# Hamilton Jacobi Equations Viscosity Solutions And Applications

## Transcript Of Hamilton Jacobi Equations Viscosity Solutions And Applications

HAMILTON–JACOBI EQUATIONS: VISCOSITY

SOLUTIONS AND APPLICATIONS

HUNG VINH TRAN Department of Mathematics University of Wisconsin Madison Van Vleck hall, 480 Lincoln Drive, Madison, WI 53706, USA

Second draft, Summer 2019

1

Contents

1 Introduction to viscosity solutions for Hamilton–Jacobi equations

7

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Vanishing viscosity method for ﬁrst-order Hamilton–Jacobi equations . . . . . 10

3 Existence of viscosity solutions via the vanishing viscosity method . . . . . . . 16

4 Consistency and stability of viscosity solutions . . . . . . . . . . . . . . . . . . . 18

5 The comparison principle and uniqueness result for static problem . . . . . . 19

6 The comparison principle and uniqueness result for Cauchy problem . . . . . 23

7 Introduction to the classical Bernstein method . . . . . . . . . . . . . . . . . . . 27

8 Introduction to Perron’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9 Lipschitz estimates for Cauchy problems using Perron’s method . . . . . . . . 34

10 Finite speed of propagation for Cauchy problems . . . . . . . . . . . . . . . . . 36

11 Rate of convergence of the vanishing viscosity process for static problems via

the doubling variables method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

12 Rate of convergence of the vanishing viscosity process for static problems via

the nonlinear adjoint method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2 First-order Hamilton–Jacobi equations with convex Hamiltonians

49

1 Introduction to the optimal control theory . . . . . . . . . . . . . . . . . . . . . . 49

2 Dynamic Programming Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Static Hamilton–Jacobi equation for the value function . . . . . . . . . . . . . . 55

4 Legendre’s transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 The optimal control formula from the Lagrangian viewpoint . . . . . . . . . . 60

6 A further hidden structure of convex ﬁrst-order Hamilton–Jacobi equations . 66

7 Maximal subsolutions and their representation formulas . . . . . . . . . . . . . 72

8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Periodic homogenization theory for Hamilton–Jacobi equations

83

1 Introduction to periodic homogenization theory . . . . . . . . . . . . . . . . . . 83

2 Cell problems and periodic homogenization of static Hamilton–Jacobi equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3 Periodic homogenization for Cauchy problems . . . . . . . . . . . . . . . . . . . 90

4 Some ﬁrst properties of the effective Hamiltonian . . . . . . . . . . . . . . . . . 94

2

5 Further properties of the effective Hamiltonian in the convex setting . . . . . 98 6 Some representation formulas of the effective Hamiltonian in nonconvex set-

tings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7 Rates of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8 Nonuniqueness of solutions to the cell problems . . . . . . . . . . . . . . . . . . 124 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4 Almost periodic homogenization theory for Hamilton–Jacobi equations

129

1 Introduction to almost periodic homogenization theory . . . . . . . . . . . . . 129

2 Vanishing discount problems and identiﬁcation of the effective Hamiltonian . 131

3 Nonexistence of sublinear correctors . . . . . . . . . . . . . . . . . . . . . . . . . 133

4 Homogenization for Cauchy problems . . . . . . . . . . . . . . . . . . . . . . . . 135

5 Properties of the effective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . 137

6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5 First-order convex Hamilton–Jacobi equations in a torus

141

1 New representation formulas for solutions of the discount problems . . . . . . 141

2 New representation formula for the effective Hamiltonian and applications . 146

3 Cell problems, backward characteristics, and applications . . . . . . . . . . . . 151

4 Optimal rate of convergence in periodic homogenization theory . . . . . . . . 156

5 Equivalent characterizations of Lipschitz viscosity subsolutions . . . . . . . . . 160

6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6 Introduction to weak KAM theory

165

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

2 Lagrangian methods in weak KAM theory . . . . . . . . . . . . . . . . . . . . . . 166

3 Mather measures and Mather set . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4 Nonlinear PDE methods in weak KAM theory . . . . . . . . . . . . . . . . . . . . 179

5 The projected Aubry set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7 Further properties of the effective Hamiltonians in the convex setting

201

1 Strict convexity of the effective Hamiltonian in certain directions . . . . . . . 201

2 Asymptotic expansion at inﬁnity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

3 The classical Hedlund example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4 A generalization of the classical Hedlund example . . . . . . . . . . . . . . . . . 209

5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Appendix

213

1 Sion’s minimax theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

2 Existence and regularity of minimizers for action functionals . . . . . . . . . . 215

3 Characterization of the Legendre transform . . . . . . . . . . . . . . . . . . . . . 219

4 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

5 Sup-convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Solutions to some exercises

237

3

Bibliography

253

4

Preface

This is a second draft of my book of viscosity solutions and applications written in 2019. In this book, I intend to cover ﬁrst the basic well-posedness theory of viscosity solutions for ﬁrst-order equations. This is, by now, quite standard and there have been quite some great books on this matter since 1980s. Nevertheless, it is important to have some key topics covered here in a self-contained way for the use throughout the book. It is not of our intention here to cover extensively about well-posedness of viscosity solutions for various different kinds of PDEs.

