Geodesic
Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set
Journal of convex analysis, Tome 25 (2018) no. 1, pp. 319-337
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We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact planar set, subject to attractive Robin boundary conditions. Under either a constraint of fixed perimeter or area, we show that the maximiser within the class of exteriors of convex sets is always the exterior of a disk. We also argue why the results fail without the convexity constraint and in higher dimensions.
Classification : 35P15, 58J50
Mots-clés : Robin Laplacian, negative boundary parameter, exterior of a convex set, lowest eigenvalue, spectral isoperimetric inequality, spectral isochoric inequality, parallel coordinates 
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     author = {D. Krejcir{\'\i}k and V. Lotoreichik},
     title = {Optimisation of the {Lowest} {Robin} {Eigenvalue} in the {Exterior} of a {Compact} {Set}},
     journal = {Journal of convex analysis},
     pages = {319--337},
     year = {2018},
     volume = {25},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2018_25_1_JCA_2018_25_1_a17/}
}
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D. Krejcirík; V. Lotoreichik. Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set. Journal of convex analysis, Tome 25 (2018) no. 1, pp. 319-337. http://geodesic.mathdoc.fr/item/JCA_2018_25_1_JCA_2018_25_1_a17/