ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS Syafiq Johar
[email protected]
Contents 1 Introduction 1.1 Homogeneous Divergence Form Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 More General Divergence Form Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 H¨ older Continuity and Harnack’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . .
1 2 3 3
2 Global Estimates and Existence of Solutions 2.1 Lax-Milgram Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Schauder Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 4 5 6
3 The 3.1 3.2 3.3
. . . . .
6 6 7 7 7 7
4 Boundary Regularity 4.1 Boundary H 2 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Regularity in the Normal Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 7 8
5 Eigenvalues and Eigenvectors 5.1 Spectrum of Self-Adjoint Linear Compact Operators . . . . . . . . 5.1.1 Lemmas Required to Prove the Spectral Theory . . . . . . 5.2 Asymptotics of Eigenvalues in Self-Adjoint Case . . . . . . . . . . 5.2.1 Lemmas Required to Prove the Asymptotics of Eigenvalues
8 8 9 9 9
1
Maximum Principle Weak Maximum Principle . Strong Maximum Principle Non-linear Examples . . . . 3.3.1 Example 1 . . . . . . 3.3.2 Example 2 . . . . . .
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Introduction
Consider the differential operator for φ ∈ C m (Ω): X Lφ = aα Dα φ. |α|≤m
The adjoint operator L∗ : C ∞ (Ω) → C ∞ (Ω) is defined by: X L∗ ψ = (−1)|α| Dα (aα ψ). |α|≤m
1
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Proposition 1.1. Given p ∈ (1, ∞) and f ∈ Lp (Ω), then u ∈ Lp (Ω) is a weak solution of Lu = f as above if and only if ˆ f φ ≤ C||L0 φ||Lq (Ω) Ω
for all φ ∈ Cc∞ (Ω), with 1 =
1 q
+ p1 .
Definition 1.1 (Harmonic distributions). We say the solutions u ∈ D0 (Ω) is (sub / / super) harmonic if −∆u (≤ / = / ≥) 0. Proposition 1.2. If u ∈ C 2 (Ω), then for all x ∈ Ω: 2(d + 2) 1 (∆u)(x) = lim R→0 R2 |BR |
ˆ (u(y) − u(x)) dy. BR (x)
¯ is (sub / / super) harmonic and BR (x) ⊂ Ω, Proposition 1.3 (Mean value theorem). If u ∈ C 2 (Ω) then: ˆ 1 u(y) dy. u(x) (≤ / = / ≥) |BR | BR (x) ˆ 1 u(x) (≤ / = / ≥) u(y) dσ. |∂BR | ∂BR (x) Proposition 1.4. If u ∈ L1loc (Ω) is a weak harmonic solution, then u ∈ C ∞ (Ω) (redefining u on a set of measure zero if necessary). Proposition 1.5 (Harnack’s inequality). Suppose that Ω0 is an open connected bounded domain within Ω, an open domain. Suppose that u ≥ 0 and satisfies the mean value property on any ball in Ω, then: max u ≤ C(Ω, Ω0 , d) min u. 0 0 Ω
Ω
Proposition 1.6 (Caccioppoli’s inequality). Given x0 ∈ Ω, let 0 < ρ0 < ρ < d(x0 , ∂Ω). If ∆u = 0 in Ω, then: ˆ ˆ ˆ 1 1 2 2 |u| ≤ |u|2 . |∇u| ≤ 0 )2 0 )2 (ρ − ρ (ρ − ρ Bρ (x0 )\Bρ0 (x0 ) Bρ (x0 ) Bρ0 (x ) 0
Corollary 1.1. For any ρ0 < ρ and any s ∈ N, if ∆u = 0 in Bρ (x0 ) then: ||u||H s (Bρ0 (x0 )) ≤ C(ρ, ρ0 , s)||u||L2 (Bρ (x0 )) . In particular, if u ∈ H 1 (Ω), then u ∈ C ∞ (Ω). Proposition 1.7. Given ρ > 0 and un is a sequence of harmonic functions on Bρ . Assume that there exists a C such that ∀n, we have ||un ||L2 (Bρ ) ≤ C. Then, for all ρ > 0, there exists a subsequence um and u ∈ H k (Bρ ) such that: un → u in H k (Bρ0 ). Furthermore, ∆u = 0 in Bρ .