Then, I aim at discussing in deep the homogenization theory for Hamilton–Jacobi equations. Although this has always been a very active research topic since 1980s until this moment (2019), there has not been any standard textbook covering this. I am hopeful that this book will serve as a gentle introductory reference on this subject. Various connections between homogenization and other research subjects are discussed as well.

Afterwards, dynamical properties, Aubry–Mather theory, and weak Kolmogorov–Arnold– Moser (KAM) theory are studied. These appear naturally in the study of ﬁrst-order Hamilton– Jacobi equations when the Hamiltonian is convex in the momentum variable. I will introduce both dynamical and PDE approaches to study these theories. Then, I will discuss connections between homogenization and dynamical system, and optimal rate of convergence in homogenization theory as well.

Let me emphasize that this is a textbook, not a research monograph. My hope is that it can be used by advanced undergraduate students, ﬁrst and second year graduate students, and new researchers entering the ﬁelds of Hamilton–Jacobi equations and viscosity solutions as a learning tool. In this case, the readers can follow the ﬂow of the book from the beginning (Chapters 1 and 2), then jump to the topics that the readers aim at. Besides, I intend to keep the contents of various topics covered here as independent as possible so that other interested readers are able to jump directly to a subject of interests in the book.

My intention here when writing this book is to present the essential ideas in the clearest possible ways, and thus, in various places, the assumptions/conditions imposed are not sharp. In many cases, the readers can improve the assumptions/conditions imposed right away. I will refer to a list of research articles and monographs at the end of each chapter that provide more general pictures of the situations.

The homework problems given in this book are of various level of difﬁculties. Most of the times, the exercises in corresponding sections are helpful for further understandings

5

of relevant methods, ideas and techniques. Few of the problems are open ended and are related to some active research directions. I would like to thank my Ph.D. student, Son Tu, who provided me the ﬁrst draft of some of these notes based on a graduate topic course (Math 821) that I taught in Fall 2016 at UW Madison. Solutions to some problems were provided by him as well. I have been sitting on the notes for a long time before putting some real effort to have this second draft. Besides, I have also used some parts of my lecture notes taught at a topic course at University of Tokyo, Tokyo, Japan (September 2014), two topic courses at University of Science, Ho Chi Minh city, Vietnam (July 2015, July 2017) to form parts of this book. I would like to thank Professors Yoshikazu Giga, Hiroyoshi Mitake (University of Tokyo), Huynh Quang Vu (University of Science, Ho Chi Minh city) for their hospitalities. I would like to thank my wife, Van Hai Van, and my daughter, An My Ngoc Tran, for their constant wonderful supports during the writing of this book. Besides, I am extremely grateful for the friendships and the supports from Wenjia Jing, Hiroyoshi Mitake, Yifeng Yu. In this second draft, Nam Le suggested me to include the characterization of the Legendre transform. I thank Nam for this very useful suggestion. I appreciate Yeon-Eung Kim, Yuchen Mao, Son Tu, Son Van, Lizhe Wan for pointing out various typos and giving some good suggestions in this draft. I am supported in part by NSF grant DMS-1664424 and NSF CAREER grant DMS-1843320 during the writing of this second draft.

6

CHAPTER 1

Introduction to viscosity solutions for Hamilton–Jacobi equations

1 Introduction

Basic notions. Let u : n → following.

be a smooth function. We have some basic notions as

• Du(x) = ∇u(x) =

∂u ∂x

( x ),

.

.

.

,

∂u ∂x

(x)

.

1

n

• D2u(x) = Hessian of u at x =

∂

2

u

2

(

x

)

∂ x1

...

∂ x∂12∂ux2 (x ) . . . ∂ x∂12∂uxn (x ) ... ... ... .

∂ x∂n2∂ux1 ( x ) ∂ x∂n2∂ux2 ( x ) . . .

