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Since there are many directions one can take when studying this equation, I am curious:

Given a function $f \in C^2$ defined on some open set, what information is given by $\nabla^2 f \geq 0$? Please let me know if I am missing something important for the question to make sense.

Thanks in advance.

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    What kind of information are you interested in? For instance they are (I'm assuming you mean functions with a non-negative Laplacian) subharmonic, and in particular they satisfy the maximum principle. – Jose27 Oct 03 '18 at 05:09
  • See here: https://math.stackexchange.com/questions/50274/intuitive-interpretation-of-the-laplacian/50285#50285 – Christian Blatter Oct 03 '18 at 08:46

1 Answers1

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It depends on your definition of $\nabla^2 f\ge 0$: If it means that the Hessian matrix of $f$ is positive semidefinite, then $f$ is a convex function.

daw
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