1.1
Homogeneous Divergence Form Equations
We now consider the divergence form problem: −div(A(x)∇u) = 0 in Ω
(1)
where A is a matrix valued map with entries in L∞ (Ω) for which there exists 0 < λ < Λ∞ such that for a.e. x ∈ Bρ (x0 ) and a.e. y ∈ R and any ζ ∈ Rd : λ|ζ|2 ≤ A(x, y)ζ · ζ ≤ Λ|ζ|2 . 2
Lemma 1.1. If u ∈ H 1 (Ω) is a solution of (1), then there exists δ(λ, d, ||A||∞ ) > 0 such that u ∈ 1,2+δ Wloc (Ω). Lemma 1.2 (Enhanced Caccioppoli’s inequality). Let u ∈ H 1 (Ω) be a solution of (1) on Ω, and suppose that A is symmetric. Let p > 1 and Bρ (x0 ) ⊂ Ω be such that |u|p+1 ∈ L1 (Bρ (x0 )). Then, for any η ∈ Cc∞ (Bρ (x0 )), we have: p+1 |u| 2 η ∈ H 1 (Bρ (x0 )). ˆ ˆ 2 p+1 Λ 2 η) ≤ |∇η|2 |u|p+1 . ∇(|u| λ Bρ Bρ Lemma 1.3. Suppose u ∈ Lq (B2R ) for some q > 1 and that there exists λ > 1 such that for any ball Bρ0 ⊂ Bρ ⊆ B2R there holds: ||u||Lλq (Bρ0 (x0 )) ≤
Cq N ρ − ρ0
q2 ||u||Lq (Bρ (x0 ))
where C, N > 0 are positive constants. Then, for all q ≤ s ≤ ∞: ||u||Ls (BR (x0 )) ≤
K(λ, C, q) 2λ
R q(λ−1)
||u||Lq (B2R (x0 )) .
1 Theorem 1.1 (L2 -like mean inequality). If u ∈ Hloc (Ω) is a solution of (1) on Ω, then for all x0 ∈ Ω such that BR (x0 ) ⊂ Ω and for almost every x ∈ B R (x0 ), there holds: 2
d2 Λ |u(x)| ≤ C(d) λ
1.2
1 |BR |
ˆ
! 12 u2 dx
.
BR (x0 )
More General Divergence Form Equations
Consider now the more general inhomogeneous divergence form in Ω: X X Dβ ((−1)|β| aαβ Dα u) = (−1)|β| Dβ fβ |α|,|β|≤m
(2)
|β|≤m
with fβ ∈ L2loc (Ω), supα,β,x aα,β (x) ≤ M a.e. in Ω (i.e. aα,β ∈ L∞ (Ω)) and there exists a constant λ > 0 such that for a.e. x ∈ Ω and any ζ ∈ Rd : X λ|ζ|2 ≤ aα,β ζα ζβ . |α|=|β|=m m Lemma 1.4 (Adapted Caccioppoli’s inequality). If u ∈ Hloc (Ω) is a weak solution of (2), then for every 0 0 < ρ < ρ such that Bρ (x0 ) ∈ Ω, there holds: 0 X ρ M , , ρ, d ||u||L2 (Bρ (x0 )) + ||f ||L2 (Bρ (x0 ) . ||u||Lm (Bρ0 (x0 )) ≤ C ρ λ |β|≤m
1.3
H¨ older Continuity and Harnack’s Inequality
We consider again the equation −div(A(x)∇u) = 0. Lemma 1.5. Suppose that u ∈ H 1 (B2R ) is a positive supersolution of (1) such that there exists ε > 0 so that |{x ∈ BR : u ≥ 1}| ≥ ε|BR |. Then, there exists a constant 0 < C ε, d, M λ < 1 such that: inf u > C.
BR 2
3
Proposition 1.8. Define the oscillation of u as oscΩ u = supΩ u − inf Ω u. If u is a weak solution of (1), then there exists a constant 21 < γ d, M λ < 1 such that: osc u ≤ γ osc u.
BR
B2R
2
Theorem 1.2 (Local H¨ older regularity). The weak solution u of (1) satisfies:
sup x,y∈B R
M |u(x) − u(y)| ≤ C d, |x − y|α λ
1 |BR |
2
for some 0 < α d,
M λ
ˆ
! 21 u2 dx
BR (x0 )
< 12 .
