Regularity results in free boundary problems

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Regularity results in free boundary

problems

GOHAR ALEKSANYAN

Doctoral Thesis

Stockholm, Sweden 2016

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TRITA-MAT-A 2016:10

ISRN KTH/MAT/A-16/10-SE

ISBN 978-91-7729-161-9

KTH

Institutionen för Matematik

100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan

fram-lägges till offentlig granskning för avläggande av Filosofie doktorsexamen i

matematik fredagen den 2 december 2016 kl 13:00 i sal D3, Kungl Tekniska

högskolan, Lindstedtsvägen 5, Stockholm.

c

Gohar Aleksanyan, 2016

Tryck: Universitetsservice US AB

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iii

In memory of my father and teacher,

Hakob Aleksanyan

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v

Abstract

This thesis consists of three scientific papers, devoted to the

regu-larity theory of free boundary problems. We use iteration arguments to

derive the optimal regularity in the optimal switching problem, and to

analyse the regularity of the free boundary in the biharmonic obstacle

problem and in the double obstacle problem.

In Paper A, we study the interior regularity of the solution to the

optimal switching problem. We derive the optimal C

1,1

_{-regularity of}

the minimal solution under the assumption that the zero loop set is the

closure of its interior.

In Paper B, assuming that the solution to the biharmonic obstacle

problem with a zero obstacle is sufficiently close-to the one-dimensional

solution

1₆

(x

n

)

3+

, we derive the C

1,α

-regularity of the free boundary,

under an additional assumption that the noncoincidence set is an

NTA-domain.

In Paper C we study the two-dimensional double obstacle problem

with polynomial obstacles p

1

≤ p

2

_{, and observe that there is a new}

type of blow-ups that we call double-cone solutions. We investigate

the existence of double-cone solutions depending on the coefficients of

p

1

, p

2

, and show that if the solution to the double obstacle problem with

obstacles p

1

= −|x|

2

and p

2

= |x|

2

is close to a double-cone solution,

then the free boundary is a union of four C

1,α

-graphs, pairwise crossing

at the origin.

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vi

Sammanfattning

Denna avhandling består av tre vetenskapliga artiklar som ägnas

åt regularitetsteori för frirandsproblem. Vi använder iterationsargument

för att härleda den optimala regulariteten hos optimalväxlingsproblemet

och för att analysera regulariteten av den fria randen i det biharmoniska

hinderproblemet och i dubbelhinderproblemet.

I Artikel A studerar vi den lokala regulariteten hos lösningen till

op-timalväxlingsproblemet. Under antagandet att nolloopmängden är

till-slutningen av sitt inre visar vi C

1,1

-regularitet för den minimala

lös-ningen.

I Artikel B, antaget att lösningen till det biharmoniska

hinderpro-blemet med ett nollhinder ligger tillräckligt nära den endimensionell

lösning, härleder vi C

1,α

-regularitet för den fria randen, under ett extra

antagande att positivitetsmängden är ett NTA-område.

I Artikel A och Artikel B utför vi analys i

R

n

för n ≥ 2, medan i

Arti-kel C studerar vi ett tvådimensionellt hinderproblem. Det huvudsakliga

objektet för undersökningen i Artikel C är så kallade

dubbelkonlösning-ar. Vi visar att om lösningen till dubbelhinderproblemet med hinder

p

1

= −|x|

2

och p

2

= |x|

2

är nära en dubbelkonlösning, då består den

fria randen en union av fyra C

1,α

-grafer, parvis korsande i origo.

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Acknowledgements

First and foremost I would like to thank my advisor, John Andersson, for

the mathematics I have learned from him. This thesis exists due to him. John

has been a great source of knowledge and inspiration to me. I am grateful for

everything.

Many thanks go to my co-advisor, Erik Lindgren, for his careful

proofread-ing of the articles included in the thesis and for all the discussions. I want

to thank Diogo Aguiar Gomes and Henrik Shahgholian for introducing me to

the optimal switching problem, and for fruitful discussions on the topic.

During the years of my doctoral studies I found many friends among the

fellow graduate students and young scientists. They have been very kind and

supportive; it is my pleasure and my duty to thank Mariusz Hynek, Jevgenija

Pavlova, my cheerful officemate Ludvig af Klinteberg, Gustav Sædén Ståhl

and many others.

I am grateful to my family for their love. Thank you for letting me travel

the world and impatiently awaiting each of my arrivals at Zvartnots

interna-tional airport. My special thanks go to my brother, Sargis Aleksanyan, for all

our conversations about life and mathematics.

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1 Common notations

N

the set of natural numbers

R

the set of real numbers

R

n

_{the n-dimensional Euclidiean space}

x · y

Pn_i=1

x

i

y

i

, the scalar product of vectors x, y ∈

R

n

B

r

(x)

{y ∈

R

n

: |y − x| < r}, the open ball in

R

n

∂A

the boundary of the set A ⊂

R

n

A

the closure of the set A ⊂

R

n

A

0

_{the interior of the set A ⊂}

_R

n

A

c

the complement of the set A ⊂

R

n

a

+

_(a

+

)

max(a, 0), for a ∈

R

a

−

(a

−

)

max(−a, 0), for a ∈

R

∇

the gradient vector, ∇u :=

_∂x∂u

1

, ...,

∂u ∂xn

for a scalar function u

∆

the Laplace operator, ∆u :=

Pn_i ∂_∂x2u2

i

for a scalar function u

∆

2

_{the biharmonic operator, ∆}

2

_{u := ∆(∆u)}

D

2

u

the Hessian matrix for u

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Part I

Introduction and summary

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5 Introduction

This thesis consists of three articles, and all of them are devoted to the

regu-larity theory of free boundary problems.

The aim of Part I is to provide a brief overview of the obstacle problem

and of the free boundary problems that are investigated in Part II. In Part I

we also summarize the results obtained in the thesis, outlining the background

and techniques used. The core of the thesis is Part II, which is a combination

of three papers, named Papers A, B and C.

