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Math Help - Maximum Principle

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    Junior Member HAL9000's Avatar
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    Question Maximum Principle

    The maximum principle for harmonic functions asserts:
    Let u \in C^2(U) \cap C(\bar{U}) be harmonic in U \subset \mathbb{R}^n.
    Then
    1) (Weak maximum principle)

    \max_{\bar{U}}u=\max_{\partial U}u;

    2) (Strong maximum principle)

    If U is connected and there exists a point x_0 \in U such that

    u(x_0)=\max_{\bar{U}}u,

    then u=const in U.

    I wonder if I got this right. The weak principle means that the maximal value on \bar{U} of a harmonic function is necessarily attained in a boundary point, i.e. there are no solely interior points which give maximal values for a harmonic function. Although it may be that there are points x \in \partial U and y \in U such that

    u(y)=\max_{\bar{U}}u=u(x),

    in which case the shape of U may cause the function to be constant (i.e. in case U is connected and so 2) holds); but if U is any open set nothing specific is known about u, i.e. it may rather be non-constant.

    I wonder why the conclusion of the strong version holds just for U and not for \bar{U}. I mean 1) holds anyway and u \in C(\bar{U}). Doesn't this, together with 2), imply that u is constant on the boundary? The proof of the strong version, however, works only with U, so that it's no big deal to accept it as it is.
    Last edited by HAL9000; January 16th 2013 at 08:48 AM.
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