Lemma 1.6 (Doubling property). Given K > 2, there exists CK such that if u ∈ H 1 (B2KR ) satisfies Lu ≥ 0 in BKR , u ≥ 0 and inf BR u ≥ 1, then inf B KR u > CK . 2
Theorem 1.3 (Harnack’s inequality). There exists a constant C d, M λ non-negative solution of Lu = 0. Then: M inf u ≥ C d, sup u. BR λ BR
2
such that if u ∈ H 1 (B4R ) is a
Global Estimates and Existence of Solutions
The typical problem we will consider are of the forms: Lu := −div(A∇u)
=
f in Ω
u =
0 on ∂Ω.
Lu := −div(A∇u + B1 u) + B2 · ∇u + cu = u =
(3) f in Ω 0 on ∂Ω.
(4)
We are also going to consider the operator Lτ defined as Lτ u = Lu + τ u.
2.1
Lax-Milgram Theorem
Theorem 2.1 (Lax-Milgram Theorem). Let L be a continuous (i.e. bounded) linear map from a Hilbert space H to H ∗ . Assume also that L is elliptic i.e. there exists λ > 0 such that for all u ∈ H, |hLu, uiH ∗ ×H | ≥ λ||u||2H . Then, L is an isomorphism between H and H ∗ . Letting H = H01 (Ω) for a bounded Lipschitz domain Ω in Rd . Since H01 (Ω) is the closure of Cc∞ (Ω), we identify the dual of H01 (Ω) with the set of distributions f on Ω so that |hf, φiD0 ×D | ≤ C||φ||H 1 for all φ ∈ Cc∞ (Ω). We equip the space H −1 with the operator norm. By density, this holds for all φ ∈ H01 (Ω) hence: |hf, φiH −1 ×H01 | ≤ ||f ||H −1 (Ω) ||φ||H01 (Ω) . Thus H −1 (Ω) can be identified with (H01 (Ω))∗ . Lemma 2.1. If f ∈ H −1 (Ω), there exists functions f0 , f1 , . . . , fd ∈ L2 (Ω) such that for all φ ∈ H01 (Ω): ˆ hf, φi =
f0 φ + Ω
Furthermore, for all q ≥
2d d+2 ,
d ˆ X i=1
fi ∂i f.
Ω
Lq (Ω) is continuously embedded in H −1 (Ω). p
Suppose that for (4), A is elliptic and bounded, B1 , B2 ∈ Lp (Ω) with p > d, c ∈ L 2 (Ω) and τ ∈ R. 4
Lemma 2.2. The map Lτ maps H01 (Ω) → H −1 (Ω) continuously. Lemma 2.3. There exists a constant γ(λ, ||B1 ||Lp , ||B2 ||Lp , ||c− ||L p2 , p, Ω) ≥ 0 such that for all u ∈ H01 (Ω): λ |hLτ u, ui| ≥ ||u||2H 1 (Ω) + (τ − γ)||u||2L2 (Ω) . 0 3 When c ≥ 0 and B1 = B2 = 0, then γ = 0. Theorem 2.2 (Application of Lax-Milgram to elliptic PDEs). For τ ≥ γ as above, Lτ is an isomorphism between H01 (Ω) and H −1 (Ω). In particular, for any f ∈ H −1 (Ω), there exists a unique u ∈ H01 (Ω) such that −div(A∇u) = f in D0 (Ω). Theorem 2.3. Let Ω be a bounded regular domain in Rd for d ≥ 2. Given 1 < p < ∞ and k ∈ {−1, 0, 1, 2, . . .}, ∆ is an isomorphism W 2+k,p (Ω) ∩ W01,p (Ω) → W k,p (Ω) i.e. given f ∈ W k,p (Ω), there exists a unique u ∈ W 2+k,p (Ω) ∩ W 1,p (Ω) such that −∆u = f in D0 (Ω) such that u = 0 on ∂Ω. Furthermore, we have: ||u||W k+2,p (Ω) ≤ C(p, k, Ω)||f ||W k,p (Ω) .
