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Zbl 0881.35028
Di Fazio, G.; Palagachev, D.K.
Oblique derivative problem for elliptic equations in non-divergence form with $VMO$ coefficients. (English)
[J] Commentat. Math. Univ. Carol. 37, No.3, 537-556 (1996). ISSN 0010-2628

The authors prove $L^p$ estimates and $W^{2,p}$ existence results for the oblique derivative problem with elliptic equation of second order in non divergent form with bounded leading coefficients in VMO space. More precisely, they consider the solution $u$ of the problem $$ \Cal L u \equiv \sum _{i,j=1}^{n} a^{i,j}(x)\frac {\partial ^2 u}{\partial x_i \partial x_j} = f(x)\ \ \text {a.e. on } \ \Omega , \tag 1 $$ $$ \Cal B u \equiv \sum _{i=1}{n} l_i(x) \frac {\partial u}{\partial x_i} + \sigma (x) u = \phi (x) \ \ \text { on } \partial \Omega \tag 2 $$ under the assumption of symmetry and uniform ellipticity of (1) and a nondegeneracy condition $$ l(x) \nu (x) > 0 , \sigma (x) < 0 \ \ \text {on} \partial \Omega .$$ Here $\Omega $ is a bounded domain with $C^{1,1}$ boundary, $\nu $ is the unit inner normal to the boundary, $l, \sigma $ are Lipschitz functions. The main results are summarized in theorems 1, 2. \par { Theorem 1:} Let $u \in W^{2,q}(\Omega ), 1 < q \leq p < \infty , \Cal L u \in L^p(\Omega ), \ \ \Cal B u \in W^{1 - 1/p, p}(\partial \Omega )$. Then $u \in W^{2,p}(\Omega )$ and the norm of $u$ in this space can be estimated by norms of $\Cal L u$ and $\Cal B u$ in corresponding spaces. \par { Theorem 2:} Let $1 < p < \infty $. Then the problem (1), (2) admits a unique strong solution $u \in W^{2,p}(\Omega )$ for every $f \in L^p(\Omega )$ and $\phi \in W^{1 - 1/p,p}(\partial \Omega )$. \par The problem (1), (2) with smooth data has already been well studied. Thus the main purpose of this paper is to study the weakening of assumptions on leading coefficients $a^{i,j}$ to the class VMO of functions with vanishing mean oscillations. The approach is similar to the proof of analogous results for Dirichlet boundary value problem given by {\it F. Chiarenza, M. Frasca} and {\it P. Longo.} The main step in proving Theorem 1 consists in estimates near the boundary, where the representation of solution's second derivatives by singular integral operators is used. Theorem 2 then folows by homotopy arguments.
[J.Stará (Praha)]

MSC 2000:: *35J25 Second order elliptic equations, boundary value problems

Keywords: oblique derivative; elliptic equation; non divergence form; VMO coefficients; strong solution; singular integral estimates

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