Remez inequality and propagation of smallness for solutions of - PowerPoint PPT Presentation

Remez inequality and propagation of smallness for solutions of second order elliptic PDEs Part I. Classical Remez inequality, analytic propagation of smallness Eugenia Malinnikova NTNU March 2018 E. Malinnikova Propagation of smallness for elliptic PDEs

Chebyshev polynomials Definition The n -th Chebyshev polynomial of the fist kind is the polynomial T n of degree n which satisfies the identity T n ( cos t ) = cos nt . Clearly T 1 ( x ) = x , T 2 ( x ) = 2 x 2 − 1 and the trigonometric formula cos ( n + 1 ) t + cos ( n − 1 ) t = 2 cos nt cos t implies the recursive formula for T n T n + 1 ( x ) = 2 xT n ( x ) − T n − 1 ( x ) . E. Malinnikova Propagation of smallness for elliptic PDEs

Properties • Leading coefficient: T n ( x ) = 2 n − 1 x n + ... + c n • Alternating min-max: We fix n and let x k = cos ( k π/ n ) , k = 0 , ..., n . Then T n ( x k ) = ( − 1 ) k , − 1 = x n < x n − 1 < ... < x k < ... < x 1 < x 0 = 1. • Extremal property: − 1 ≤ x ≤ 1 | T n ( x ) | = 1 ≤ 2 n − 1 max max − 1 ≤ x ≤ 1 | P n ( x ) | for any P n ( x ) = x n + a n − 1 x n − 1 + ... + a 0 (monic polynomial of degree n .) √ √ x 2 − 1 ) n + ( x − x 2 − 1 ) n • Formula 2 T n ( x ) = ( x + E. Malinnikova Propagation of smallness for elliptic PDEs

Remez inequality Theorem (Remez, 1936) Let E be a measurable subset of an interval I and | E | = m. Then for any polynomial P n of degree n � 1 + 2 ( | I | − m ) � max x ∈ I | P n ( x ) | ≤ T n max x ∈ E | P n ( x ) | m The equality is attained when P n ( x ) = CT n ( 2 x / m ) , I = ( − m / 2 , m / 2 + a ) and E = ( − m / 2 , m / 2 ) . Corollary � n � 4 | I | max x ∈ I | P n ( x ) | ≤ max x ∈ E | P n ( x ) | | E | E. Malinnikova Propagation of smallness for elliptic PDEs

Tool: Lagrange interpolation formula If P is a polynomial of degree ≤ n and y 0 , ..., y n are distinct points then n y − y k � � P ( y ) = P ( y j ) y j − y k j = 1 k � = j Proof of Remez inequality Renormalize to have | E | = 2, I = [ − 1 , 1 + a ] . Then find y j ∈ E such that | y j − y k | ≥ | x j − x k | (extremal points for Chebyshev polynomial) and | 1 + a − y j | ≥ | 1 + a − x j | and compare interpolation formulas for P n ( 1 + a ) and T n ( 1 + a ) . E. Malinnikova Propagation of smallness for elliptic PDEs

Turan-Nazarov inequality for exponential sums Let F n ( x ) = � n k = 0 a k e i λ k x . Theorem (Nazarov, 1993) Let E be a measurable subset of an interval I and | E | = m. (i) If all λ k ∈ R then � n � C | I | x ∈ I | F n ( x ) | ≤ x ∈ E | F n ( x ) | max max | E | (ii) If λ k ∈ C we define s = max | Im λ k | , then � n � C | I | x ∈ I | F n ( x ) | ≤ e s | I | x ∈ E | F n ( x ) | max max | E | E. Malinnikova Propagation of smallness for elliptic PDEs

Reformulation of Remez inequality The Remez inequality is equivalent to � 1 / n � max x ∈ E | P n ( x ) | | E | ≤ 4 | I | . max x ∈ I | P n ( x ) | We rewrite it as | E δ | ≤ 4 | I | δ 1 / n , where E δ = { x ∈ I : | P n ( x ) | < δ max x ∈ I | P n ( x ) |} . E. Malinnikova Propagation of smallness for elliptic PDEs

Classical results of Cartan and Polya Let P n ( z ) = z n + ... be a monic polynomial of degree n . Lemma (Cartan, 1928) Let F s = { z ∈ C : | P n ( z ) | ≤ s n } and let α > 0 then there are disks B j ( z j , r j ) such that � r α j ≤ e ( 2 s ) α F s ⊂ ∪ j B j , j For α = 2 one obtains an estimate for the measure of the set | F s | . The sharp result here is due to Polya (1928) and it says that | F s | ≤ π s 2 . E. Malinnikova Propagation of smallness for elliptic PDEs

