11 Classical isoperimetric inequality [De Giorgi] St. Petersburg, July 2010 13 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
11 Classical isoperimetric inequality [De Giorgi] ∀ E ⊂ IR n . H n − 1 ( ∂ ∗ E ) ≥ nω 1 /n | E | 1 /n ′ n St. Petersburg, July 2010 13 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
11 Classical isoperimetric inequality [De Giorgi] ∀ E ⊂ IR n . H n − 1 ( ∂ ∗ E ) ≥ nω 1 /n | E | 1 /n ′ n Here: ∂ ∗ E stands for the essential boundary of E , • St. Petersburg, July 2010 13 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
11 Classical isoperimetric inequality [De Giorgi] ∀ E ⊂ IR n . H n − 1 ( ∂ ∗ E ) ≥ nω 1 /n | E | 1 /n ′ n Here: ∂ ∗ E stands for the essential boundary of E , • | E | = H n ( E ), the Lebesgue measure of E , • H n − 1 is the ( n − 1)-dimensional Hausdorff measure (the surface area). • St. Petersburg, July 2010 13 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
11 Classical isoperimetric inequality [De Giorgi] ∀ E ⊂ IR n . H n − 1 ( ∂ ∗ E ) ≥ nω 1 /n | E | 1 /n ′ n Here: ∂ ∗ E stands for the essential boundary of E , • | E | = H n ( E ), the Lebesgue measure of E , • H n − 1 is the ( n − 1)-dimensional Hausdorff measure (the surface area). • In other words, the ball has the least surface area among sets of fixed volume . St. Petersburg, July 2010 13 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
12 In general the isoperimetric function λ M : [0 , H n ( M ) / 2] → [0 , ∞ ) of M (introduced by V.G.Maz’ya) is defined as λ M ( s ) = inf {H n − 1 ( ∂ ∗ E ) : s ≤ H n ( E ) ≤ H n ( M ) / 2 } , (5) for s ∈ [0 , H n ( M ) / 2]. St. Petersburg, July 2010 14 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
12 In general the isoperimetric function λ M : [0 , H n ( M ) / 2] → [0 , ∞ ) of M (introduced by V.G.Maz’ya) is defined as λ M ( s ) = inf {H n − 1 ( ∂ ∗ E ) : s ≤ H n ( E ) ≤ H n ( M ) / 2 } , (5) for s ∈ [0 , H n ( M ) / 2]. Isoperimetric inequality on M : H n − 1 ( ∂ ∗ E ) ≥ λ M ( H n ( E )) ∀ E ⊂ M, H n ( E ) ≤ H n ( M ) / 2 . St. Petersburg, July 2010 14 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
13 The geometry of M is related to λ M , and, in particular, to its asymptotic behavior at 0. St. Petersburg, July 2010 15 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
13 The geometry of M is related to λ M , and, in particular, to its asymptotic behavior at 0. For instance, if M is compact, then λ M ( s ) ≈ s 1 /n ′ as s → 0. St. Petersburg, July 2010 15 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
13 The geometry of M is related to λ M , and, in particular, to its asymptotic behavior at 0. For instance, if M is compact, then λ M ( s ) ≈ s 1 /n ′ as s → 0. Here, f ≈ g means that ∃ c, k > 0 such that cg ( cs ) ≤ f ( s ) ≤ kg ( ks ) . St. Petersburg, July 2010 15 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
13 The geometry of M is related to λ M , and, in particular, to its asymptotic behavior at 0. For instance, if M is compact, then λ M ( s ) ≈ s 1 /n ′ as s → 0. Here, f ≈ g means that ∃ c, k > 0 such that cg ( cs ) ≤ f ( s ) ≤ kg ( ks ) . n Moreover, n ′ = n − 1. St. Petersburg, July 2010 15 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
14 Approach by isocapacitary inequalities. St. Petersburg, July 2010 16 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
14 Approach by isocapacitary inequalities. Standard capacity of E ⊂ M : � � |∇ u | 2 dx : u ∈ W 1 , 2 ( M ) , C ( E ) = inf M � ” u ≥ 1” in E, and u has compact support . St. Petersburg, July 2010 16 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
