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In questa Nota gli autori presentano alcuni risultati riguardanti il comportamento alla frontiera di domini non cilindrici delle soluzioni positive dell'equazione del calore. Una conseguenza è che due soluzioni positive qualunque, che si annullano su una parte della frontiera laterale, tendono a zero con lo stesso ordine.
The convexity of level sets of solutions to the mean curvature equation is a long standing open problem. In this paper we give a counterexample to it.
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