An Min-Max Problem in Riemannian Geometry
问题 1. Let $\Omega$ be a connected open subset of $\R^n$ and $u:\Omega\to\R$ be a $C^\infty$ function. Call $\Delta$ the Laplace operator
\[
\Delta u=\sum_{i=1}^n\frac{\partial^2 u}{\partial (x^i)^2}.
\]
If $u$ satisfies $\Delta u\geq0$, prove that $u$ cannot achieve a strict local maximum at any interior point of $\Omega$.
\[
\Delta u=\sum_{i=1}^n\frac{\partial^2 u}{\partial (x^i)^2}.
\]
If $u$ satisfies $\Delta u\geq0$, prove that $u$ cannot achieve a strict local maximum at any interior point of $\Omega$.
解答 1.1. Here is the first answer…
解答 1.2. Here is another answer…
问题 2. Let $(M,g)$ be a complete Riemannian manifold without boundary (possibly noncompact) and $u\mathpunct{:}M\to\R$ be a $C^\infty$ function. Call $\Delta$ the Laplace operator of $g$
\[
\Delta u=g^{ij}\frac{\partial^2 u}{\partial x^i\partial x^j}-g^{ij}\Gamma_{ij}^k\frac{\partial u}{\partial x^k}.
\]
where $\Gamma_{ij}^k$ are the Christoffel symbols of the Levi-Civita connection of $g$. If $u$ satisfies $\Delta u\geq 0$, prove that $u$ cannot achieve a strict local maximum at any point of $M$.
\[
\Delta u=g^{ij}\frac{\partial^2 u}{\partial x^i\partial x^j}-g^{ij}\Gamma_{ij}^k\frac{\partial u}{\partial x^k}.
\]
where $\Gamma_{ij}^k$ are the Christoffel symbols of the Levi-Civita connection of $g$. If $u$ satisfies $\Delta u\geq 0$, prove that $u$ cannot achieve a strict local maximum at any point of $M$.
解答 2.1. The first answer for problem 2
解答 2.2. The second answer for problem 2
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