14 Approach by isocapacitary inequalities. Standard capacity of E ⊂ M : � � |∇ u | 2 dx : u ∈ W 1 , 2 ( M ) , C ( E ) = inf M � ” u ≥ 1” in E, and u has compact support . Capacity of a condenser ( E ; G ), E ⊂ G ⊂ M : � � |∇ u | 2 dx : u ∈ W 1 , 2 ( M ) , C ( E ; G ) = inf M � ” u ≥ 1” in E ” u ≤ 0” in M \ G . St. Petersburg, July 2010 16 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
15 Isocapacitary function (introduced by V.G.Maz’ya) ν M : [0 , H n ( M ) / 2] → [0 , ∞ ) ν M ( s ) = inf { C ( E, G ) : E ⊂ G ⊂ M , s ≤ H n ( E ) and H n ( G ) ≤ H n ( M ) / 2 } for s ∈ [0 , H n ( M ) / 2]. St. Petersburg, July 2010 17 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
15 Isocapacitary function (introduced by V.G.Maz’ya) ν M : [0 , H n ( M ) / 2] → [0 , ∞ ) ν M ( s ) = inf { C ( E, G ) : E ⊂ G ⊂ M , s ≤ H n ( E ) and H n ( G ) ≤ H n ( M ) / 2 } for s ∈ [0 , H n ( M ) / 2]. Isocapacitary inequality: C ( E, G ) ≥ ν M ( H n ( E )) ∀ E ⊂ G ⊂ M , H n ( G ) ≤ H n ( M ) / 2 . St. Petersburg, July 2010 17 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
15 Isocapacitary function (introduced by V.G.Maz’ya) ν M : [0 , H n ( M ) / 2] → [0 , ∞ ) ν M ( s ) = inf { C ( E, G ) : E ⊂ G ⊂ M , s ≤ H n ( E ) and H n ( G ) ≤ H n ( M ) / 2 } for s ∈ [0 , H n ( M ) / 2]. Isocapacitary inequality: C ( E, G ) ≥ ν M ( H n ( E )) ∀ E ⊂ G ⊂ M , H n ( G ) ≤ H n ( M ) / 2 . If M is compact and n ≥ 3, then n − 2 ν M ( s ) ≈ s as s → 0. n St. Petersburg, July 2010 17 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
16 The isoperimetric function and the isocapacitary function of a manifold M are related by � H n ( M ) / 2 1 dr for s ∈ (0 , H n ( M ) / 2). ν M ( s ) ≤ (6) λ M ( r ) 2 s St. Petersburg, July 2010 18 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
16 The isoperimetric function and the isocapacitary function of a manifold M are related by � H n ( M ) / 2 1 dr for s ∈ (0 , H n ( M ) / 2). ν M ( s ) ≤ (6) λ M ( r ) 2 s A reverse estimate does not hold in general. St. Petersburg, July 2010 18 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
16 The isoperimetric function and the isocapacitary function of a manifold M are related by � H n ( M ) / 2 1 dr for s ∈ (0 , H n ( M ) / 2). ν M ( s ) ≤ (6) λ M ( r ) 2 s A reverse estimate does not hold in general. Roughly speaking, a reverse estimate only holds when the geometry of M is sufficiently regular. St. Petersburg, July 2010 18 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
17 Both the conditions in terms of ν M , and those in terms of λ M , for eigenfunction estimates in L q ( M ) or L ∞ ( M ) to be presented are sharp in the class of manifolds M with prescribed asymptotic behavior of ν M and λ M at 0. St. Petersburg, July 2010 19 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
17 Both the conditions in terms of ν M , and those in terms of λ M , for eigenfunction estimates in L q ( M ) or L ∞ ( M ) to be presented are sharp in the class of manifolds M with prescribed asymptotic behavior of ν M and λ M at 0. Each one of these approaches has its own advantages. St. Petersburg, July 2010 19 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
17 Both the conditions in terms of ν M , and those in terms of λ M , for eigenfunction estimates in L q ( M ) or L ∞ ( M ) to be presented are sharp in the class of manifolds M with prescribed asymptotic behavior of ν M and λ M at 0. Each one of these approaches has its own advantages. The isoperimetric function λ M has a transparent geometric character, and it is usually easier to investigate. St. Petersburg, July 2010 19 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
