Geodesic
A characterization result for the existence of a two-phase material minimizing the first eigenvalue
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1215-1226

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Given two isotropic homogeneous materials represented by two constants 0<α<β in a smooth bounded open set Ω⊂RN, and a positive number κ<|Ω|, we consider here the problem consisting in finding a mixture of these materials αχω+β(1−χω), ω⊂RN measurable, with |ω|≤κ, such that the first eigenvalue of the operator u∈H01(Ω)→−div((αχω+β(1−χω))∇u) reaches the minimum value. In a recent paper, [6], we have proved that this problem has not solution in general. On the other hand, it was proved in [1] that it has solution if Ω is a ball. Here, we show the following reciprocate result: If Ω⊂RN is smooth, simply connected and has connected boundary, then the problem has a solution if and only if Ω is a ball.

DOI : 10.1016/j.anihpc.2016.09.006
Classification : 49J20
Keywords: Two-phase material, Control in the coefficients, Eigenvalue, Non-existence 
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     author = {Casado-D{\'\i}az, Juan},
     title = {A characterization result for the existence of a two-phase material minimizing the first eigenvalue},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1215--1226},
     publisher = {Elsevier},
     volume = {34},
     number = {5},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.09.006},
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Casado-Díaz, Juan. A characterization result for the existence of a two-phase material minimizing the first eigenvalue. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1215-1226. doi: 10.1016/j.anihpc.2016.09.006

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