Abstract
In this paper, we report on some recent results dealing with geometrical properties of solutions of some semilinear elliptic equations in bounded smooth convex domains. We investigate the quasiconcavity, i.e. the fact that the superlevel sets of a positive solution are convex or not.We actually construct a counterexample to this fact in two dimensions, showing that the solutions under consideration do not always inherit the convexity of the domain. We report on the results in [23].
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Hamel, F., Nadirashvili, N., Sire, Y. (2020). Geometric properties of superlevel sets of semilinear elliptic equations in convex domains. In: de Gier, J., Praeger, C., Tao, T. (eds) 2018 MATRIX Annals. MATRIX Book Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-38230-8_16
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