17 Both the conditions in terms of ν M , and those in terms of λ M , for eigenfunction estimates in L q ( M ) or L ∞ ( M ) to be presented are sharp in the class of manifolds M with prescribed asymptotic behavior of ν M and λ M at 0. Each one of these approaches has its own advantages. The isoperimetric function λ M has a transparent geometric character, and it is usually easier to investigate. The isocapacitary function ν M is in a sense more appropriate: it not only implies the results involving λ M , but leads to finer conclusions in general. Typically, this is the case when manifolds with complicated geometric configurations are taken into account. St. Petersburg, July 2010 19 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
18 Estimates for eigenfunctions. St. Petersburg, July 2010 20 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
18 Estimates for eigenfunctions. If u is an eigenfunction of the Laplacian, then, by definition, u ∈ W 1 , 2 ( M ). St. Petersburg, July 2010 20 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
18 Estimates for eigenfunctions. If u is an eigenfunction of the Laplacian, then, by definition, u ∈ W 1 , 2 ( M ). Hence, trivially, u ∈ L 2 ( M ). St. Petersburg, July 2010 20 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
18 Estimates for eigenfunctions. If u is an eigenfunction of the Laplacian, then, by definition, u ∈ W 1 , 2 ( M ). Hence, trivially, u ∈ L 2 ( M ). Problem: given q ∈ (2 , ∞ ], find conditions on M ensuring that any eigenfunction u of the Laplacian on M belongs to L q ( M ). St. Petersburg, July 2010 20 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
19 L q bounds for eigenfunctions Theorem 1: Assume that s lim ν M ( s ) = 0 . (7) s → 0 St. Petersburg, July 2010 21 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
19 L q bounds for eigenfunctions Theorem 1: Assume that s lim ν M ( s ) = 0 . (7) s → 0 Then for any q ∈ (2 , ∞ ) there exists a constant C such that � u � L q ( M ) ≤ C � u � L 2 ( M ) (8) for every eigenfunction u of the Laplacian on M . St. Petersburg, July 2010 21 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
20 The assumption s lim ν M ( s ) = 0 (9) s → 0 is essentially minimal in Theorem 1. St. Petersburg, July 2010 22 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
20 The assumption s lim ν M ( s ) = 0 (9) s → 0 is essentially minimal in Theorem 1. Sharpness of condition (9) Theorem 2: For any n ≥ 2 and q ∈ (2 , ∞ ], there exists an n -dimensional Riemannian manifold M such that ν M ( s ) ≈ s near 0 , (10) ∈ L q ( M ). and the Laplacian on M has an eigenfunction u / St. Petersburg, July 2010 22 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
21 Conditions in terms of the isoperimetric function for L q bounds for eigenfunctions can be derived via Theorem 2. St. Petersburg, July 2010 23 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
21 Conditions in terms of the isoperimetric function for L q bounds for eigenfunctions can be derived via Theorem 2. Corollary 2 Assume that s lim λ M ( s ) = 0 . (11) s → 0 Then for any q ∈ (2 , ∞ ) there exists a constant C such that � u � L q ( M ) ≤ C � u � L 2 ( M ) for every eigenfunction u of the Laplacian on M . St. Petersburg, July 2010 23 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
21 Conditions in terms of the isoperimetric function for L q bounds for eigenfunctions can be derived via Theorem 2. Corollary 2 Assume that s lim λ M ( s ) = 0 . (11) s → 0 Then for any q ∈ (2 , ∞ ) there exists a constant C such that � u � L q ( M ) ≤ C � u � L 2 ( M ) for every eigenfunction u of the Laplacian on M . Assumption (12) is minimal in the same sense as the analogous assumption in terms of ν M . St. Petersburg, July 2010 23 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