2.2
Fredholm Alternative
Now we want to examine the case where τ < γ in Theorem 2.2. We need to use the Fredholm alternative. Recall the definition of compact operators: Definition 2.1 (Compact operators). Let X and Y be Banach spaces. A bounded linear operator K : ∞ X → Y is called compact provided if each bounded sequence {uk }∞ k=1 ⊂ X, the sequence {Kuk }k=1 ⊂ Y ∞ is precompact in Y . In other words, there exists a subsequence {uj }∞ j=1 such that {Kuj }j=1 converges. Remark 2.1. If H is a Hilbert space and K : H → H is compact, then so is the adjoint K ∗ : H → H. Theorem 2.4 (Fredholm alternative). Let K : H → H be a compact linear operator on a Hilbert space H. Then we have the following: 1. N (I − K) ⊂ X is finite dimensional. 2. R(I − K) is closed in Y . 3. R(I − K) = N (I − K ∗ )⊥ . 4. N (I − K) = {0} iff R(I − K) = H. 5. dim(N (I − K)) = dim(N (I − K ∗ )). Remark 2.2. The above theorem asserts that in particular, only one of the following can hold: 1. For each f ∈ H, the equation u − Ku = f has a unique solution. 2. The homogeneous equation u − Ku = 0 has nontrivial solutions u 6= 0. In addition, should (2) holds, the space of solutions of homogeneous problem is finite dimensional and the nonhomogeneous equation u − Ku = f has solutions iff f ∈ N (I − K ∗ )⊥ . Theorem 2.5 (Spectrum of compact operators). Let dim(H) = ∞ and K : H → H is compact. Then 1. 0 ∈ σ(K). 2. σ(K) \ {0} is finite or σ(K) \ {0} is a sequence tending to 0. 5
Theorem 2.6 (Application of Fredholm alternative to elliptic PDEs). There exists an at most countable set Σ ⊂ R such that for all τ ∈ [−∞, γ] \ Σ (where γ is from Lemma 2.3) and any f ∈ H −1 (Ω), the Lτ problem for (4) has a unique solution u ∈ H01 (Ω). If τ ∈ Σ, then there exists v ∈ H01 (Ω), v 6≡ 0 such that v solves the adjoint problem of (4): −div(AT ∇v + B2 v) + B1 · ∇v + cv + τ v = 0 in D0 (Ω). For each τ ∈ Σ, the associated vector space for v, denoted N ∗ (τ ), is of finite dimension. Moreover, dim(N ∗ (τ )) is also the dimension of the vector space of solutions in H01 (Ω) of: −div(A∇u + B1 u) + B2 · ∇u + cu + τ u = 0. Finally, for any τ ∈ R, a solution to the Lτ problem for (4) exists if and only if: hf, viH −1 ×H01 (Ω) = 0 for all v ∈ N ∗ (τ ).
2.3
The Schauder Method
Theorem 2.7. Consider the problem (3). Let Ω be a bounded regular domain in Rd for d > 2 and let dp . For any 1 ≤ q ≤ p∗ , there exists η(Ω, p, q) > 0 such that if: 1 < p < d. Let p∗ = d−p X |Aij (x) − δij | ≤ η ||A − Id ||∞ = sup x∈Ω
i,j
then, for any f ∈ Lp (Ω), there exists a unique weak solution u ∈ W01,q (Ω) to (3). Furthermore, there exists C(p, q, Ω) such that: ||u||W 1,q (Ω) ≤ C(p, q, Ω)||f ||Lp . 0
3
The Maximum Principle
Consider two forms (divergence and non-divergence) of the elliptic PDEs: LD u := −div(A∇u + B1 u) + B2 · ∇u + cu. X LN D u := − Aij Dij u + B · ∇u + cu.
(5) (6)
i,j
3.1
Weak Maximum Principle
Theorem 3.1 (Weak maximum principle for divergence form). Let Ω be a bounded open connected set and assume A, B1 , B2 , c ∈ L∞ (Ω) and additionally that c − div(B1 ) ≥ 0 in D0 (Ω). If u ∈ H 1 (Ω) is a subsolution to the divergence form LD as in (5) (i.e. LD u ≤ 0 in D0 ). Then: sup u ≤ sup u+ . Ω
∂Ω p
Corollary 3.1. If Ω is open bounded and connected, B1 , B2 ∈ Lp and c ∈ L 2 with p > d, provided that c − div(B1 ) ≥ 0 in D0 (Ω), there is a unique solution in H01 (Ω) to LD u = f in D0 (Ω). Theorem 3.2 (Weak maximum principle for non-divergence form). Let Ω be a bounded open set and ¯ is a subsolution to the non-divergence form LN D as in (6) (i.e. LN D u ≤ 0). c ≥ 0. If u ∈ C 2 (Ω) ∩ C(Ω) Then: max u ≤ max u+ . ¯ Ω
∂Ω
6
3.2
Strong Maximum Principle
Theorem 3.3 (Strong maximum principle for divergence form). Under the same hypothesis for weak maximum principle for divergence form, if for some u ∈ H 1 (Ω) such that LD u ≤ 0 and for some ball B ⊂ Ω, supB u = supΩ u ≥ 0, then u is a constant on Ω. Furthermore, if u 6≡ 0, then div(B1 ) − c = 0. Lemma 3.1 (Hopf’s Lemma). Let B be a ball in Rd . Suppose u ∈ C 2 (B) and LN D u ≤ 0 with c ≥ 0. Suppose that u attains its maximum at x0 ∈ ∂B, and u(x0 ) > u(x) in B. Suppose also that u is continuous at x0 and that cu(x0 ) ≥ 0 in Ω (so either c ≡ 0 or u(x0 ) ≥ 0). Then, ∂n u(x0 ) > 0. Theorem 3.4 (Strong maximum principle for non-divergence form). Suppose that Ω is connected, open ¯ satisfies LN D u ≤ 0 in Ω with c ≥ 0. If there exists x0 ∈ Ω such and bounded. Assume u ∈ C 2 (Ω) ∩ C(Ω) that u(x0 ) = maxΩ u and either c ≡ 0 or u(x0 ) ≥ 0, then u is constant in Ω. If c 6≡ 0 in Ω, then u ≡ 0.
