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\title[Harmonic functions on self-similar structures]
{\large A short note on harmonic functions on self-similar structures}  

\author[B. E. Breckner]{BRIGITTE E. BRECKNER}

\date{}
\begin{abstract}
A sufficient condition is given concerning the harmonic structure on a post critically finite self-similar structure $K$ that ensures that harmonic functions are not zero divisors in the algebra of real-valued continuous functions on $K$.
\end{abstract}

\begin{amssubject}
28A80, 49J52.
\end{amssubject}

\begin{keyword}
Self-similar structure, harmonic function, Sierpinski fractal.
\end{keyword}



\maketitle

\let\thefootnote\relax\footnote{ \ }\hspace{-5mm}\let\thefootnote\relax\footnote{\hspace{-13pt}DOI: 10.24193/mathcluj.2020.2.05}




\begin{thebibliography}{10}


\bibitem{Br1}
B.E. Breckner, {\it Real-valued functions of finite energy on the Sierpinski gasket}, Mathematica, {\bf (55) 78} (2013), {\it 2}, 142--158.
\bibitem{Br2}
B.E. Breckner,  {\it A short note on harmonic functions and zero divisors on the Sierpinski fractal}, Arch. Math. (Basel), {\bf 106} (2016), 183--188.
\bibitem{BrCh}
B.E. Breckner and R. Chill, {\it The Laplace operator on the Sierpinski gasket with Robin boundary conditions}, Nonlinear Anal. Real World Appl., {\bf 38} (2017), 245--260.
\bibitem{BrVa1}
B.E. Breckner and  Cs. Varga,  {\it Multiple solutions of Dirichlet problems on the Sierpinski gasket}, J. Optim. Theory Appl., {\bf 167} (2015),  {\it 3}, 842--861.
\bibitem{BrVa2}
B.E. Breckner and Cs.  Varga,
{\it An application of a fixed-point theorem to Neumann problems on the Sierpinski fractal}, Fixed Point Theory, {\bf19} (2018), {\it 2}, 475--486.
\bibitem{Ga1}
M. Galewski, {\it Multiple Solutions to a Dirichlet problem on the Sierpinski gasket}, Taiwanese J. Math., {\bf 20} (2016), {\it 5}, 1079--1092.
\bibitem{Ga2}
M. Galewski, {\it Optimization problems on the Sierpinski gasket}, Electron. J. Differential Equations, {\bf 2016}, {\it 105}, 1--11.
\bibitem{Ki1}
J. Kigami, {\it A harmonic calculus on the Sierpinski spaces}, Japan J. Appl. Math., {\bf 6} (1989),  259--290.
\bibitem{Ki2}
J. Kigami, {\it Harmonic calculus on p.\,c.\,f. self-similar sets}, Trans. Am. Math. Soc., {\bf 335} (1993), 721--755.
\bibitem{Ki} 
J. Kigami, {\it Analysis on Fractals}, Cambridge University Press, Cambridge, UK, 2001.
\bibitem{MoRa}
G. Molica Bisci and V. R\u{a}dulescu, {\it A characterization for elliptic problems on fractal sets}, Proc. Amer. Math. Soc., {\bf143} (2015), {\it 7}, 2959--2968.
\bibitem{MoReSe}
G. Molica Bisci, D. Repov\v{s} and R.  Servadei, {\it Nonlinear problems on the Sierpinski gasket}, J. Math. Anal. Appl., {\bf 452} (2017), {\it 2}, 883--895.
\bibitem{stribook}
R.S. Strichartz, {\it Differential Equations on Fractals. A Tutorial}, Princeton University Press, Princeton, NJ, 2006.
\end{thebibliography}

\vspace{10pt}

\hspace{-4mm}{\small{Received December 3, 2019}}\\
\hspace{-4mm}{\small{Accepted March 25, 2020}}

\vspace{-25.3pt}
\ \hfill \
\begin{tabular}{c}
{\small\em Babe\c{s}-Bolyai University}\\
{\small\em Faculty of Mathematics and Computer Science}\\
{\small\em  Str. M. Kog\u{a}lniceanu nr. 1}\\
{\small\em 400084 Cluj-Napoca, Romania} \\
{\small\em E-mail: {\tt brigitte@math.ubbcluj.ro}}\\
\end{tabular}
\end{document}