∂

2

u

2

(

x

)

∂ xn

• The Laplacian ∆u(x) = tr(D2u(x)) =

n

∂

2

u

2

(

x

)

is

the

trace

of

D2u( x ).

i=1 ∂ xi

For u : n × [0, ∞) → smooth, we write

• Du(x, t) = Dx u(x, t) and ut (x, t) = ∂∂ ut (x, t). • D2u(x, t) = Dx2u(x, t), and ∆u(x, t) = ∆x u(x, t).

The following equations are of interests.

Cauchy problem. We consider the initial value problem

ut(x, t) + F (x, Du(x, t), D2u(x, t)) = 0

in n × (0, ∞),

u(x, 0)

= u0(x) on n, (C)

where u : n × [0, ∞) → is the unknown. Here, the initial data u0 is given.

7

Static (Stationary) problem. Given λ ≥ 0, we consider the equation:

λu + F (x, Du, D2u) = 0 in n.

(Sλ)

Here u : n → is the unknown. In both problems, F : n × n × n → is a given function, where n is the set of all symmetric matrices of size n. These problems come from a lot of

sources such as

• Hamilton–Jacobi equations (classical mechanics, n-body problems);

• Optimal control theory;

• Differential games (two players zero-sum differential games);

• Front propagation (level set method).

Next, we present few examples that lead to either a Cauchy problem or a static problem.

Example 1.1 (First-order front propagation). Consider a surface Γt ⊂ n under the law of motions at time t > 0 with the initial proﬁle Γ0. The goal is to study how {Γt}t≥0 evolves.

• The simplest example is Γ0 is the unit sphere, and every point is moving inward with constant (vector) speed 1, then Γt is remain a sphere for t ∈ [0, 1), and eventually shrinks into a point at t = 1, located at the center.

• If each point on the surface Γt is moving with variable velocity, then the situation becomes more complicated. Osher, Sethian [102] introduced the level set method (numerically) to study this problem. The rigorous treatment was developed later by Evans, Spruck [44] and Chen, Giga, Goto [25], independently.

Magically, we assume that Γt is the 0-level set of some function u(x, t), that is, Γt = {x ∈ n : u(x, t) = 0} .

We set u(x, t) > 0 in the region enclosed by Γt and u(x, t) < 0 elsewhere. Assume u and Γt are smooth, and the given velocity at x ∈ Γt is V (x) = a(x)n(x), where n(x) is the inward normal vector to Γt at x. Let us then try to ﬁnd a PDE for u(x, t) based on this given law of motions.

Figure 1.1: Front propagation of {Γt}t≥0.

For a particle x(0) ∈ Γ0, we keep track with its position x(t) ∈ Γt for t ≥ 0 under this front propagation problem. First of all, we have

x (t) = a(x(t)) n(x(t)) = a(x(t)) Du(x(t), t) . |Du(x(t), t)|

8

Moreover, in light of the fact that u(x(t), t) = 0,

d u(x(t), t) = ut (x(t), t) + Du(x(t), t) · x (t) = 0,

dt

which implies Thus, we obtain a PDE

ut (x(t), t) + a(x(t)) |Du(x(t), t)| = 0. ut + a(x)|Du| = 0 in n × (0, ∞),

which is a ﬁrst-order Hamilton–Jacobi equation.

Example 1.2 (Level set mean curvature ﬂow). Let κ(x) be the mean curvature at x ∈ Γt of the surface Γt. For example, if Γt is a sphere of radius R(t), then for x ∈ Γt, κ(x) = Rn−(t1) . Again, we assume that Γt is the 0-level set of some function u(x, t), that is,

Γt = {x ∈ n : u(x, t) = 0} .

Set u(x, t) > 0 in the region enclosed by Γt and u(x, t) < 0 elsewhere. Assume u and Γt are smooth, and the given velocity at x ∈ Γt is V (x) = κ(x)n(x), where n(x) is the inward normal vector to Γt at x. As above, for a particle x(0) ∈ Γ0, we keep track with its position x(t) ∈ Γt for t ≥ 0 under this mean curvature ﬂow motion. It is clear that

ut(x(t), t) + Du(x(t), t) · x (t) = 0,

where

x (t) = κ(x(t))n(x(t)) = −div Du(x(t), t) |Du(x(t), t)|

Thus the level set mean curvature ﬂow equation of interest is

Du(x(t), t) |Du(x(t), t)| .

Du ut = |Du|div

|Du|

in n × (0, ∞).

