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 ∈ A, there exists α = α(E, B1, B2) ∈ (0, 1) such that

||u|| ≤ ||u||α ||u||1−α . ∞,B1 ∞,E ∞,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 §1): 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, |f (z)| ≤ M in the unit disk D and that |f (z)| ≤ m on a curve γ that connects the origin to ∂D. How large can |f (z0)| 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,   ∆u = 0, on Ω, (1)  u = f , on ∂Ω,

where f ∈ C(∂Ω). Then, for each x ∈ Ω there is a probability measure ω(x, Ω) on ∂Ω, called harmonic measure for each x, such that the solution to (1) is

u(x) = f (y)dω(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 ∈ 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 \ 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 ω(z, E; Ω). For E ﬁxed, ω(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 ω(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, ω(z, α; D) verifies

l´ım ω(z, α; D) = 1 and l´ım ω(z, α; D) = 0. z→α z→β

The fact that ω(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 ω(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, ω(z, α, G) ≤ ω(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 ω 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 |f (z)| ≤ M in Ω and l´ım sup |f (z)| ≤ m < M when z ∈ Ω and z→ζ ζ ∈ E ⊂ ∂Ω. Then, for any z ∈ Ω,

log |f (z)| ≤ ω(z, E; Ω) log m + (1 − ω(z, E; Ω)) log M,

or similarly, |f (z)| ≤ mω(z,E;Ω)M1−ω(z,E;Ω).

A quantitative expression of the unique determination of analytic functions by their boundary values. Answer to the Milloux-Carleman problem: by the two-constants ω 1−ω theorem, |f (z0)| ≤ m M , where ω = ω(z0, γ, D \ γ).

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 16 / 37 Origin of the propagation of smallness property Two-constants and Hamard three-circles theorems

Theorem (n constants theorem)

Suppose that f is analytic in a Jordan domain D whose boundary is the union of n distinct Jordan curves α1, α2, . . . , αn. Suppose that for each j there is a constat Mj such that if z approaches any point of αj , then the limits of f (z) do not exceed Mj in absolute value. Then, for each z ∈ D,

n X log |f (z)| ≤ ω(z, αj ; D) log Mj . j=1

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 17 / 37 Origin of the propagation of smallness property Two-constants and Hamard three-circles theorems

Jacques Hadamard published this result without a proof in 1896 in Sur les fonctions entières. Then, in 1935, he presented a proof in Selecta: Jubilé Scientique de M. Jacques Hadamard. Meanwhile, other proofs were given by Otto Blumenthal and Georg Faber in 1907.

Theorem (Hadamard three-circles theorem) Let f (z) be an analytic function in |z| < R and M(r) = m´ax |f (reiθ)|. If θ 0 < r1 ≤ r ≤ r2 < R, then,

log r2 − log r log r − log r1 log M(r) ≤ log M(r1) + log M(r2). log r2 − log r1 log r2 − log r1

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 18 / 37 Origin of the propagation of smallness property Two-constants and Hamard three-circles theorems

J. Hadamard

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 19 / 37 Origin of the propagation of smallness property Two-constants and Hamard three-circles theorems

Proof (Hadamard three-circles theorem): We consider the ring D = {z ∈ C : r1 < |z| < r2} and its boundary will be formed by the arcs α = {z : |z| = r1} and β = {z : |z| = r2}, that is, ∂D = α ∪ β. We define ω(z, α; D) = log r2−log r as the harmonic measure on our log r2−log r1 ring D. By the harmonic measure definition, ω(z, β; D) = 1 − ω(z, α; D) = 1 − log r2−log r = log r−log r1 . log r2−log r1 log r2−log r1 We define M1 = M(r1) and M2 = M(r2), and substitute them in the two-constants theorem: log r2 − log r log r − log r1 log |f (z)| ≤ log M(r1) + log M(r2). log r2 − log r1 log r2 − log r1

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 20 / 37 Origin of the propagation of smallness property P-L theorem and H. three-lines lemma Phragmén-Lindelöf theorem and Hadamard three-lines lemma

A generalization of the maximum modulus principle. Lars Edvard Phragmén and Ernst Leonard Lindelöf, Sur une extension d’un principe classique de l’analyse et sur quelques propriétés des fonctions monogènes dans le voisinage d’un point singulier, 1908. Diﬀerent kinds of domains, such as strips, sectors or half-planes. We will consider the closed strip S = {z ∈ C : 0 ≤ Re z ≤ 1}, the open strip S˚ = {z ∈ C : 0 < Re z < 1} and the boundary of S, ∂S = {z ∈ C : Re z ∈ {0, 1}}.

