Boundary regularity for some nonlinear elliptic degenerate equations |
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Authors: | Haïm Brezis Pierre-Louis Lions |
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Institution: | 1. Department of Mathematics, University of Paris VI, F-75230, Paris, France
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Abstract: | We consider the nonlinear elliptic degenerate equation (1) $$ - x^2 \left( {\frac{{\partial ^2 u}}{{\partial x^2 }} + \frac{{\partial ^2 u}}{{\partial y^2 }}} \right) + 2u = f(u)in\Omega _a ,$$ where $$\Omega _a = \left\{ {(x,y) \in \mathbb{R}^2 ,0< x< a,\left| y \right|< a} \right\}$$ for some constanta>0 andf is aC ∞ functions on ? such thatf(0)=f′(0)=0. Our main result asserts that: ifu∈C \((\bar \Omega _a )\) satisfies (2) $$u(0,y) = 0for\left| y \right|< a,$$ thenx ?2 u(x,y)∈C ∞ \(\left( {\bar \Omega _{a/2} } \right)\) and in particularu∈C ∞ \(\left( {\bar \Omega _{a/2} } \right)\) . |
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