Abstract
Let u be harmonic in a simply connected domainG ⊂ ℝ2 and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth functionu 1 and a second order uniformly elliptic operatorL with smooth coefficients in ℝ2, satisfying:u 1=u inK, Lu 1=0 in ℝ2. A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved.
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Esposito, A. An elliptic continuation result for harmonic functions in two dimensions. Rend. Circ. Mat. Palermo 53, 437–442 (2004). https://doi.org/10.1007/BF02875736
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DOI: https://doi.org/10.1007/BF02875736