Let $f:ℝ²→ℝ$ be an arbitrary harmonic function. A level curve in two dimensions is a curve on which the value of a function $f(x,y)$ is a constant. My question is: Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$ (this mean that there exists a unique real number $a$ such that for all $y$, $f(x,y)=0$ implies $x=a$). The same question for complex functions from the complex plane into itself.
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1$\begingroup$ Does this mean that there exists an $a$ such that for all $y$, $f(x,y)=0$ implies $x=a$? Or does it mean that for all $y$, there exists an $a$ such that $f(x,y)=0$ implies $x=a$? $\endgroup$– Steven LandsburgCommented Mar 29, 2014 at 13:57
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$\begingroup$ @StevenLandsburg: It is the first sentence. $\endgroup$– SafwaneCommented Mar 29, 2014 at 14:09
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Modulo an elementary transformation you are asking for harmonic functions which are zero on a line and nowhere else. Such a function is linear. This is Theorem I in Short proofs of three theorems on harmonic functions by H.P. Boas and R.P. Boas (published in Proc. Amer. Math. Soc. 102 (1988), 906-908).
answered Mar 29, 2014 at 15:27