Home > Analysis, Partial Differential Equations > Problem regarding subharmonic functions

Problem regarding subharmonic functions


Suppose that a function f: \Bbb{R}^2 \to \Bbb{R} of class C^2 satisfies the inequality f_{xx}+f_{yy} \geq 0 at every point of \Bbb{R}^2. Suppose also that all its critical points are non-degenerate, i.e. the matrix of second order derivatives at the critical point has non-zero determinant. Prove that f cannot have local maxima.

Hint: A function with the property that \Delta f\geq 0 is called subharmonic. For the subharmonic functions, the maximum principle holds, i.e. the maximum of a subharmonic function cannot be achieved in the interior of the domain. If a subharmonic function has a local maximum then it is locally constant, and therefore has degenerate critical points. This solves our problem.

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