22 Estimate for the growth of constant in the L q ( M ) bound for eigenfunctions in terms of the eigenvalue. St. Petersburg, July 2010 24 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
22 Estimate for the growth of constant in the L q ( M ) bound for eigenfunctions in terms of the eigenvalue. Proposition Assume that s lim ν M ( s ) = 0 . (12) s → 0 Define r for s ∈ (0 , H n ( M ) / 2]. Θ( s ) = sup ν M ( r ) r ∈ (0 ,s ) St. Petersburg, July 2010 24 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
22 Estimate for the growth of constant in the L q ( M ) bound for eigenfunctions in terms of the eigenvalue. Proposition Assume that s lim ν M ( s ) = 0 . (12) s → 0 Define r for s ∈ (0 , H n ( M ) / 2]. Θ( s ) = sup ν M ( r ) r ∈ (0 ,s ) Then � u � L q ( M ) ≤ C � u � L 2 ( M ) for any q ∈ (2 , ∞ ) and for every eigenfunc- tion u of the Laplacian on M associated with the eigenvalue γ , where C 1 C ( ν M , q, γ ) = , � 1 2 − 1 � Θ − 1 ( C 2 /γ ) q and C 1 = C 1 ( q, H n ( M )) and C 2 = C 2 ( q, H n ( M )). St. Petersburg, July 2010 24 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
23 Example. Assume that there exists β ∈ [( n − 2) /n, 1) such that ν M ( s ) ≥ Cs β . Then there exists a constant C = C ( q, H n ( M )) such that q − 2 2 q (1 − β ) � u � L 2 ( M ) � u � L q ( M ) ≤ Cγ for every eigenfunction u of the Laplacian on M associated with the eigenvalue γ . St. Petersburg, July 2010 25 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
24 Consider now the case when q = ∞ , namely the problem of the boundedness of the eigenfunctions. St. Petersburg, July 2010 26 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
24 Consider now the case when q = ∞ , namely the problem of the boundedness of the eigenfunctions. Theorem 3: boundedness of eigenfunctions Assume that � ds ν M ( s ) < ∞ . (13) 0 St. Petersburg, July 2010 26 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
24 Consider now the case when q = ∞ , namely the problem of the boundedness of the eigenfunctions. Theorem 3: boundedness of eigenfunctions Assume that � ds ν M ( s ) < ∞ . (13) 0 Then there exists a constant C such that � u � L ∞ ( M ) ≤ C � u � L 2 ( M ) (14) for every eigenfunction u of the Laplacian on M . St. Petersburg, July 2010 26 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
25 The condition ds � ν M ( s ) < ∞ (15) 0 is essentially sharp in Theorem 4. St. Petersburg, July 2010 27 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
25 The condition ds � ν M ( s ) < ∞ (15) 0 is essentially sharp in Theorem 4. This is the content of the next result. St. Petersburg, July 2010 27 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
25 The condition ds � ν M ( s ) < ∞ (15) 0 is essentially sharp in Theorem 4. This is the content of the next result. Recall that f ∈ ∆ 2 near 0 if there exist constants c and s 0 such that f (2 s ) ≤ cf ( s ) if 0 < s ≤ s 0 . (16) St. Petersburg, July 2010 27 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
26 sharpness of condition (15) Theorem 4: Let ν be a non-decreasing function, vanishing only at 0, St. Petersburg, July 2010 28 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
26 sharpness of condition (15) Theorem 4: Let ν be a non-decreasing function, vanishing only at 0, such that s lim ν ( s ) = 0 , (17) s → 0 St. Petersburg, July 2010 28 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