Theorem (Phragmén-Lindelöf principle) Let f be an analytic function on S˚ and bounded and continuous on S. Then, sup |f (z)| ≤ m´ax sup |f (it)|, sup |f (1 + it)| . z∈S t∈R t∈R

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 21 / 37 Origin of the propagation of smallness property P-L theorem and H. three-lines lemma

L. E. Phragmén E. L. Lindelöf

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 22 / 37 Origin of the propagation of smallness property P-L theorem and H. three-lines lemma

Lemma (Hadamard three-lines lemma) Let f be an analytic function on the open strip S˚ = {z ∈ C : 0 < Re z < 1}, continuous and bounded in its closure, such that |f (z)| ≤ B0 when Re z = 0 and |f (z)| ≤ B1 when Re z = 1, for some 0 < B0, B1 < ∞. Then,

1−x x |f (z)| ≤ B0 B1 , when Re z = x, for any 0 ≤ x ≤ 1.

It can be proved using Phragmén-Lindelöf principle. Hadamard or G. Doetsch in Über die obere Grenze des absoluten Betrages einer analytischen Funktion auf Geraden in 1920?

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 23 / 37 Origin of the propagation of smallness property P-L theorem and H. three-lines lemma

Lars Ahlfors

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 24 / 37 Applications to Control Theory 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 25 / 37 Applications to Control Theory Higher dimensions: three-balls and three-regions theorems Higher dimensions

Evgenii Landis proved a three-spheres theorem for harmonic n functions in R and also a more general one for solutions of second order elliptic equations. Some problems of the qualitative theory of second order elliptic equation, 1963.

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 26 / 37 Applications to Control Theory Higher dimensions: three-balls and three-regions theorems

Nicola Garofalo and Fang-Hua Lin, Jacob Korevaar and Jan L. H. Meyers, Raymond Brummelhuis or Igor Kukavica. J. Korevaar and J. L. H. Meyers, Logarithmic convexity for supremum norms of harmonic functions, 1994. They mainly proved that the L2-version of the three-spheres theorem n was valid for harmonic functions in R with the same coeﬃcients as the logarithm terms that appear in the Hadamard theorem. The two main theorems J. Korevaar and J. L. H. Meyers proved in 1994.

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 27 / 37 Applications to Control Theory Higher dimensions: three-balls and three-regions theorems

Theorem (Three-balls theorem) Suppose 0 < ρ < r < R and n ≥ 2. Then, there exists a constant α ∈ (0, 1), depending only on ρ/R, r/R and n, such that for all n complex-valued harmonic functions u on the ball B(0, R) in R ,

α 1−α ||u||r ≤ ||u||ρ ||u||R , where ||u||t = sup |u(x)| on the ball B(0, t).

Some authors have obtained three-balls type theorems with an additional constant C = C(ρ/R, r/R, n),

α 1−α ||u||r ≤ C||u||ρ ||u||R . (2) The ﬁrst author that proved this kind of inequality could have been E. M. Landis in 1963. He actually proved an inequality (2) for solutions of second order linear elliptic partial diﬀerential equations.

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 28 / 37 Applications to Control Theory Higher dimensions: three-balls and three-regions theorems

The second theorem J. Korevaar and J. L. H. Meyers proved in 1994: a propagation of smallness property for arbitrary harmonic functions in n R with n ≥ 2. We can observe that the sets where they applied the propagation of smallness property are not concentric. Theorem

n Let Ω be a domain in R , Ω0 ⊂ Ω a nonempty subdomain and E ⊂ Ω a nonempty compact subset. Then, there is a constant α = α(E, Ω0, Ω) ∈ (0, 1] such that for all complex-valued harmonic functions u on Ω, α 1−α ||u||E ≤ ||u||Ω0 ||u||Ω , where ||u||A = sup |u(x)|. x∈A

The argument they used to prove this theorem is called covering by balls’ argument.

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 29 / 37 Applications to Control Theory Null-control for some parabolic problems Null-control for some parabolic problems

One application of the propagation of smallness property among many. We can prove the null-controllability of some parabolic equations when the control is over measurable space-time subsets, in the interior or in the boundary. The null-controllability problems we present, are for some time-independent parabolic evolutions with controls acting over measurable sets D = ω × [0, T ] and J = γ × [0, T ] with positive measure, where ω ⊂ Ω and γ ⊂ ∂Ω are measurable subsets.