In Paper A we investigate the optimal regularity of solutions to the optimal

switching problem (see also Chapter 3). The problem is related both to the

classical obstacle problem (Chapter 1) and to the double obstacle problem

(Chapter 2), while the regularity of solutions to the optimal switching problem

is not covered by the known regularity theory for any of the above mentioned

problems. In fact, we are dealing with a system of obstacle-type equations,

where the obstacles depend on the solution.

Paper A also includes an example of a homogeneous global solution to the

two dimensional double obstacle problem, where the free boundary is a cross.

This example shows a new type of behavior of the free boundary for the double

obstacle problem, which does not happen in the case of the obstacle problem.

Paper C analyses the blow-ups of the solution to the double obstacle problem

in dimension two, with an emphasis on the so-called double-cone solutions.

In Paper B we use a linearization argument and show that if the solution

to the biharmonic obstacle problem is close to a halfspace solution, then the

free boundary is locally a C

1,α

-graph. The method does not require the use

of comparison principles and regularity of the solution. For the biharmonic

obstacle problem there are no optimal growth or nondegeneracy properties

known. The comparison principle for harmonic functions plays an important

role in the analysis of solutions to the obstacle problem. While there are

no comparison principles for biharmonic functions, as one can easily see by

looking at the following one-dimensional example; f (x) = x

2

in

R, f is a

nonnegative biharmonic function, obtaining a minimum at an interior point

x = 0. Hence we make an additional assumption that the noncoincidence

set is a non-tangentially accessible (NTA) domain. Chapter 4 contains a brief

introduction to the biharmonic obstacle problem and an overview of Paper B.

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Chapter 1 The obstacle problem

The obstacle problem is a motivating example in the theory of variational

in-equalities and free boundary problems. In this chapter we give the variational

formulation of the problem and a short survey on the known regularity

the-ory of the obstacle problem. This presentation is far from being exhaustive.

We have selected the properties of the obstacle problem which have a strong

relation to the problems studied in the main text.

The chapter is based on the article [5]. The text is also supported by the

book [11].

Let Ω ⊂

R

n

be a given domain with a smooth boundary, and let ϕ ∈ C

2

(Ω)

be a given function, called an obstacle. We are looking for the minimizer

u ∈ W

1,2

_{(Ω) to the Dirichlet functional}

J [u] =

Z

Ω

|∇u(x)|

2

_dx,

_(1.1)

over admissible functions u ≥ ϕ in Ω, with boundary condition u = g on

∂Ω, where g is a continuous function, satisfying g ≥ ϕ on ∂Ω. Since J is a

strictly convex functional over a convex admissible set, there exists a unique

minimizer. The minimizer u is called the solution to the obstacle problem.

A variational argument easily verifies that u is a superharmonic function in

Ω, and therefore it has a lower semi-continuous equivalent, which we call u.

Denote by

Ω

u

:= {x ∈ Ω; u(x) > ϕ(x)},

(1.2)

then Ω

u

is an open set, called the noncoincidence set. Denote the free

bound-ary for the obstacle problem by

Γ

u

:= ∂Ω

u

∩ Ω.

(1.3)

We see that Γ

u

depends on the solution u which is not known a priori,

ex-plaining the usage of the word ‘free’.

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8 CHAPTER 1. THE OBSTACLE PROBLEM

In the set Ω

u

, the function u does not touch the obstacle ϕ, and it locally

minimizes the Dirichlet energy, (1.1). Therefore u is a harmonic function in

Ω

u

. Hence we obtain the following Euler-Lagrange equation for the obstacle

problem,

min(−∆u, u − ϕ) = 0.

(1.4)

It has been shown (see for instance [5]) that the solution to the obstacle

problem u is as regular as the obstacle ϕ up to C

1,1

_{, which is the best regularity}

that the solution may achieve, since ∆u jumps from zero to ∆ϕ across the

free boundary.

Next we turn to to the regularity of the free boundary in the obstacle

problem. First let us rewrite equation (1.4) in a more convenient form. Denote

by v := u − ϕ, then according to (1.4), v is a nonnegative function solving

∆v = −∆ϕχ

{v>0}

,

(1.5)

where χ

A

is the characteristic function of a set A ⊂

R

n

. It follows from (1.4)

that ∆ϕ ≤ 0 in the coincidence set {u = ϕ}. Taking ∆ϕ ≡ −1, we study the

normalized obstacle problem,

∆u = χ

{u>0}

.

(1.6)

It is easy to verify that u minimizes the following energy functional

˜

J [u] =

Z Ω

1

2 |∇u(x)|

2

_{+ u(x)dx,}

_(1.7)

over nonnegative functions with given boundary condition.

Let u be the solution to the normalized obstacle problem in Ω ⊂

R

n

_,

B

1

(0) ⊂⊂ Ω. For a fixed x

0

∈ Γ

u

∩B

1

and r < 1 small, satisfying B

r

(x

0

) ⊂ B

1

,

consider the following rescalings,

u

x0,r

(x) :=

u(rx + x

0

)

r

2

.

(1.8)

The abbreviation u

r

:= u

0,r

is used. Let us note that the rescalings u

x0,r

solve

the obstacle problem (1.6) in the ball B

1

r

(0).

By using comparison principle, we can show the following quadratic growth

estimate

sup

Br(x0)

u ≤ C

n

r

2

,

(1.9)

where x

0

∈ Γ

u

, B

2r

(x

0

) ⊂ Ω, and C

n

is a dimensional constant. Hence the

rescalings u

x0,r

are bounded,

sup

B1

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9 for small r > 0.

Furthermore, at free boundary points the solution cannot decay faster than

quadratically. Let, x

0

∈ Ω

u

∩ Ω, then

sup

Br(x0)

u ≥

1 2n

r

2

_,

_(1.11)

which can be derived by applying the maximum principle for the harmonic

function v = u −

_2n1

|x − x

0

|

2

in Ω

u

∩ B

r

(x

0

). Inequality (1.11) implies that

rescalings u

x0,r

possess the following nondegenaracy property,

sup

B1

u

x0,r

≥

1 2n

.

(1.12)

From (1.10) we can obtain that ku

x0,r

k

C1,1(B1/2)

is uniformly bounded.

