2016/09/13 by Antonio R�os, David Ruiz, Ros, Antonio +3 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1609.03739
In this paper we construct nontrivial exterior domains \Ω \⊂\n\ℝN, for all N\≥ 2, such that the problem
left
ll -
Deltaν +u -up=0,
u gt;0 amp;
mboxin
;
Omega, 1mm]\n
u= 0 amp;
mboxon
;
partial
Omega, [1mm]\n
frac
partial u
partial
nu =
mboxcte amp;
mboxon
;
partial\n
Omega,
right. admits a positive bounded solution. This result gives a\nnegative answer to the Berestycki-Caffarelli-Nirenberg conjecture on\noverdetermined elliptic problems in dimension 2, the only dimension in which\nthe conjecture was still open. For higher dimensions, different counterexamples\nhave been found in the literature; however, our example is the first one in the\nform of an exterior domain.\n