In this paper we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates (r, θ, z) , we investigate the shape of possibly sign-changing solutions whose derivative in θ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of θ or strictly monotone with respect to θ. In the special case of a planar domain, the result holds in a circular sector as well as in an annular one, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case where its opening is larger than π.

Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains

Greco A.
2023-01-01

Abstract

In this paper we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates (r, θ, z) , we investigate the shape of possibly sign-changing solutions whose derivative in θ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of θ or strictly monotone with respect to θ. In the special case of a planar domain, the result holds in a circular sector as well as in an annular one, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case where its opening is larger than π.
Eigenvalues; Monotonicity; Morse index; Semilinear PDEs; Symmetry
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11584/347415
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