Hence through a subsequence u

x0,r

converges to a function u

0

in C

1,α

_(B

1/2

),

where 0 < α < 1. Let us emphasize that, at this point we do not have any

tools which would guarantee the existence of the lim

r→0+

u

x0,r

in any function

space. Any limit of the rescaling u

x0,r

over a sequence r

j

→ 0+ as j → ∞ is

called a blow-up of the solution u. The blow-ups are defined in

R

n

.

Further-more, the blow-ups are global solutions, i.e. solutions in

R

n

to the normalized

obstacle problem (1.6). It follows from the nondegenaracy property (1.12) and

the C

1,α

_{-convergence that the blow-ups are not identically zero.}

The analysis of the possible blow-ups and the proof of the uniqueness of

the blow-up are far from being easy, and we cannot include it in this short

overview of the obstacle problem. Let us only state the following important

theorem.

Theorem 1.1. Let u be the solution to the normalized obstacle problem, (1.6),

and let x

0

∈ Γ

u

be a free boundary point. Then the blow-up of u at x

0

is unique,

that is the limit u

0

(x) = lim

j→∞

u(rjx+x0)

r2

j

does not depend on the sequence

r

j

→ 0+. Furthermore, there are only two types of possible blow-ups; either

u

0

is a half-space solution,

u

0

(x) =

1

2 (x · e)

2 +

, where e ∈

R

n

_{is a unit vector,}

_(1.13)

or u

0

is a polynomial solution,

u

0

(x) =

1

2 (x · Ax), where A is an n × n nonnegative

symmetric matrix with trA = 1.

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10 CHAPTER 1. THE OBSTACLE PROBLEM

If u

x0,r

→

1 2

(x · e)

2

+

, then the vector e is the approximate unit normal to Γ

u

at x

0

, and x

0

is called a regular point. It has been shown that almost every

free boundary point is regular, and that the set of regular points is a relatively

open subset of Γ

u

. Furthermore, in a neighborhood of a regular point, the free

boundary is a C

1,α

_{-graph, moreover, it is real analytic.}

If the blow-up of u at x

0

is a polynomial, then x

0

is called a singular point.

It has been shown that the set of singular points is a closed set contained in

a lower dimensional C

1

_manifold.

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Chapter 2 The double obstacle problem

Let Ω be a bounded open set in

R

n

_{with smooth boundary. The solution to}

the double obstacle problem in Ω is the minimizer of the functional

J (u) =

Z

Ω

|∇u(x)|

2

_dx

over functions u ∈ W

1,2

_{(Ω), ψ}

1

_{≤ u ≤ ψ}

2

_{, satisfying the boundary condition}

u = g on ∂Ω. For the problem to be well defined we assume that ψ

1

≤ ψ

2

_in

Ω, and ψ

1

_{≤ g ≤ ψ}

2

_{on ∂Ω. The functions ψ}

1

_{and ψ}

2

_{are called respectively}

the lower and the upper obstacles.

If u < ψ

2

, then u solves the obstacle problem with ψ

1

, and therefore

−∆u ≥ 0 by (1.4). Also observe that if u > ψ

1

_{, then −u solves the obstacle}

problem with −ψ

2

_{, and ∆u ≥ 0 by (1.4). Therefore we may conclude that u}

satisfies the following inequalities,

ψ

1

≤ u ≤ ψ

2

_{, ∆u ≥ 0 if u > ψ}

1

_{and ∆u ≤ 0 if u < ψ}

2

_.

_(2.1)

It has been shown that the solution to the double obstacle problem is locally

C

1,1

_{under the assumption ψ}

i

_{∈ C}

2

_{(Ω), see for instance [3, 8]. Hence we can}

rewrite (2.1) as follows,

ψ

1

≤ u ≤ ψ

2

_{and ∆u = ∆ψ}

1

_χ

{u=ψ1_}

+ ∆ψ

2

χ

_{u=ψ2_}

− ∆ψ

1

χ

_{ψ1_=ψ2_}

a.e.,

where χ

A

is the characteristic function of a set A ⊂

R

n

. Let us observe that

∆u = ∆ψ

1

_{in the set {u = ψ}

1

_{} ∩ {u < ψ}

2

_{}, hence (2.1) implies that ∆ψ}

1

_{≤ 0,}

and ψ

1

_{is a superharmonic function. By a similar argument, we obtain that}

ψ

2

is a subharmonic function in the set {u = ψ

2

} ∩ {u > ψ

1

_}.

Since the optimal regularity of the solution is known, our main interest

becomes the analysis of the free boundary. Denote by

Ω

1

:= {u > ψ

1

}, Ω

2

:= {u < ψ

2

}, and Ω

12

:= Ω

1

∩ Ω

2

,

(2.2)

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12 CHAPTER 2. THE DOUBLE OBSTACLE PROBLEM

then Ω = Ω

1

∪ Ω

2

∪ Λ, where Λ := {x ∈ Ω : ψ

1

(x) = ψ

2

(x)}. We call Ω

12

the noncoincidence set. It follows from (2.1) that ∆u ≥ 0 and ∆u ≤ 0 in Ω

12

.

Hence u is a harmonic function in the noncoincidence set, as in the ”single”

obstacle problem.

Define the free boundary for the double obstacle problem

Γ

u

:= ∂Ω

12

∩ Ω,

(2.3)

and let

Γ

i

:= ∂Ω

i

∩ Ω, i = 1, 2, then Γ

u

⊂ Γ

1

∪ Γ

2

.

(2.4)

Assume that

ψ

1

≤ ψ

2

_{, and ψ}

i

_{∈ C}

2

_(Ω).

(2.5)

If ψ

1

_{< ψ}

2

_{, then Γ}

1

∩ Γ

2

= ∅, and x

0

∈ Γ

u

implies that x

0

∈ Γ

1

or x

0

∈ Γ

2

. Let

x

0

∈ Γ

1

, then there exists a small ball B

r

(x

0

) such that B

r

(x

0

)∩{u = ψ

2

} = ∅.

Hence u solves the obstacle problem with obstacle ψ

1

in the ball B

r

(x

0

), and

the known regularity theory for the obstacle problem can be applied to analyse

the free boundary in a neighborhood of x

0

. Therefore we assume that

Λ = {x ∈ Ω : ψ

1

(x) = ψ

2

(x)} 6= ∅.

