The Limit Equation of a Singularly Perturbed System
In this chapter we establish the limit equations of the singularly perturbed elliptic and parabolic systems arising in the study of Bose–Einstein condensation. The proof relies on a stationary condition and a monotonicity formula.
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The Limit Equation of a Singularly Perturbed System
Abstract In this chapter we establish the limit equations of the singularly perturbed elliptic and parabolic systems arising in the study of Bose–Einstein condensation. The proof relies on a stationary condition and a monotonicity formula.
7.1 The Elliptic Case In this section, we consider the limit equation of the following system as κ → +∞: −Δui = fi (ui ) − κui
bij u2j ,
in B1 (0),
(7.1)
j =i
where bij > 0 are constants which satisfy bij = bj i , 1 ≤ i, j ≤ M. B1 (0) is the unit ball in Rn (n ≥ 1). Typical model for fi (ui ) is fi (u) = ai u − up with constants ai > 0, p > 1. We only consider positive solutions, that is, those ui ≥ 0 in its domain for all i. We will denote the solution corresponding to κ as uκ = (u1,κ , u2,κ , . . . , uM,κ ). Solutions to the above problem are critical points of the following functional: 1 κ Jκ (u) = |∇ui |2 + bij u2i u2j − Fi (ui ), (7.2) 4 Ω 2 i
i=j
i
u
where Fi (u) = 0 fi (t) dt. As described in the introduction, the expected limit equation of (7.1) as κ → +∞ should be ⎧ −Δui ≤ fi (ui ), ⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎪ ⎨−Δ ui − uj ≥ fi (ui ) − fj (uj ), ⎪ ⎪ ⎪ ⎪ ui ≥ 0, ⎪ ⎪ ⎪ ⎩ ui uj = 0,
j =i
in B1 (0), in B1 (0), (7.3)
j =i
in B1 (0), in B1 (0).
In this section, we will verify the above conjecture. K. Wang, Free Boundary Problems and Asymptotic Behavior of Singularly Perturbed Partial Differential Equations, Springer Theses, DOI 10.1007/978-3-642-33696-6_7, © Springer-Verlag Berlin Heidelberg 2013
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The Limit Equation of a Singularly Perturbed System
Theorem 7.1.1 As κ → +∞, any bounded solution uκ of (7.1) converges to u, which satisfies the system (7.3). Remark 7.1.2 If fi (u) = ai u−up where ai > 0, p > 1 are constants, uκ are positive solutions of (7.1) with suitable boundary conditions, ui,κ are uniformly bounded in κ. For fixed κ < +∞, because the solution uκ is bounded in B1 (0), standard elliptic estimates show that it is smooth. Since it is the critical point of the functional (7.2), we can consider the following domain variation of uκ . Take a compactly supported vector field Y ∈ C0∞ (B1 (0)); define usκ (x) = uκ x + sY (x) . It is well defined for |s| small and smooth in B1 (0). Now by the definition of critical points, we have
d s
= 0. Jκ uκ ds s=0 From this condition, by a well-known computation (analogues to the derivation of monotonicity formula in harmonic map or Yang-Mills field), we get
κ 2 2 1 2 |∇ui,κ | + ui,κ uj,κ − Fi (ui,κ ) div Y 4 Ω 2 i=j
i
−
i
∇Y (∇ui,κ , ∇ui,κ ) = 0.
(7.4)
i
Written in coordinates, the divergence of a vector field is div Y =
∂Yi , ∂xi
i=1,...,n
and for a function v ∇Y (∇v, ∇v) =
i,j =1,...,n
∂Yi ∂v ∂v . ∂xj ∂xi ∂xj
Note that, with our assumption on fi , from the above identity we can derive the following famous monotonicity formula (for another approach, see [4]): ∃ a constant C > 0 independent of κ, such that 1 κ 2 2 eCr r 2−n |∇ui,κ |2 + ui,κ uj,κ (7.5) 4 Br 2 i
is nondecreasing in r.
i=j
7.1 The Ellipti
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