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- W2962999755 abstract "We prove m-dimensional symmetry results, that we call m-Liouville theorems, for stable and monotone solutions of the following nonuniformly elliptic equation −div((x ′ )∇u(x)) = �(x ′ )f(u(x)) for x = (x ′ ,x ′′ ) ∈ R d × R s = R n , where 0 ≤ m < n and 0 < �, are smooth functions and f ∈ C 1 (R). The interesting fact is that the decay assumptions on the weight function (x ′ ) play the fundamental role in deriving m-Liouville theorems. We show that under certain assumptions on the sign of the nonlinearity f, the above equation satisfies a 0-Liouville theorem. More importantly, we prove that for the double-well potential nonlinearities, i.e. f(u) = u−u 3 , the above equation satisfies a (d+1)-Liouville theorem. This can be considered as a higher dimensional counterpart of the celebrated conjecture of De Giorgi for the Allen-Cahn equation. The remarkable phenomenon is that the tanh function that is the profile of monotone and bounded solutions of the Allen-Cahn equation appears towards constructing higher dimensional Liouville theorems." @default.
- W2962999755 created "2019-07-30" @default.
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- W2962999755 date "2013-11-27" @default.
- W2962999755 modified "2023-09-23" @default.
- W2962999755 title "Higher-Dimensional Solutions for a Nonuniformly Elliptic Equation" @default.
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- W2962999755 doi "https://doi.org/10.1093/imrn/rnt249" @default.
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