(2.6)

Let x

0

∈ Γ

u

be a free boundary point, if x

0

∈ Λ, then in a neighborhood

/

of x

0

, we have ψ

1

< ψ

2

, this case has already been discussed above.

Assume that x

0

∈ Γ

u

∩ Λ. First let us observe that x

0

cannot be an interior

point of Λ. Otherwise, if x

0

∈ Λ

0

, then there exists B

r

(x

0

) ⊂ Λ, which is the

same as ψ

1

= ψ

2

in B

r

(x

0

). Since ψ

1

≤ u ≤ ψ

2

, we obtain u = ψ

1

= ψ

2

in B

r

(x

0

), hence x

0

∈ Γ

/

u

. So we are interested in the behavior of the free

boundary at the points x

0

∈ ∂Λ ⊂ Γ

u

. First we want to understand the

behavior of the free boundary when x

0

∈ Λ is an isolated point. Paper C

partially answers this question in dimension two with polynomial obstacles.

2.1 Summary of Paper C

In Paper C we study the following normalized double obstacle problem in

dimension two,

∆u = λ

1

χ

{u=p1_}

+ λ

₂

χ

_{u=p2_}

(2.7)

with polynomial obstacles p

1

_{≤ p}

2

_{, where λ}

1

= ∆p

1

≤ 0 and λ

2

= ∆p

2

≥ 0

are constants. We assume that the obstacles p

1

_{and p}

2

_{meet at a single point}

x

0

, i.e. p

1

(x) = p

2

(x) iff x = x

0

.

The work is inspired by the following example in

R

2

_,

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2.1. SUMMARY OF PAPER C

13 It is easy to see that u

0

solves the double obstacle problem (2.1), with obstacles

p

1

_{(x) = −x}

2

1

− x

22

and p

2

(x) = x

21

+ x

22

. Observe that Λ = {0}, and the free

boundary consists of two lines crossing at the origin. The function u

0

is a

motivational example for double-cone solutions, which are the novelty of this

paper.

Let u be the solution to the normalized double obstacle problem (2.7).

Assume that x

0

= 0, we can always come to this situation with a change

of variables. By subtracting a first order polynomial from p

1

_{, p}

2

_{and u, and}

taking into account that u ∈ C

1,1

_{, we get}

u(0) = p

1

(0) = p

2

(0) = 0, and ∇u(0) = ∇p

1

(0) = ∇p

2

(0) = 0.

Hence

p

1

(x) = a

1

x

21

+ 2b

1

x

1

x

2

+ c

1

x

22

and p

2

_{(x) = a}

2

x

21

+ 2b

2

x

1

x

2

+ c

2

x

22

.

Furthermore, we may assume that b

1

= b

2

= 0, by rotating the coordinate

system and subtracting a harmonic polynomial from p

1

_{, p}

2

_{and from u, thus}

obtaining

p

1

(x) = a

1

x

21

+ c

1

x

22

and p

2

_{(x) = a}

2

x

21

+ c

2

x

22

.

(2.9)

Observe that p

1

_{, p}

2

_{are second order homogeneous polynomials; which means}

p

1

(rx) = r

2

p

1

(x), and p

2

(rx) = r

2

p

2

(x).

(2.10)

As in the classical obstacle problem, any limit of

u(rx)_r2

as r → 0+, is called

a blow-up of the solution u to the double obstacle problem at the origin.

We show that the blow-ups of a solution to the normalized double obstacle

problem are homogeneous of degree two functions via Weiss’ monotonicity

formula. This result is not surprising at all, since we already saw in Theorem

1.1 that the blow-ups of the solution to the obstacle problem are homogeneous

degree two functions.

Knowing that the blow-ups are homogeneous global solutions, we make

a complete characterization of possible blow-ups in dimension n = 2.

In

particular we see that there exist blow-ups of a new type. We call these

solutions double-cone solutions, since the noncoincidence set is a union of two

cones with a common vertex at the origin. We show that there exist

double-cone solutions if and only if the following polynomial

P = P (x

1

, x

2

) ≡ p

1

(x

1

, x

2

) + p

2

(x

2

, x

1

) = (a

1

+ c

2

)x

21

+ (a

2

+ c

1

)x

22

(2.11)

has zeroes other than x = 0. If P ≡ 0, there are infinitely many double-cone

solutions, and if P 6≡ 0, but P = 0 on a line, there are finitely many

double-cone solutions. This result is quite surprising and unexpected. In the next

paragraph we describe how (2.11) affects the stability of the free boundary.

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14 CHAPTER 2. THE DOUBLE OBSTACLE PROBLEM

Let ε be an arbitrary number, |ε| << 1. Then for polynomials p

1

(x) =

−x

2

1

− x

22

, p

2

(x) = x

21

+ x

22

there exist infinitely many double-cone solutions.

While when we look at the double obstacle problem with p

1

_{= −x}

2

1

− x

22

and

˜

p

2

= (1 − ε)x

2

1

+ (1 + ε)x

22

there are only four double-cone solutions, and

for p

1

_{= −x}

2

1

− x

22

and ¯

p

2

= (1 + ε)x

21

+ (1 + ε)x

22

there are none. This

property reveals the instability of the double obstacle problem in the sense

that changing the obstacles slightly, may change the solution and the free

boundary significantly.

In this paper we also study the regularity of the free boundary in the case

when P ≡ 0. Since in this case we have infinitely many rotational invariant

double-cone solutions, we can use a flatness improvement argument to show

that the free boundary is locally a union of four C

1,α

_{-graphs. We have already}

used a similar argument in Paper B, see also Chapter 4.

The case P ≡ 0, can be reduced to the case p

1

_{= −|x|}

2

_{, and p}

2

_{= |x|}

2

_{. A}

general double-cone solution in this case can be written in polar coordinates

as follows,

µ = µ

φ1,φ2

(r, θ) :=

            

r

2

,

if − φ

2

≤ 2θ ≤ φ

1

r

2

cos(2θ − φ

1

),

if φ

1

≤ 2θ ≤ π + φ

1

r

2

cos(2θ + φ

2

),

if − π − φ

2

≤ 2θ ≤ −φ

2

−r

2

_,

_otherwise,

(2.12)

where 0 ≤ φ

1

, φ

2

≤ π are parameters describing a double-cone solution µ.