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 30 / 37 Applications to Control Theory Null-control for some parabolic problems

We focus on a parabolic equation over a smooth and bounded n domain Ω in R and for a time interval (0, T ), for a distributed control f in the interior case, or for a boundary control h in the boundary case.  ∂ u − 4u = f (x, t)χ (x), in Ω × (0, T ),  t ω u = 0, on ∂Ω × [0, T ],  u(0) = u0, in Ω, and  ∂ u − 4u = 0, in Ω × (0, T ),  t u = h(x, t)χγ(x), on ∂Ω × [0, T ],  u(0) = u0, in Ω, where ω ⊂ Ω is an interior control region and γ ⊂ ∂Ω is a boundary control region.

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 31 / 37 Applications to Control Theory Null-control for some parabolic problems

Theorem (Interior null-control)

Let n ≥ 2. Then, ∂t − 4 is null-controllable at all positive times with distributed controls acting over a measurable set ω ⊂ Ω with positive Lebesgue measure when

4 = ∇ · (A(x)∇ · ) + V (x), is a self-adjoint elliptic operator, the coeﬃcients matrix A is smooth in Ω, V is bounded in Ω and both are real-analytic in an open neighborhood of ω. The same holds when n = 1, 1 4 = [∂ (a(x)∂ ) + b(x)∂ + c(x)] ρ(x) x x x and a, b, c and ρ are measurable functions in Ω = (0, 1).

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 32 / 37 Applications to Control Theory Null-control for some parabolic problems

Theorem (Boundary null-control)

Let n ≥ 2. Then, ∂t − 4 is null-controllable at all times T > 0 with boundary controls acting over a measurable set γ ⊂ ∂Ω with positive surface measure when

4 = ∇ · (A(x)∇ · ) + V (x) is a self-adjoint elliptic operator, the coeﬃcients matrix A is smooth in Ω, V is bounded in Ω and A, V and ∂Ω are real-analytic in an open neighborhood of γ in Ω.

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 33 / 37 Applications to Control Theory Null-control for some parabolic problems

The results in the previous theorems follow from: A straightforward application of the linear construction of the control function developed in by Gilles Lebeau and Luc Robbiano in 1995. For interior null-control theorem, a finite number of applications of the next propagation of smallness estimate from measurable sets established by Sergio Vessella in 1999. A suitable covering argument for the domain Ω (in the interior null-control theorem) and ∂Ω (in the boundary null-control theorem). For boundary null-control theorem, a successive iteration of a finite number of applications of the three-spheres type inequalities associated with the obvious extension of S. Vessella’s theorem for real n+1 analytic functions defined over real analytic hypersurfaces in R

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 34 / 37 Applications to Control Theory Null-control for some parabolic problems

Theorem (S. Vessella’s theorem) n Assume that f : B2 ⊂ R −→ R is an analytic function that veriﬁes

α −|α| n |∂ f (x)| ≤ M|α|!ρ , when x ∈ B2, α ∈ N , for some M > 0, 0 < ρ ≤ 1 and E ⊂ B 1 a measurable set of positive 2 measure. Then, positive constants N = N(ρ, |E|/|B1|) and θ = θ(ρ, |E|/|B1|), 0 < θ < 1, exist such that

θ 1−θ ∞ kf kL (B1) ≤ N |f | dx M . E

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 35 / 37 Applications to Control Theory Null-control for some parabolic problems

We can substitute the classic elliptic Carleman inequalities for the simple application of the propagation of the smallness bound or interpolation inequality described in the S. Vessella’s theorem, which doesn’t need Carleman inequalities. Lemma n Let Ω be a Lipschitz domain in R , R > 0 and assume that BR (x0) ∩ Ω is star-shaped with center at some x0 ∈ Ω. Then,

log r3 θ 1−θ r2 kuk 2 ≤ kuk 2 kuk , with θ = , (3) L (Br (x0)∩Ω) L (Br (x )∩Ω) L2(B (x )∩Ω) r3 2 1 0 r3 0 log r1 when ∆u = 0 in BR (x0) ∩ Ω, u = 0 on BR (x0) ∩ ∂Ω and 0 < r1 < r2 < r3 ≤ R. I. Kukavica, Unique continuation on the boundary for Dini domains, 1998.

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 36 / 37 Applications to Control Theory Null-control for some parabolic problems

Thank you very much for your attention!!

Jone Apraiz (UPV/EHU) Control en tiempos de crisis 04/05/2020 37 / 37