Of course, the Cauchy problem (C) is a general form of both above examples. From the PDE viewpoints, we focus on the following main issues

1. Well-posedness theory: Existence, uniqueness and stability of solutions;

2. Study ﬁne properties of solutions such as large time behavior, homogenization, dynamical properties.

Example 1.3 (one dimensional eikonal equation).

|u (x)| = 1 u(−1) = u(1) = 0.

in (−1, 1),

It is not hard to see that there are inﬁnitely many almost everywhere solutions to this equation. To design such a solution, one just need to draw its graph which is zero at the two endpoints ±1, and always has slope ±1 in between. Here are some simple but important observations.

9

1. This eikonal equation has no classical solution (C1 solution).

2. If u is an a.e. solution, then so is −u. In a sense, if we want to select only one solution (well-posedness goal), then we have to breakdown the symmetry. Besides, we might need to be careful with stability then.

3. Clearly, we need to impose a bit more in order to get less solutions. This is typically the case in the theories of viscosity solution, renormalized solutions, etc.

2 Vanishing viscosity method for ﬁrst-order Hamilton–Jacobi equations

Let us look at the following simple Cauchy problem for Hamilton–Jacobi equation

ut + H(Du) = 0

u(x, 0)

= u0(x)

in n × (0, ∞), on n,

(1.1)

where H : n → is the given Hamiltonian, and u0 is the given initial data. Assume that H and u0 are smooth enough. One way to study the solution of (1.1) is using the idea of vanishing viscosity procedure. For each > 0, we consider

ut + H(Du ) = ∆u

u (x, 0)

= u0(x)

in n × (0, ∞), on n.

(1.2)

Under some appropriate assumptions on H and u0, (1.2) is a parabolic equation, which has a unique smooth solution u . The question is what happens as → 0. Do we have u → u for some function u and in some sense? If it is the case, do we have that u solves (1.1) in some sense? This is the idea of a selection principle, which often appears when one introduces some approximation processes to a nonlinear PDE.

Evans [36] ﬁrst showed that this procedure leads to u → u locally uniformly on n×[0, ∞), and u solves (1.1) in the viscosity sense, which will be deﬁned later. Later on, Crandall and Lions [32] proved the uniqueness of viscosity solutions to (1.1), thus, established the ﬁrm foundation for the theory of viscosity solutions to ﬁrst-order equations. Roughly speaking, the procedure is carried out as following.

• Equation (1.2) is a parabolic equation, and thus, it has maximum principle;

• Hamiltonian H(p) is nonlinear in p in general (e.g., H(p) = |p|2), so there is no way to use integration by parts technique to deﬁne weak solutions;

• There is a priori estimate for {u } >0: There exists a constant C > 0 independent of such that ut L∞( n×[0,∞)) + Du L∞( n×[0,∞)) ≤ C .

We will supply a proof of this later. Thus, {u (x, t)} >0 is equi-continuous and thus by the Arzelà-Ascoli theorem, there exists j 0 such that u j → u locally uniformly on

n × [0, ∞). We hence hope that u solves (1.1) naturally in some sense that ﬁts well

with the context of maximum principle.

10

SOLUTIONS AND APPLICATIONS

HUNG VINH TRAN Department of Mathematics University of Wisconsin Madison Van Vleck hall, 480 Lincoln Drive, Madison, WI 53706, USA

Second draft, Summer 2019

1

Contents

1 Introduction to viscosity solutions for Hamilton–Jacobi equations

7

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Vanishing viscosity method for ﬁrst-order Hamilton–Jacobi equations . . . . . 10

3 Existence of viscosity solutions via the vanishing viscosity method . . . . . . . 16

4 Consistency and stability of viscosity solutions . . . . . . . . . . . . . . . . . . . 18

5 The comparison principle and uniqueness result for static problem . . . . . . 19

6 The comparison principle and uniqueness result for Cauchy problem . . . . . 23

7 Introduction to the classical Bernstein method . . . . . . . . . . . . . . . . . . . 27

8 Introduction to Perron’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9 Lipschitz estimates for Cauchy problems using Perron’s method . . . . . . . . 34

10 Finite speed of propagation for Cauchy problems . . . . . . . . . . . . . . . . . 36

11 Rate of convergence of the vanishing viscosity process for static problems via

the doubling variables method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

12 Rate of convergence of the vanishing viscosity process for static problems via

the nonlinear adjoint method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2 First-order Hamilton–Jacobi equations with convex Hamiltonians

49

1 Introduction to the optimal control theory . . . . . . . . . . . . . . . . . . . . . . 49

2 Dynamic Programming Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Static Hamilton–Jacobi equation for the value function . . . . . . . . . . . . . . 55

4 Legendre’s transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 The optimal control formula from the Lagrangian viewpoint . . . . . . . . . . 60