26 sharpness of condition (15) Theorem 4: Let ν be a non-decreasing function, vanishing only at 0, such that s lim ν ( s ) = 0 , (17) s → 0 but ds � ν ( s ) = ∞ . (18) 0 St. Petersburg, July 2010 28 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
27 Assume in addition that ν ∈ ∆ 2 near 0 and ν ( s ) is equivalent to a non-decreasing function near 0, (19) n − 2 s n for some n ≥ 3. St. Petersburg, July 2010 29 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
27 Assume in addition that ν ∈ ∆ 2 near 0 and ν ( s ) is equivalent to a non-decreasing function near 0, (19) n − 2 s n for some n ≥ 3. Then, there exists an n -dimensional Riemannian manifold M fulfilling ν M ( s ) ≈ ν ( s ) as s → 0, (20) and such that the Laplacian on M has an unbounded eigenfunction. St. Petersburg, July 2010 29 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
27 Assume in addition that ν ∈ ∆ 2 near 0 and ν ( s ) is equivalent to a non-decreasing function near 0, (19) n − 2 s n for some n ≥ 3. Then, there exists an n -dimensional Riemannian manifold M fulfilling ν M ( s ) ≈ ν ( s ) as s → 0, (20) and such that the Laplacian on M has an unbounded eigenfunction. n − 2 Assumption (19) is consistent with the fact that ν M ( s ) ≈ s near 0 if n the geometry of M is nice (e.g. M compact), St. Petersburg, July 2010 29 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
27 Assume in addition that ν ∈ ∆ 2 near 0 and ν ( s ) is equivalent to a non-decreasing function near 0, (19) n − 2 s n for some n ≥ 3. Then, there exists an n -dimensional Riemannian manifold M fulfilling ν M ( s ) ≈ ν ( s ) as s → 0, (20) and such that the Laplacian on M has an unbounded eigenfunction. n − 2 Assumption (19) is consistent with the fact that ν M ( s ) ≈ s near 0 if n the geometry of M is nice (e.g. M compact), and that ν M ( s ) → 0 faster n − 2 than s otherwise. n St. Petersburg, July 2010 29 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
28 Owing to the inequality � H n ( M ) / 2 1 dr for s ∈ (0 , H n ( M ) / 2), ν M ( s ) ≤ λ M ( r ) 2 s Theorem 4 has the following corollary in terms of isoperimetric inequalities. St. Petersburg, July 2010 30 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
29 Corollary 3 Assume that � s λ M ( s ) 2 ds < ∞ . (21) 0 St. Petersburg, July 2010 31 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
29 Corollary 3 Assume that � s λ M ( s ) 2 ds < ∞ . (21) 0 Then there exists a constant C such that � u � L ∞ ( M ) ≤ C � u � L 2 ( M ) (22) for every eigenfunction u of the Laplacian on M . St. Petersburg, July 2010 31 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
29 Corollary 3 Assume that � s λ M ( s ) 2 ds < ∞ . (21) 0 Then there exists a constant C such that � u � L ∞ ( M ) ≤ C � u � L 2 ( M ) (22) for every eigenfunction u of the Laplacian on M . Assumption (21) is sharp in the same sense as the analogous assumption in terms of ν M . St. Petersburg, July 2010 31 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
30 Estimate for the growth of constant in the L ∞ ( M ) bound for eigenfunctions in terms of the eigenvalue. St. Petersburg, July 2010 32 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
30 Estimate for the growth of constant in the L ∞ ( M ) bound for eigenfunctions in terms of the eigenvalue. Proposition Assume that ds � ν M ( s ) < ∞ . 0 Define � s dr for s ∈ (0 , H n ( M ) / 2]. Ξ( s ) = ν M ( r ) 0 St. Petersburg, July 2010 32 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
30 Estimate for the growth of constant in the L ∞ ( M ) bound for eigenfunctions in terms of the eigenvalue. Proposition Assume that ds � ν M ( s ) < ∞ . 0 Define � s dr for s ∈ (0 , H n ( M ) / 2]. Ξ( s ) = ν M ( r ) 0 Then � u � L ∞ ( M ) ≤ C � u � L 2 ( M ) for every eigenfunction u of the Laplacian on M associated with the eigenvalue γ , where C 1 C ( ν M , γ ) = , � 1 � Ξ − 1 ( C 2 /γ ) 2 and C 1 and C 2 are absolute constants. St. Petersburg, July 2010 32 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
