Gelfand problem on a large spherical cap
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Gelfand problem on a large spherical cap Yoshitsugu Kabeya1 · Vitaly Moroz2 Received: 5 August 2020 / Accepted: 4 November 2020 © Orthogonal Publisher and Springer Nature Switzerland AG 2020
Abstract We study the behaviour of the minimal solution to the Gelfand problem on a spherical cap under the Dirichlet boundary conditions. The asymptotic behaviour of the solution is discussed as the cap approaches the whole sphere. The results are based on the sharp estimate of the torsion function of the spherical cap in terms of the principle eigenvalue which we derive in this work. Keywords Gelphand problem · Spherical cap · Torsion function · Eigenvalues · Spherical harmonics Mathematics Subject Classification 35J60 · 33C55 · 35R01
1 Introduction We consider the nonlinear problem { − Δ𝕊N u = 𝜆f (u) u=0
in Ω, on 𝜕Ω,
(1.1)
where 𝜆 > 0 is a parameter, Δ𝕊N denotes the Laplace–Beltrami operator on the unit sphere 𝕊N ⊂ ℝN+1 ( N ≥ 1 ) and Ω ⊂ 𝕊N is a sub-domain in 𝕊N with a smooth boundary 𝜕Ω ≠ � . The principal Dirichlet eigenvalue of −Δ𝕊N in Ω is denoted by 𝜆1 (Ω) > 0 , and 𝜑1,Ω denotes the corresponding positive Dirichlet eigenfunction normalized as ‖𝜑1,Ω ‖2 = 1 . By wΩ we denote the torsion function of Ω , that is the unique solution of the Dirichlet problem * Vitaly Moroz [email protected] Yoshitsugu Kabeya [email protected]‑u.ac.jp 1
Department of Mathematical Sciences, Osaka Prefecture University, Gakuencho, Sakai 599‑8531, Japan
2
Department of Mathematics, Swansea University, Fabian Way, Swansea SA1 8EN, Wales, UK
13
Vol.:(0123456789)
Y. Kabeya, V. Moroz
{
− ΔwΩ = 1 in Ω, wΩ = 0 on 𝜕Ω.
(1.2)
By the standard elliptic regularity, wΩ ∈ C2 (Ω). We shall assume that the nonlinearity f ∈ C2 (ℝ) is a convex monotone increasing function with f (0) > 0 that satisfies the assumption
lim
s→∞
f (s) = +∞. s
(1.3)
Typical examples include
f (s) = exp(s),
f (s) = (1 + s)p
(p > 1).
Problem (1.1) with this type of nonlinearities is usually referred to as the Gelfand problem. It was introduced by Frank–Kamenetskii as a model of thermal explosion in a combustion vessel [9], and became known in the mathematical community due to the chapter written by Barenblatt in a survey by Gelfand [10, Chapter 15]. We denote { } f (s) f (s) , s∗ ∶= min s > 0 ∶ = a∗ . a∗ ∶= min (1.4) s>0 s s By convexity and since f (0) > 0 , we observe that a∗ > 0 and s∗ > 0 . Denote { } 𝜆∗Ω ∶= sup 𝜆 > 0 ∶ (1.1) has a classical positive solution . The following proposition is standard. Proposition 1.1 For each 𝜆 ∈ (0, 𝜆∗Ω ) , problem (1.1) admits a unique minimal classical positive solution u𝜆 . Moreover, (i) the following estimate holds,
𝜆 (Ω) 1 ≤ 𝜆∗Ω ≤ 1 . a∗ ‖wΩ ‖∞ a∗
(1.5)
(ii) For 𝜆 = 𝜆∗Ω problem (1.1) admits a weak extremal solution u∗ > 0 defined as
u∗ (x) ∶= lim∗ u𝜆 (x). 𝜆→𝜆Ω
(1.6)
(iii) For 𝜆 > 𝜆∗Ω problem (1.1) admits no weak solutions. For the precise definition of the weak solution see (2.4). In the case when Ω is a bounded smooth domain in ℝN , Proposition 1.1 was essentially proved in Gelfand [10], Keller and C
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