The main result in Paper C is the following theorem.

Theorem 2.1 (Theorem 4.7 in Paper C). Let u be the solution to the

two-dimensional double obstacle problem with obstacles p

1

= −x

2₁

− x

2

and p

2

=

x

2

1

+ x

22

. Assume that ku − µk

L2_(B

2)

= δ is sufficiently small, where µ is a

double-cone solution, and is not a halfspace solution. Then in a small ball B

r0

the free boundary consists of four C

1,γ

- graphs meeting at the origin, denoted

by Γ

₁+

, Γ

₁−

, Γ

₂+

, Γ

₂−

. Neither Γ

1

= Γ

1+

∪ Γ

−

1

nor Γ

2

= Γ

2+

∪ Γ

−

2

has a normal

at the origin. The curves Γ

₁+

and Γ

₂+

cross at a right angle, the same is true

for Γ

₁−

and Γ

₂−

.

We show that if the solution is close to a double-cone solution in B

1

,

then the blow-up at the origin is unique. Furthermore, employing the known

regularity theory for the free boundary in the obstacle problem, we derive

that the free boundary Γ for the double obstacle problem is a union of four

C

1,γ

-graphs meeting at the origin. Neither Γ

1

nor Γ

2

is flat at the origin, and

(25)

2.1. SUMMARY OF PAPER C

15 x

1

x

2

Γ

₂+

Γ

₁+

Γ

₂−

Γ

₁−

∆u = 0

u = p

2

u = p

1

∆u = 0

S

1

S

2

x

0 ν(0) ν(x0) ν(0)

Figure 2.1: The behavior of the free boundary, with obstacles touching at a single point

We mentioned that if P 6≡ 0, but P = 0 on a line, there are finitely many

double-cone solutions. We aim to describe the behavior of the free boundary

in this case in a future publication.

(26)

(27)

Chapter 3 The optimal switching problem

Let Ω ⊂

R

n

_{be a bounded domain with smooth boundary. Consider a power}

plant with several modes (states) of energy production in Ω. When running

the production mode 1 ≤ i ≤ m, we pay a running cost f

i

(x) depending also

on the present position x ∈ Ω. We can switch from mode i to another mode

1 ≤ j ≤ m in order to minimize the cost of energy production, but we have

to pay a switching cost ψ

ij

(x) depending also on the present state x. Assume

that ψ

ii

_{= 0, which is reasonable; if we do not switch, there is no need to pay}

a switching cost.

The solution to the optimal switching problem is a vector valued function

u = (u

1

_{, u}

2

_{, ..., u}

m

_{) and it represents the cost of an optimal strategy for energy}

production. As it has been discussed in the literature [10], [4], the negative

of the solution to the optimal switching problem, −u, solves the following

system:

min(−L

i

u

i

+ f

i

, min

j

(u

i

_{− u}

j

_{+ ψ}

ij

_{)) = 0, in Ω,}

_(3.1)

where L

i

_{is an elliptic operator, corresponding to the mode i. Equation (3.1)}

says that if min

j

(u

i

− u

j

+ ψ

ij

) > 0, it is not optimal to switch to any other

mode j 6= i, and we continue running mode i.

For the optimal switching problem to be well defined, we need to impose

the nonnegative loop condition: Let i

0

, i

1

, . . . , i

l

= i

0

be any loop of length

l, i.e. including l number of states. Assume that (u

1

_{, u}

2

_{, ..., u}

m

_{) is a solution}

to system (3.1), then u

i

_{− u}

j

_{+ ψ}

ij

_{≥ 0 for any i, j ∈ {1, 2, ..., m}, and after}

summing the equations over the loop, we get

l X

j=1

ψ

ij−1,ij

_{≥ 0.}

_(3.2)

Condition (3.2) is a necessary assumption for the existence of a solution to

(3.1). Without (3.2) we can make the energy production cost arbitrary small

(28)

18 CHAPTER 3. THE OPTIMAL SWITCHING PROBLEM

by looping. For example, if l = 2 and ψ

12

(x) + ψ

21

_{(x) < 0 for some x, then}

at the point x we can switch from mode 1 to mode 2, and immediately switch

back to mode 1, thus paying a negative cost. This way we can decrease the

cost as much as we want by switching many times. Therefore the assumption

ψ

12

_{+ ψ}

21

_{≥ 0 is needed in order to obtain a finite cost function.}

The uniqueness and C

1,1

_{-regularity of the solution to system (3.1) have}

been studied in the literature under the assumption that the switching costs

ψ

i

_{are nonnegative constants, [7], [10], [4].}

In Paper A we consider a system, arising in a model optimal switching

problem with only two states, i. e. m = 2, and L

1

= L

2

_{= ∆,}

 



min(−∆u

1

_{+ f}

1

_{, u}

1

_{− u}

2

_{+ ψ}

1

_{) = 0}

min(−∆u

2

_{+ f}

2

_{, u}

2

_{− u}

1

_{+ ψ}

2

_{) = 0,}

(3.3)

with given Dirichlet boundary conditions u

i

_{= g}

i

_{on ∂Ω. The switching costs}

ψ

1

_{, ψ}

2

_{are given functions, satisfying the nonnegative loop assumption (3.2),}

which is the same as

ψ

1

+ ψ

2

≥ 0,

(3.4)

since m = 2. Denote by

L := {x ∈ Ω; ψ

1

(x) + ψ

2

(x) = 0},

and call it a zero loop set.

3.1 Summary of Paper A

In Paper A our aim is to investigate if solutions to (3.3) are C

1,1

_{. Assume that}

f

1

, f

2

∈ C

α

_{, and ψ}

1

_{, ψ}

2

_{∈ C}

2,α

_{, for some 0 < α < 1.}

_(3.5)

Then there exist v

1

_{, v}

2

_{∈ C}

2,α

loc

solving the Poisson equation ∆v

i

= f

i

in Ω.