3.3 3.3.1
Non-linear Examples Example 1
Consider the following problem: −∆u + a|∇u|2 + b = ∂n u =
0 in Ω 0 on ∂Ω
(7)
where a and b are in C 1 (Ω) and a−1 (0) = ∅. Proposition 3.1. Assuming that Ω is bounded and satisfies the interior sphere condition on its boundary, any regular solution to (7) satisfies: |∇u|2 < C(a, b, |∇a|, |∇b|). 3.3.2
Example 2
Consider the following problem for A ∈ L∞ (Ω; Rd×d ) and λ ≥ 0: −div(A(x)∇u) + λu =
H(x, ∇u) + f (x)
(8)
2
A(x)ζ · ζ
≥
α|ζ|
|H(x, ζ)|
≤
b(x)(1 + |ζ|)
|H(x, ζ) − H(x, ξ)|
≤
b(x)|ζ − ξ|
with b ∈ Ld (Ω). Proposition 3.2. Suppose that v ∈ H01 (Ω) satisfies −div(A∇v) + λv ≤ b(x)|∇v| in Ω. Then, v ≤ 0 in Ω.
4 4.1
Boundary Regularity Boundary H 2 Regularity
Proposition 4.1. Let Ω = B1 (0) ∩ {xd > 0} and u ∈ H01 (Ω) be a weak solution of: ˆ ˆ A∇u · ∇v = (f − b · ∇u − cu)v for all v ∈ H01 (Ω) Ω
+ BR
7
¯ with Aζ · ζ ≥ λ and ||A||∞ + ||b||∞ + ||c||∞ ≤ M . Suppose additionally that A ∈ C 1 (Ω). Then, 2 1 u ∈ H (Ω) ∩ H0 (Ω) and:
||u||H 2 (Ω)
4.2
M ≤ C Ω, M, λ
(||f ||L2 (Ω) + ||u||L2 (Ω) ).
Regularity in the Normal Direction
Proposition 4.2. Suppose that Ω is a bounded Lipschitz domain. There exists a constant C(Ω) such that if F ∈ L2 (Ω; Rd ) satisfies div(F ) ∈ L2 (Ω). Then, F · n, where n is the unit outward pointing normal, 1 can be defined as an element of H − 2 (∂Ω) which satisfies: ||F · n||
1
H − 2 (∂Ω)
≤ C(Ω)(||div(F )||L2 (Ω) + ||F ||L2 (Ω) )
¯ Rd ) such that h = n on ∂Ω where n is Lemma 4.1. If Ω is a C 2 domain, there exists a map h ∈ C 1 (Ω; the outward pointing unit normal. Lemma 4.2. If f ∈ L2 (Ω) and u ∈ H 1 (Ω) is a weak solution of −∆u = f in D0 (Ω), then ∂n u ∈ L2 (∂Ω) and: ˆ ˆ (∂n u)2 ≤ 2 (∂τ u)2 + C(h)(||f ||2H 1 (Ω) + ||u||2H 1 (Ω) ). ∂Ω
5
∂Ω
Eigenvalues and Eigenvectors
Proposition 5.1. Let Ω be a connected bounded domain with a Lipschitz boundary. Let τ be large enough so that c + τ − div(B1 ) > 0. If f ∈ Cc0,α (Ω) is such that f ≥ 0 and f 6≡ 0, then u ∈ H01 (Ω) is the solution of (5) such that u > 0 in Ω. We say that the linear compact operator K : C 0,β (Ω) → C 0,β (Ω) (with an appropriate β < α) associated with L−1 τ with zero boundary condition is strongly positive. Proposition 5.2. Under the hypothesis of the previous proposition, K has positive spectral radius r(K) > 0 and r(K) is a simple eigenvalue whose eigenfunction may be chosen to be positive. All other (real or complex) eigenvalues satisfy |µ| < r(K). Proposition 5.3 (Self-adjoint elliptic differential operator). Assume that AT = A and B1 = B2 = B. Then, for any u, v ∈ H01 (Ω), we have: hLτ u, viH −1 (Ω)×H01 (Ω) = hLτ v, uiH −1 (Ω)×H01 (Ω) . In other words, the operator Lτ is self-adjoint.
