value problem for second order quasilinear elliptic operators,} Proc. Sixth Int. Coll. Differ. Equations, Plovdiv'95, VSP International Science Publishers, Zeist, The Netherlands, 1996, 197--208. \bibitem[Poi]{Poi} {H.~Poincar\'e,} {\em Th\'eorie de Mar\'ees, Lecons de M\'echanique C\'eleste,} Tome III, Gauthiers--Villars, Paris, 1910. \bibitem[P-Pa]{PPa} {P. Popivanov and D. K. Palagachev,} {\em Boundary value problem with a tangential oblique derivative for second order quasilinear elliptic operators}, {Nonl. Anal.} {\bf 21} (1993), 123--130. \bibitem[P-K1]{PK1} {P. Popivanov and N. Koutev,} {\em Sur le probl\`eme avec une tangentielle d\'eriv\'ee oblique pour une classe des \'equations elliptiques quasilin\'eaires du deuxi\`eme ordre}, {C. R. Acad. Sci. Paris S\'er. I Math.} {\bf 304} (1987), 383--385. \bibitem[P-K2]{PK2} {P. Popivanov and N. Kutev,} {\em The tangential oblique derivative problem for nonlinear elliptic equations}, {Commun. Partial Differ. Equations} {\bf 14} (1989), 413--428. \bibitem[Sm]{Sm} {H. F. Smith,} {\em The subelliptic oblique derivative problem}, {Commun. Partial Differ. Equations} {\bf 15} (1990), 97--137. \bibitem[To]{To} {F. Tomi,} {\em \"Uber semilineare elliptische Differentialgleichungen zweiter Ordnung}, {Math. Z.} {\bf 111} (1969), 350--366. \bibitem[Tr]{Tr} {F. Tr\`eves,} {\em A new method of proof of the subelliptic estimates}, {Commun. Pure Appl. Math.} {\bf 24} (1971), 71--115. \bibitem[W1]{W1} {B. Winzell,} {\em The oblique derivative problem I}, {Math. Ann.} {\bf 229} (1977), 267--278. \bibitem[W2]{W2} {B. Winzell,} {\em Subelliptic estimates for the oblique derivative problem}, {Math. Scandinav.} {\bf 43} (1978), 169--176. \bibitem[W3]{W3} {B. Winzell,} {\em The oblique derivative problem II}, {Ark. Mat.} {\bf 17} (1979), 107--122. \bibitem[W4]{W4} {B. Winzell,} {\em A boundary value problem with an oblique derivative}, {Commun. Partial Differ. Equations} {\bf 6} (1981), 305--328. \end{thebibliography} \end{document}