Denote u

i

0

:= u

i

− v

i

, then u

i0

is as regular as u

i

up to C

2,α

, and (u

10

, u

20

) solves

the following system

 



min(−∆u

1

_{, u}

1

_{− u}

2

_{+ ϕ}

1

_{) = 0,}

min(−∆u

2

_{, u}

2

_{− u}

1

_{+ ϕ}

2

_{) = 0,}

(3.6)

where ϕ

1

_{= v}

1

_{− v}

2

_{+ ψ}

1

_{and ϕ}

2

_{= v}

2

_{− v}

1

_{+ ψ}

2

_{are the new switching cost}

functions preserving the loop condition, since ϕ

1

_{+ ϕ}

2

_{≡ ψ}

1

_{+ ψ}

2

_.

We see that u

1

solves the obstacle problem with the obstacle u

2

− ϕ

1

_{, and}

(29)

3.1. SUMMARY OF PAPER A

19 the obstacle u

1

− ϕ

2

_{. Thus (3.7) is a system of obstacle-type equations, where}

the obstacles depend on the solution (u

1

_{, u}

2

_{). Hence the regularity of the}

solution to system (3.6) cannot be obtained directly by the known regularity

theory for the obstacle problem. We see that (u

1

, u

2

) cannot have a better

regularity than C

1,1

_{which is the best regularity for the obstacle problem.}

If

L = ∅, that is ϕ

1

_+ϕ

2

_{> 0, then for any x ∈ Ω, u}

1

_(x)−u

2

_(x)+ϕ

1

_{(x) > 0}

or u

2

_{(x) − u}

1

_{(x) + ϕ}

2

_{(x) > 0. Otherwise, if u}

1

_{(x) − u}

2

_{(x) + ϕ}

1

_{(x) = 0 and}

u

2

(x) − u

1

_{(x) + ϕ}

2

_{(x) = 0, we obtain ϕ}

1

_{(x) + ϕ}

2

_{(x) = 0, contradicting}

_{L = ∅.}

Observe that if u

1

_{− u}

2

_{+ ϕ}

1

_{> 0, then u}

1

_{is a harmonic function, and u}

2

_solves

the obstacle problem with the obstacle u

1

_{− ϕ}

2

_{. Hence we may conclude that}

u

1

is an infinitely differentiable function, and u

2

∈ C

_loc1,1

.

Thus, if

L = ∅, we obtain (u

1

_{, u}

2

_{) ∈ C}

1,1

loc

by the regularity theory for the

obstacle problem. This result is interesting, but not enough, since we want to

know if the C

1,1

_{-regularity holds in case}

_{L 6= ∅.}

In

L we have that ϕ

1

_{= −ϕ}

2

_{, hence we obtain from (3.6) that u}

1

_−u

2

_+ϕ

1

_≥

0 and u

2

_{− u}

1

_{− ϕ}

1

_{≥ 0, which implies u}

1

_{− u}

2

_{+ ϕ}

1

_{≡ 0 in L . Take any}

subharmonic function u

1

_{∈ W}

2,p

_{, for n < p < ∞ and let u}

2

_{= u}

1

_{+ ϕ}

1

_{, then}

(u

1

, u

2

) solves (3.6) in

L . Hence system (3.6) may not have a unique solution

if

L 6= ∅, and (3.6) admits solutions that are not C

1,1

_{. Instead we consider}

the following system,

      

min(−∆u

1

_{, u}

1

_{− u}

2

_{+ ϕ}

1

_{) = 0,}

min(−∆u

2

_{, u}

2

_{− u}

1

_{+ ϕ}

2

_{) = 0,}

min(−∆u

1

_{, −∆u}

2

_{) = 0,}

(3.7)

which already has a unique solution. By using comparison principles, we show

that the solution to system (3.7) is the smallest among the solutions to (3.6).

The third equation in system (3.7) has a natural implication in the optimal

switching setting; it says that we always run one of the modes i = 1 or i = 2,

even when we can switch for free. Let us also observe that the extra (third)

equation in (3.7) holds automatically if

L = ∅.

Now let us study the relation between the system (3.7) and the double

obstacle problem. It is easy to see that U = u

1

− u

2

_{solves the following}

double obstacle problem

−ϕ

1

_{≤ U ≤ ϕ}

2

_{, −∆U ≤ 0 a.e. if U > −ϕ}

1

_{, −∆U ≥ 0 a.e. if U < ϕ}

2

_(3.8)

with obstacles −ϕ

1

_{≤ ϕ}

2

_{. Assuming that ϕ}

i

_{∈ C}

2,α

_{, by the known regularity}

theory for the double obstacle problem, we obtain that U ∈ C

_loc1,1

. Thus the

difference u

1

− u

2

_{∈ C}

1,1

_{, while the question if u}

1

_{∈ C}

1,1

loc

and u

2

∈ C

1,1 loc

is not

covered by the regularity theory of the double obstacle problem.

(30)

20 CHAPTER 3. THE OPTIMAL SWITCHING PROBLEM

Let us sketch the main results in Paper A. Consider system (3.7), and let

Ω

1

:= {−∆u

1

> 0}, Ω

2

:= {−∆u

2

> 0}, and Ω

12

:= Ω \ Ω

1

∪ Ω

2

be disjoint

open sets. Then

−∆u

1

_{= ∆ϕ}

1

_{> 0, −∆u}

2

_{= 0 in Ω}

1

−∆u

2

_{= ∆ϕ}

2

_{> 0, −∆u}

2

_{= 0 in Ω}

2

−∆u

1

_{= 0, −∆u}

2

_{= 0 in Ω}

12

.

(3.9)

In the set Ω \ Ω

1

we get −∆u

1

= 0, and therefore u

2

solves the obstacle

problem

min(−∆u

2

, u

2

− u

1

_{+ ϕ}

2

_{) = 0,}

with a C

2,α

_{-obstacle u}

1

_{− ϕ}

2

_{. Hence u}

2

_{∈ C}

1,1

loc

(Ω \ Ω

1

). By a similar argument

we obtain u

1

_{∈ C}

1,1

loc

(Ω \ Ω

2

).

