In particular, what is known for harmonic
functions? The proof of standard Maximum Principle for
Solutions of Elliptic Systems is based on
Proposition 1: If $B$ is positive definite matrix and $A$ non- positive
definite matrix, then $tr(AB)\leq 0$, where tr denotes trace.
In general $AB$ is not non positive definite matrix.
Are there new techniques which are not essentially based on Proposition 1?
The interested reader can see for example:
[1] D. Khavinson, An extremal problem for harmonic
functions in the ball, Canadian Math. Bulletin 35(2) (1992), 218-220.
Sahand Ghaseminejad Liasi
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