I need a counterexample can prove that sequence of harmonic functions may not convergences to a harmonic function

Question

Firstly, we know that if the sequence is uniformly convergences to a function then it must be harmonic. As the title,just given the sequence convergences to function, can we get it will be a harmonic function? If not, i need a example.

Answer 1

I assume that you mean pointwise convergence (not convergence in some function space). If so, there are ways to construct (using Runge's theorem) a sequence of holomorphic functions that converges pointwise to (say) a non-continuous function. Taking the real part gives you a counterexample for harmonic functions.

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More details: let $\Omega$ be the unit disc $\mathbb{C}$ ($z = x+iy$) and define a sequence of compact sets $K_j$, $L_j$ ($j = 2, 3, \ldots$) where

$$ K_j = \{ xy = 0 \} \cap D_j, $$ and $$ L_j = \{ |x| > 2^{-j}, |y| > 2^{-j} \} \cap D_j $$ where $D_j = \{ |z| < 1-2^{-j} \}$. (Draw a picture!) Then the complement of $K_j \cup L_j$ is connected, so by Runge's theorem, there is a polynomial $p_j$ where $$ | p_j(z) - 0 | < 2^{-j} \quad\text{for $z \in L_j$} $$ and $$ | p_j(z) - 1 | < 2^{-j} \quad\text{for $z \in K_j$}. $$ It's straightforward to see that $p_j$ converges pointwise on $\Omega$ to a function that is $1$ on the coordinate axes and $0$ otherwise. Take the real part of $p_j$ to get a sequence of harmonic functions that converges pointwise to the same discontinuous limit function.