It remains to study the regularity of the solution in a neighborhood of the

set ∂Ω

1

∩ ∂Ω

2

∩ Ω. Note that ∂Ω

1

∩ ∂Ω

2

⊂ L , since u

1

− u

2

+ ϕ

1

= 0 in Ω

1

and u

2

_{− u}

1

_{+ ϕ}

2

_{= 0 in Ω}

2

.

Let us study the regularity of (u

1

, u

2

) in the interior of

L , denoted by

L

0

_{. We have already observed before that u}

2

_{= u}

1

_{+ ϕ}

1

_in

_{L . Hence we}

obtain from the third equation in system (3.7) that min(−∆u

1

_{, −∆u}

2

_{) =}

min(−∆u

1

_{, −∆u}

1

_{− ∆ϕ}

1

_{) = 0, therefore u}

1

_{solves the following equation,}

−∆u

1

_{= (∆ϕ}

1

₎

+

_in

_L

0

_,

_(3.10)

and u

2

= u

1

_{+ ϕ}

1

_in

_{L . By the classical regularity theory for elliptic}

equa-tions, solutions to equation (3.10) are locally C

2,α

_{, if ∆ϕ}

1

_{∈ C}

α

_{. Thus in a}

neighborhood of the points x ∈ ∂Ω

1

∩ ∂Ω

2

∩ L

0

, the solution (u

1

, u

2

) ∈ C

2,α

.

The main difficulty is to study the regularity of (u

1

, u

2

_{) at the points}

x

0

∈ ∂Ω

1

∩ ∂Ω

2

∩ ∂L , called ‘meeting’ points.

Let u be a twice continously differentiable function in B

1

(x

0

), having

Hölder continuous second order derivatives, i. e. u ∈ C

2,α

(B

1

(x

0

)), for some

0 < α < 1. Let p

x0

be the second degree Taylor polynomial for u at x

0

, then

sup

x∈Br(x0)

|u(x) − p

x0

(x)| ≤ C

n

kD

2

uk

Cα

r

2+α

,

(3.11)

where C

n

is a dimensional constant. Estimate (3.11) can be obtained easily

by using the mean-value property for continously differentiable functions and

that D

2

_{u ∈ C}

α

_.

On the other hand if u is a continuous function in Ω, and for every x

0

∈ Ω

there exists a polynomial p

x0

such that

sup

x∈Br(x0)

|u(x) − p

x0

(x)| ≤ M r

(31)

3.1. SUMMARY OF PAPER A

21 for a uniform constant M > 0, then u is twice continously differentiable, and

kD

2

_uk

Cα

≤ c

_n

M .

Our main result in Paper A is the following theorem.

Theorem 3.1 (Theorem 4 in Paper A). Assume ϕ

1

_{, ϕ}

2

_{∈ C}

2,α

_{, and}

_{L = L}

0

_.

Then the solution to the system (3.7), (u

1

_{, u}

2

_{) is C}

2,α

_{-regular on ∂Ω}

1

∩ ∂Ω

2

∩

∂

L ∩ Ω, in the sense that for every x

0

∈ ∂Ω

1

∩ ∂Ω

2

∩ ∂L ∩ Ω, there exist

second order polynomials p

1 x0

, p

2 x0

, such that

sup

x∈Br(x0)

|u

i

(x) − p

i_x₀

(x)| ≤ Cr

2+α

(3.13)

where the constant C > 0 depends only on the given data.

The idea of the proof comes from [2], where the authors derive the optimal

regularity for the no-sign obstacle problem. Let u ∈ W

2,p

, for p < ∞, if ∆u is

a bounded function, it does not imply that D

2

_{u is bounded, but it does imply}

that D

2

_{u ∈ BM O. First, we show that ∆u}

1

_{and ∆u}

2

_{are bounded functions,}

then use the BM O-argument to derive D

2

u

i

∈ L

∞

_.

Let 0 ∈ ∂Ω

1

∩ ∂Ω

2

∩ ∂L

0

, be a meeting point. Since ϕ

i

∈ C

2,α

, we get

∆ϕ

1

_{(0) + ∆ϕ}

2

_{(0) = 0. According to (3.9), ∆ϕ}

1

_{≥ 0 in Ω}

1

and ∆ϕ

2

≥ 0 in Ω

2

,

hence ∆ϕ

1

_{(0) = ∆ϕ}

2

_{(0) = 0.}

Employing the BM O-estimates for D

2

_u

1

_{and D}

2

_u

2

_{, we show that the}

polynomials

p

1_r

(x) := (u

1

)

r

+ (∇u

1

)

r

· x +

1

2 x · (D

2

_u

1

₎

r

· x and

p

2_r

(x) := (u

2

)

r

+ (∇u

2

)

r

· x +

1

2 x · (D

2

u

2

)

r

· x

converge to harmonic polynomials, denoted respectively by p

1₀

and p

2₀

. We also

describe the rate of convergence,

sup

x∈Br(0)

|p

i_r

(rx) − p

i₀

(rx)| ≤ Cr

2+α

,

where C is just a constant. Considering the following rescalings;

v

_ri

(x) =

u

i

_{(rx) − p}

i 0

(rx)

r

2+α

,

we study the corresponding system for (v

1

r

, v

r2

). Taking into account that

∆ϕi_(rx)

rα

is uniformly bounded for i = 1, 2, we show that rescalings (v

1

r

, v

2r

) are

uniformly bounded in the ball B

1

, which is equivalent to (3.13).

(32)

22 CHAPTER 3. THE OPTIMAL SWITCHING PROBLEM

It follows from Theorem A that the solution to system (3.7) is locally C

1,1

if

L = L

0

_.

In the end of the article we justify our assumption x

0

∈ ∂L

0

with a

counterexample. By considering a particular system in

R

2

, with

L = {0},

we find an explicit solution, which is not C

1,1

_{in any neighborhood of the}

origin. This example reveals that the regularity of the solution to the optimal

switching problem depends on the topological properties of the zero loop set.