Hadamard three circle theorem Theorem Suppose that f is an analytic function in the domain { r 0 < | z | < R } . Let M ( r ) = max | z | = r | f ( z ) | and r 0 < r 1 < r 2 < r 3 < R. Then M ( r 2 ) ≤ M ( r 1 ) α M ( r 3 ) 1 − α , where r 2 = r α 1 r 1 − α . 3 It follows from the maximum principle for (sub)harmonic function h ( z ) = log | z a f ( z ) | . We have r a 2 M ( r 2 ) ≤ max { r a 1 M ( r 1 ) , r a 3 M ( r 3 ) } and choose a such that r a 1 M ( r 1 ) = r a 3 M ( r 3 ) , then r a 2 M ( r 2 ) ≤ ( r a 1 M ( r 2 )) α ( r a 3 M ( r 3 )) 1 − α E. Malinnikova Propagation of smallness for elliptic PDEs

Two-constant theorem Theorem Suppose that f is a bounded analytic function in a Jordan domain Ω such that | f ( z ) | ≤ M in Ω and | f ( ζ ) | ≤ m when ζ ∈ E ⊂ ∂ Ω . Then for any z ∈ Ω | f ( z ) | ≤ m ω E ( z ) M 1 − ω E ( z ) , where ω E ( z ) is the harmonic measure of E at point z. In other words, ω E is the harmonic function with boundary values 1 on E and 0 on ∂ Ω \ E . We once again use the maximum principle and compare log | f ( z ) | to ω E ( z ) log m + ( 1 − ω E ( z )) log M . E. Malinnikova Propagation of smallness for elliptic PDEs

Propagation of smallness for real analytic functions Suppose that u is a real-analytic function in the unite ball B ⊂ R d , u extends to a holomorphic function U in O ⊂ C d such that O ∩ R d ⊃ B and | U | ≤ M in O . Suppose that E ⊂ 1 / 2 B , | E | > 0 and max E | u | ≤ m . Then 1 / 2 B | u | ≤ Cm β M 1 − β , max where β depends on O and on | E | . Theorem (Hayman, 1970) Suppose that u is a harmonic function in B that satisfies max B | u | ≤ M. Then there exists a holomorphic function U in √ √ B C ( 1 / 2 ) such that U ( x ) = u ( x ) when x ∈ B R ( 1 / 2 ) and √ | U ( z ) | ≤ C ( | z | ) M when | z | < 1 / 2 . E. Malinnikova Propagation of smallness for elliptic PDEs

Łojasiewicz inequality Suppose that f is a non-zero real analytic function in B ⊂ R n , Z f = f − 1 ( 0 ) ,. Then Z f has dimension n − 1, Z f = ∪ n − 1 j = 0 A j , where A j is a countable union of j -dimensional manifolds. Let Z f ∩ B 1 � = ∅ . Then for any compact subset K ⊂ B there exists c > 0 and β such that | f ( x ) | ≥ c dist ( x , Z f ) β , x ∈ K , β is called the Łojasiewicz exponent of f (in K ). In particular | f |} ⊂ K ∩ ( Z f ) + B ( 0 , c 1 δ 1 /β ) , E δ = { x ∈ K : | f ( x ) | < δ max B where D + B ( 0 , ǫ ) is the ǫ -neighborhood of a set D . E. Malinnikova Propagation of smallness for elliptic PDEs

Second order elliptic equations We study operators of the form Lf = div ( A ∇ f ) , where A ( x ) = [ a ij ( x )] 1 ≤ i , j ≤ d is a symmetric matrix with Lipschitz entries and Λ − 1 � v � 2 ≤ ( A ( x ) v , v ) ≤ Λ � v � 2 uniformly in x . We will study local properties of solutions to the equation Lf = 0 and changing the coordinates assume that L is a small perturbation of the Laplacian. E. Malinnikova Propagation of smallness for elliptic PDEs

Harnack inequality and comparison of norms Suppose that Lf = 0 in B 1 ⊂ R d and f ≥ 0 in B 1 then max B 1 / 2 f ≤ C H min B 1 / 2 f . In particular E δ ( f ) = { x ∈ B 1 / 2 : | f ( x ) | < δ max B 1 / 2 | f |} is empty when δ is sufficiently small. We will also use the following inequality (equivalence of norms) for any solution f of Lf = 0 in B 1 we have 1 � B 1 / 2 | f | 2 ≤ C 1 � | f | 2 ≤ max | f | 2 . | S 1 / 2 | | S 1 | S 1 / 2 S 1 E. Malinnikova Propagation of smallness for elliptic PDEs