31 Example. Assume that there exists β ∈ [( n − 2) /n, 1) such that ν M ( s ) ≥ Cs β . Then there exists an absolute constant C such that 1 2(1 − β ) � u � L 2 ( M ) � u � L ∞ ( M ) ≤ Cγ for every eigenfunction u of the Laplacian on M associated with the eigenvalue γ . St. Petersburg, July 2010 33 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
32 Pb.: Discreteness of the spectrum of the Laplacian on M . St. Petersburg, July 2010 34 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
32 Pb.: Discreteness of the spectrum of the Laplacian on M . Consider the semi-definite self-adjoint Laplace operator ∆ on the Hilbert space L 2 ( M ) associated with the bilinear form a : W 1 , 2 ( M ) × W 1 , 2 ( M ) → IR defined as � ∇ u · ∇ v d H n ( x ) a ( u, v ) = (23) M for u, v ∈ W 1 , 2 ( M ). St. Petersburg, July 2010 34 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
32 Pb.: Discreteness of the spectrum of the Laplacian on M . Consider the semi-definite self-adjoint Laplace operator ∆ on the Hilbert space L 2 ( M ) associated with the bilinear form a : W 1 , 2 ( M ) × W 1 , 2 ( M ) → IR defined as � ∇ u · ∇ v d H n ( x ) a ( u, v ) = (23) M for u, v ∈ W 1 , 2 ( M ). 0 ( M ) = W 1 , 2 ( M ), the operator ∆ agrees with the Friedrichs • When C ∞ extension of the classical Laplace operator. St. Petersburg, July 2010 34 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
32 Pb.: Discreteness of the spectrum of the Laplacian on M . Consider the semi-definite self-adjoint Laplace operator ∆ on the Hilbert space L 2 ( M ) associated with the bilinear form a : W 1 , 2 ( M ) × W 1 , 2 ( M ) → IR defined as � ∇ u · ∇ v d H n ( x ) a ( u, v ) = (23) M for u, v ∈ W 1 , 2 ( M ). 0 ( M ) = W 1 , 2 ( M ), the operator ∆ agrees with the Friedrichs • When C ∞ extension of the classical Laplace operator. This is the case, for instance, if M is complete, and, in particular, if M is compact. St. Petersburg, July 2010 34 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
32 Pb.: Discreteness of the spectrum of the Laplacian on M . Consider the semi-definite self-adjoint Laplace operator ∆ on the Hilbert space L 2 ( M ) associated with the bilinear form a : W 1 , 2 ( M ) × W 1 , 2 ( M ) → IR defined as � ∇ u · ∇ v d H n ( x ) a ( u, v ) = (23) M for u, v ∈ W 1 , 2 ( M ). 0 ( M ) = W 1 , 2 ( M ), the operator ∆ agrees with the Friedrichs • When C ∞ extension of the classical Laplace operator. This is the case, for instance, if M is complete, and, in particular, if M is compact. • When M is an open subset of IR n , or, more generally, of a Riemannian manifold, then ∆ corresponds to the Neumann Laplacian on M . St. Petersburg, July 2010 34 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
33 A necessary and sufficient condition for the discreteness of the spectrum of ∆ can be given in terms of the isocapacitary function of M . St. Petersburg, July 2010 35 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
33 A necessary and sufficient condition for the discreteness of the spectrum of ∆ can be given in terms of the isocapacitary function of M . Discreteness of the spectrum of ∆ Theorem 5: The spectrum of the Laplacian on M is discrete if and only if s lim ν M ( s ) = 0 . (24) s → 0 St. Petersburg, July 2010 35 / A. Cianchi (Univ. Firenze) Eigenfunctions of the Laplacian 46
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