We saw that at the so-called meeting points |D

2

_u

i

_{(rx)| is of order r}

α

_{. While if}

0 is an isolated point of

L , we may have |D

2

_{u(rx)| ≈ − ln r as r → 0+. This}

is a new result, and it shows how different the optimal switching problem is

from the obstacle or double obstacle problems.

Let us also mention that Paper C was born from the example in Paper A,

although we gave the summary of Paper C in the previous chapter.

(33)

Chapter 4 The biharmonic obstacle problem

Let Ω ⊂

R

n

be a given domain, and ϕ ∈ C

2

_{(Ω), ϕ ≤ 0 on ∂Ω be a given}

function, called an obstacle. Consider the problem of minimizing the following

functional

J [u] =

Z

Ω

(∆u(x))

2

dx,

(4.1)

over all functions u ∈ W

₀2,2

(Ω), such that u ≥ ϕ. The functional J admits

a unique minimizer, called the solution to the biharmonic obstacle problem

with obstacle ϕ.

A variational argument easily verifies that ∆u is a weakly subharmonic

function. Furthermore, ∆u is a harmonic function in the noncoincidence set

{u > ϕ}. Therefore the solution satisfies the following variational inequality

∆

2

u ≥ 0, u ≥ ϕ, ∆

2

u · (u − ϕ) = 0.

It has been shown in [6] and [9] that the solution u ∈ W

_loc3,2

(Ω), ∆u ∈

L

∞_loc

(Ω), and moreover u ∈ W

_loc2,∞

(Ω).

Furthermore, in the paper [6], the

authors show that in dimension n = 2 the solution u ∈ C

2

(Ω) and that the

free boundary Γ

u

:= ∂{u = ϕ} lies on a C

1

-curve in a neighbourhood of the

points x

0

∈ Γ

u

, such that ∆u(x

0

) > ∆ϕ(x

0

).

The setting of our problem is slightly different from the one in [6] and

[9]. We consider a zero-obstacle problem with general nonzero boundary

con-ditions. We look for a minimizer to the functional (4.1) over the admissible

set

A :=

(

u ∈ W

2,2

(Ω), u ≥ 0, u = g > 0,

∂u

∂ν

= f on ∂Ω

)

.

The minimizer u exists, and it is unique. The minimizer is called the solution

to the biharmonic obstacle problem. As in the obstacle problem, we denote

the free boundary by

Γ

u

:= ∂Ω

u

∩ Ω, where Ω

u

:= {u > 0}.

23

(34)

24 CHAPTER 4. THE BIHARMONIC OBSTACLE PROBLEM

There are several important questions regarding the biharmonic obstacle

problem that remain open. For example, the optimal regularity of the solution,

the characterization of blow-ups at free boundary points, etc..

In Paper B we focus on the regularity of the free boundary for an

n-dimensional biharmonic obstacle problem, assuming that the solution is close

to the one-dimensional solution

1₆

(x

n

)

3+

. We saw in Chapter 1 that the solution

to the obstacle problem at a regular point has a unique blow-up of the form

1 2

(x

+

n

)

2

in some coordinate sytem. A flatness improvement argument shows

that if the solution to the obstacle problem is close to the halfspace solution

1 2

(x

+

n

)

2

, then the free boundary is locally a graph of a C

1,α

-function. We expect

to obtain a similar result for the biharmonic obstacle problem assuming that

the solution is close to the one-dimensional solution c(x

+

n

)

3

. We choose c = 1/6

so that ∇∆

1₆

x

3

n

= e

n

, but we could have as well as chosen a different positive

constant.

4.1 Summary of Paper B

In [1], using flatness improvement argument, the authors show that the free

boundary in the p-harmonic obstacle problem is a C

1,α

graph in a

neighbor-hood of the points where the solution is almost one-dimensional. We apply

the same technique in order to study the regularity of the free boundary in

the biharmonic obstacle problem.

In the first step we study one-dimensional solutions.

Assume that u

0

solves the biharmonic obstacle problem with zero obstacle in an open

inter-val (−a, a) ⊂

R, and let the origin be a free boundary point. By a direct

computation we see that u

0

is one of the following functions; cx

3+

, cx

3−

or cx

3

,

where c is a positive constant. If u(x) = u

0

(x

n

) = c(x

n

)

3+

, for x ∈

R

n

, then

the noncoincidence set for u is the halfspace {x ∈

R

n

, x

n

> 0}, and the free

boundary is the plane {x

n

= 0}.

Let us choose c =

1₆

, and introduce the class

B

%

κ

(ε) of solutions to the

biharmonic obstacle problem, that are close to the one-dimensional solution

1 6

(x

+ n

)

3

.

Definition 4.1. Let u ≥ 0 be the solution to the biharmonic obstacle problem

in a domain Ω ⊂

R

n

_{, B}

2

⊂⊂ Ω and assume that 0 ∈ Γ

u

is a free boundary

point. We say that u ∈

B

%

κ

(ε), if the following assumptions are satisfied:

1. u is almost one dimensional, that is

k∇

0

uk

W2,2_(B 2)

≤ ε,

(35)

4.1. SUMMARY OF PAPER B

25 2. The set Ω

u

:= {u > 0} is a non-tangentially accessible (NTA) domain

with constants r

0

= M

−1

= %.

3. There exists 2 > t > 0, such that u = 0 in B

2

∩ {x

n

< −t}.

4. We have the following normalization

kD

3

uk

L2_(B 1)

=

1

6 D

3

(x

n

)

3+ _L2_(B 1)

=

|B

1

|

1 2

2

12

:= ω

n

,

(4.2)

and we also assume that

kD

3

_uk

L2_(B 2)

< κ,

(4.3)

where κ >

1₆

D

3

_(x

n

)

3+ _L2_(B 2)

= 2

n2

ω

_n

.

Assumption 1 states that the solution is flat in the x

n

-direction. This can

be compared to the flatness assumption in [1].

We show that the precise value of the parameter t in assumption 3 is not

very important. The normalization in point 4 corresponds to our choice of

c =

1₆

, and it can always be achieved by a renormalization.

For the biharmonic obstacle problem there are no optimal growth or

non-degeneracy properties known, that could help us avoid additional assumptions