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Consider $u:\mathbf{R}\times\omega\rightarrow\mathbf{R}$, where $\omega\subset\mathbf{R}^{n-1}$ is a bounded domain. For each $y\in\omega$ and each $\lambda>0$, consider $y^\lambda=(x,2\lambda-y_1,y_2,...,y_{n-1})$ the reflection of $y$ in the plane $\{y_1=\lambda\}$. Suppose that $$w_\lambda(x,y)=u(x,y)-u(x,y^\lambda)<0.$$ If $y_1>0$ then $$\frac{\partial u}{\partial y_1}\leq0.$$

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    Are you sure this is the right formulation of the question? I think there is no function u with this property because $w_{y_1}(x,y)=u(x,y)-u(x,y^{y_1})=0$2012-12-04
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    Do you assume that $\lambda$ is such that $u(x,y^\lambda)$ is defined? We must have $y^\lambda \in \omega$.2012-12-04
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    Zach, $\lambda\in(0,\sup_\omega y_1)$ then $u(x,y^\lambda)$ is well defined. Dominik, when you take $\lambda=y_1$, you have $w_{y_1}=0$, it is ok. I know that this consequence is true, but i don't know how to argue. I accept ideas. Thanks.2012-12-05

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