Rigorous Numerical Enclosures for Positive Solutions of Lane–Emden's Equation with Sub-Square Exponents. (17th February 2022) - Tanaka, Kazuaki; Plum, Michael; Sekine, Kouta; Kashiwagi, Masahide; Oishi, Shin'ichi
Abstract: The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane–Emden's equation − Δ u = | u | p − 1 u with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution u nearby a numerically computed approximation u ̂ together with an explicit error bound, i.e., a bound for the difference between u and u ̂ . In particular, we focus on the sub-square case in which 1 < p < 2 so that the derivative p | u | p − 1 of the nonlinearity | u | p − 1 u is not Lipschitz continuous. In this case, it is problematic to apply the classical Newton-Kantorovich theorem for obtaining the existence proof, and moreover several difficulties arise in the procedures to obtain numerical integrations rigorously. We design a method for enclosing the required integrations explicitly, proving the existence of a desired solution based on a generalized Newton-Kantorovich theorem. A numerical example is presented where an explicit solution-enclosure is obtained for p = 3 / 2 on the unit square domain Ω = ( 0, 1 ) 2 .