5.1
Spectrum of Self-Adjoint Linear Compact Operators
Theorem 5.1 (Spectral theorem for self-adjoint linear compact operators). Assume that Ω is bounded. Given the operator: L := H01 (Ω)
→ H −1 (Ω)
u →
−div(A∇u + Bu) + B∇u + cu
with elliptic AT = A ∈ L∞ (Ω; Rd×d ), B ∈ L∞ (Ω; Rd ) and c ∈ L∞ (Ω; R). The set of Σ ⊂ R for which Lu = µu in D0 (Ω) for µ ∈ Σ has non-trivial weak solutions u ∈ H01 (Ω) forms an increasing sequence µ1 < µ2 ≤ · · · where µk → ∞ and the repetition of an eigenvalue µk indicates the dimension of the 8
associated eigenspace. Furthermore, for each µk , there is an associated φk ∈ H01 (Ω) such that hLφk , ψi = µk (φk , ψ)L2 (Ω) with {φ1 , φ2 , . . .} forming a complete orthonormal set in L2 (Ω) in the sense that (φm , φn )L2 (Ω) = δmn and for all ψ ∈ L2 (Ω): ∞ X ψ= φk (φk , ψ)L2 (Ω) . k=1
If ψ ∈ 5.1.1
H01 (Ω),
the same identity holds and the convergence is with respect to the H01 (Ω) norm.
Lemmas Required to Prove the Spectral Theory
Lemma 5.1. Let {φi }ki=1 for k > 1 be a (finite) sequence of linearly independent vectors in H01 (Ω) such that (φm , φn )L2 (Ω) = δmn . Let H0 = 0 and Hk = span{φ1 , . . . , φk }. There exists µk+1 > 0 such that: µk+1 =
inf
u∈H01 (Ω)\{0} u∈Hk⊥
hLu, uiH −1 ×H 1 . ||u||L2
Furthermore, this infimum is attained i.e. there exists φk+1 ∈ (H01 (Ω) \ {0}) ∩ Hk⊥ with ||φK+1 ||L2 = 1 such that µk+1 = hLτ φk+1 , φk+1 iH −1 ×H 1 . Lemma 5.2. For every k such that Lφk = µk φk in D0 (Ω), we have µk+1 ≥ µk and limk→∞ µk = ∞. 2 Furthermore, the sequence {φk }∞ k=1 is complete in L (Ω). Lemma 5.3. Assume additionally that B ∈ W 1,∞ (Ω) and −div(B) + c > 0. The first eigenvalue is of multiplicity one and the corresponding eigenfunction is the only eigenfunction which can be chosen positive.
5.2
Asymptotics of Eigenvalues in Self-Adjoint Case
Theorem 5.2 (Asymptotic behaviour of eigenvalues in self-adjoint case). There exists a constant C d, Ω, M such that the k-th eigenvalue satisfies: λ 2
2
C −1 k d ≤ µk ≤ Ck d . 5.2.1
Lemmas Required to Prove the Asymptotics of Eigenvalues
Lemma 5.4 (Min-Max Principle). If {µ1 , µ2 , . . .} are the eigenvalues given above, they satisfy: µk = min
max
hLu, uiH −1 (Ω)×H01 (Ω) ||u||L2 (Ω)
Sk u∈Sk \{0}
where Sk denotes subspaces of dimension k of H01 (Ω). Corollary 5.1. If Ω0 ⊂ Ω bounded and λ0j and λj are the j-th eigenvalues (ordered by increasing values) of L in H01 (Ω0 ) and H01 (Ω0 ) respectively, then: λ0j ≥ λj . Lemma 5.5. The asymptotic behaviour of the eigenvalues is true on the cube [0, π]d when L = −∆.
9
