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We consider, for example, the harmonic function $f(x,y)=x^2-y^2$ in $\Omega = (0,1)^2.$ Is it possible to extend $f$ on $R^2$ to a $C^2-$function which is not harmonic in $R^2$? Thanks

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Sure, just take $x^2 - y^2$ and add a non-harmonic but $C^2$ "bump" somewhere away from $\Omega$.

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