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The Origins of the Propagation of Smallness Property and Its Utility in Controllability Problems

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The Origins of the Propagation of Smallness Property and Its Utility in Controllability Problems The Origins of the Propagation of Smallness Property and its Utility in Controllability Problems. Jone Apraiz University of the Basque Country, Spain Seminario control en tiempos de crisis May 4th 2020 Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 1 / 37 Index 1 Definition of the propagation of smallness property 2 Origin of the propagation of smallness property Harmonic measure Two-constants and Hamard three-circles theorems Phragmén-Lindelöf theorem and Hadamard three-lines lemma 3 Applications to Control Theory Higher dimensions: three-balls and three-regions theorems Null-control for some parabolic problems Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 2 / 37 Definition of the propagation of smallness property Index 1 Definition of the propagation of smallness property 2 Origin of the propagation of smallness property Harmonic measure Two-constants and Hamard three-circles theorems Phragmén-Lindelöf theorem and Hadamard three-lines lemma 3 Applications to Control Theory Higher dimensions: three-balls and three-regions theorems Null-control for some parabolic problems Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 3 / 37 Definition of the propagation of smallness property Definition of the propagation of smallness property Definition (Propagation of smallness) n Given three subsets of R , E, B1 and B2, verifying E ⊂ B1 ⊂ B2 and a class of functions A ⊂ C(B2), we say that E is a propagation of smallness set for A if, for any u 2 A, there exists α = α(E; B1; B2) 2 (0; 1) such that jjujj ≤ jjujjα jjujj1−α : 1;B1 1;E 1;B2 The “information of the function” on the smaller domain is propagating or affecting over bigger domains in some sense. Also called “rate of growth result”,“quantification of the propagation of smallness” or “transfer of smallness”. Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 4 / 37 Origin of the propagation of smallness property Index 1 Definition of the propagation of smallness property 2 Origin of the propagation of smallness property Harmonic measure Two-constants and Hamard three-circles theorems Phragmén-Lindelöf theorem and Hadamard three-lines lemma 3 Applications to Control Theory Higher dimensions: three-balls and three-regions theorems Null-control for some parabolic problems Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 5 / 37 Origin of the propagation of smallness property Harmonic measure Harmonic measure The main reason why the harmonic measure is in the origins of the propagation of smallness property: It is connected to estimating the modulus of an analytic function inside a domain given certain bounds on the modulus of the function on the boundary of the domain. It is very useful for estimating the growth of analytic and harmonic functions. When and how was the harmonic measure defined for the first time? Although it could be difficult to specify, in many references we look into, it seems that all of them point to Rolf Nevanlinna in the 1920s. Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 6 / 37 Origin of the propagation of smallness property Harmonic measure Some of the references where we can find Nevanlinna’s work related to harmonic measure: Ueber die Eigenschaften einer analytischen Funktion in der Umgebung einer singulären Stelle oder Linie, with is brother Frithiof Nevanlinna, 1922. Das harmonische Mass von Punktmengen und seine Anwendung in der Funktionentheorie, 1934. Eindeutige analytische Funktionen, 1936. English translation: Analytic functions, 1970. In Nevanlinna’s Analytic functions book (mainly in chapters I, II and III’s section x1): The ideas and details of how he worked on and built the harmonic measure that we know nowadays. He starts defining the arcs in the unit circle and analyzing conformal mappings. Then, he defines the harmonic measure and studies its properties in the same circle to later extend the same idea to any bounded region on the plane. Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 7 / 37 Origin of the propagation of smallness property Harmonic measure Although it was R. Nevanlinna who gave the name to the harmonic measure, some years earlier, Torsten Carleman and Henri Milloux started to use it trying to measure the growth of analytic functions or, in some sense, studying the propagation of smallness property: T. Carleman, Sur les fonctions inverses des fonctions entières, 1921 H. Milloux, Le théorème de M. Picard, suites de fonctions holomorphes, fonctions méromorphes et fonctions entières, 1924. H. Milloux, Sur le théorème de Picard, 1925. Milloux-Carleman problem: Suppose that f is analytic, jf (z)j ≤ M in the unit disk D and that jf (z)j ≤ m on a curve γ that connects the origin to @D. How large can jf (z0)j be at a given point z0? Answer: using the two-constants theorem (attributed to T. Carleman); independently by R. Nevanlinna and Arne Beurling (1933). Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 8 / 37 Origin of the propagation of smallness property Harmonic measure T. Carleman A. Beurling Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 9 / 37 Origin of the propagation of smallness property Harmonic measure Most of the literature dealing with harmonic measure refers to it from, basically, three different, but somehow related to each other, points of view: Partial differential equations: harmonic measure defines a probability measure to define the solution for the Dirichlet problem. Let Ω ⊂ Rn+1 be a bounded connected open set such that @Ω is connected. The Dirichlet problem, 8 < ∆u = 0; on Ω, (1) : u = f ; on @Ω, where f 2 C(@Ω). Then, for each x 2 Ω there is a probability measure !(x; Ω) on @Ω, called harmonic measure for each x, such that the solution to (1) is u(x) = f (y)d!(x; Ω)(y): ˆ@Ω Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 10 / 37 Origin of the propagation of smallness property Harmonic measure Probability theory: a boundary hitting distribution of Brownian motion. Shizuo Kakutani 1944. n Ω ⊂ R is an open and bounded connected domain, E ⊂ @Ω and z 2 Ω. We set a random particle in z and let it move. The Brownian motion starting from a point z will hit on E without hitting @Ω n E before it, defines a probability measure on Ω. The measure of E at a point z with respect to Ω is usually denoted by !(z; E; Ω). For E fixed, !(z; E; Ω) is a harmonic function of z on Ω. Harmonic functions theory: We will see it soon. We can see that the desire to analyze the propagation of smallness property opened new paths and enriched other areas of mathematics. Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 11 / 37 Origin of the propagation of smallness property Harmonic measure Definition (Jordan curve) We say that a plane curve is a Jordan curve if it is simple and closed or if it is a homeomorphic image of the unit circle. Definition (Jordan domain) A Jordan domain is a domain in a plane bounded by a Jordan curve or a finite number of Jordan curves. Definition 0 0 Let Ω and Ω be two open subsets of C. If f :Ω ! Ω is one-to-one, and f and f −1 are holomorphic, then, we say that f is a conformal mapping from Ω to Ω0. Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 12 / 37 Origin of the propagation of smallness property Harmonic measure Definition (Harmonic measure) Let D ⊂ C be a domain bounded by a finite number of Jordan curves, Γ. Let Γ = α [ β, Int(α) \ Int(β) = ;, where α and β are finite sets of Jordan curves. We call harmonic measure of α with respect to D, evaluated at the point z and expressed as !(z; α; D), to the harmonic function with boundary limit 1 at points of α and boundary limit 0 at points of β, that is, !(z; α; D) verifies l´ım !(z; α; D) = 1 and l´ım !(z; α; D) = 0: z!α z!β The fact that !(z; α; D) is a harmonic function is the reason why this measure is called harmonic measure. Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 13 / 37 Origin of the propagation of smallness property Harmonic measure Carleman’s principle of monotoneity The harmonic measure !(z; α; G) increases to the domain G 0 when the region G is extended across the portion β of the boundary Γ = α + β complementary to α. That is, in G, !(z; α; G) ≤ !(z; α; G 0) and equality holds only when G ≡ G 0. Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 14 / 37 Origin of the propagation of smallness property Harmonic measure It is useful because it allows complicated configurations (G; α; z) to be replaced by simpler ones (G 0; α; z) for which the computational mastery of the relationships between the harmonic measure ! and the euclidean data on the configuration (G 0; α; z) isn’t a problem. It allows to set up majorants and minorants of the harmonic measure. The direct application of these harmonic majorants and minorants will in many cases permit theorems of a general nature to be proved in an elementary way, without bringing into play the principle of harmonic measure in its most general form. Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 15 / 37 Origin of the propagation of smallness property Two-constants and Hamard three-circles theorems Two-constants and Hamard three-circles theorems Theorem (Two-constants theorem) Suppose that f is a bounded analytic function in a Jordan domain Ω such that jf (z)j ≤ M in Ω and l´ım sup jf (z)j ≤ m < M when z 2 Ω and z!ζ ζ 2 E ⊂ @Ω.
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