6 A further hidden structure of convex ﬁrst-order Hamilton–Jacobi equations . 66

7 Maximal subsolutions and their representation formulas . . . . . . . . . . . . . 72

8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Periodic homogenization theory for Hamilton–Jacobi equations

83

1 Introduction to periodic homogenization theory . . . . . . . . . . . . . . . . . . 83

2 Cell problems and periodic homogenization of static Hamilton–Jacobi equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3 Periodic homogenization for Cauchy problems . . . . . . . . . . . . . . . . . . . 90

4 Some ﬁrst properties of the effective Hamiltonian . . . . . . . . . . . . . . . . . 94

2

5 Further properties of the effective Hamiltonian in the convex setting . . . . . 98 6 Some representation formulas of the effective Hamiltonian in nonconvex set-

tings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7 Rates of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8 Nonuniqueness of solutions to the cell problems . . . . . . . . . . . . . . . . . . 124 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4 Almost periodic homogenization theory for Hamilton–Jacobi equations

129

1 Introduction to almost periodic homogenization theory . . . . . . . . . . . . . 129

2 Vanishing discount problems and identiﬁcation of the effective Hamiltonian . 131

3 Nonexistence of sublinear correctors . . . . . . . . . . . . . . . . . . . . . . . . . 133

4 Homogenization for Cauchy problems . . . . . . . . . . . . . . . . . . . . . . . . 135

5 Properties of the effective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . 137

6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5 First-order convex Hamilton–Jacobi equations in a torus

141

1 New representation formulas for solutions of the discount problems . . . . . . 141

2 New representation formula for the effective Hamiltonian and applications . 146

3 Cell problems, backward characteristics, and applications . . . . . . . . . . . . 151

4 Optimal rate of convergence in periodic homogenization theory . . . . . . . . 156

5 Equivalent characterizations of Lipschitz viscosity subsolutions . . . . . . . . . 160

6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6 Introduction to weak KAM theory

165

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

2 Lagrangian methods in weak KAM theory . . . . . . . . . . . . . . . . . . . . . . 166

3 Mather measures and Mather set . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4 Nonlinear PDE methods in weak KAM theory . . . . . . . . . . . . . . . . . . . . 179

5 The projected Aubry set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7 Further properties of the effective Hamiltonians in the convex setting

201

1 Strict convexity of the effective Hamiltonian in certain directions . . . . . . . 201

2 Asymptotic expansion at inﬁnity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

3 The classical Hedlund example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4 A generalization of the classical Hedlund example . . . . . . . . . . . . . . . . . 209

5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Appendix

213

1 Sion’s minimax theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

2 Existence and regularity of minimizers for action functionals . . . . . . . . . . 215

3 Characterization of the Legendre transform . . . . . . . . . . . . . . . . . . . . . 219

4 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

5 Sup-convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Solutions to some exercises

237

3

Bibliography

253

4

Preface

This is a second draft of my book of viscosity solutions and applications written in 2019. In this book, I intend to cover ﬁrst the basic well-posedness theory of viscosity solutions for ﬁrst-order equations. This is, by now, quite standard and there have been quite some great books on this matter since 1980s. Nevertheless, it is important to have some key topics covered here in a self-contained way for the use throughout the book. It is not of our intention here to cover extensively about well-posedness of viscosity solutions for various different kinds of PDEs.

Then, I aim at discussing in deep the homogenization theory for Hamilton–Jacobi equations. Although this has always been a very active research topic since 1980s until this moment (2019), there has not been any standard textbook covering this. I am hopeful that this book will serve as a gentle introductory reference on this subject. Various connections between homogenization and other research subjects are discussed as well.

Afterwards, dynamical properties, Aubry–Mather theory, and weak Kolmogorov–Arnold– Moser (KAM) theory are studied. These appear naturally in the study of ﬁrst-order Hamilton– Jacobi equations when the Hamiltonian is convex in the momentum variable. I will introduce both dynamical and PDE approaches to study these theories. Then, I will discuss connections between homogenization and dynamical system, and optimal rate of convergence in homogenization theory as well.

Let me emphasize that this is a textbook, not a research monograph. My hope is that it can be used by advanced undergraduate students, ﬁrst and second year graduate students, and new researchers entering the ﬁelds of Hamilton–Jacobi equations and viscosity solutions as a learning tool. In this case, the readers can follow the ﬂow of the book from the beginning (Chapters 1 and 2), then jump to the topics that the readers aim at. Besides, I intend to keep the contents of various topics covered here as independent as possible so that other interested readers are able to jump directly to a subject of interests in the book.

My intention here when writing this book is to present the essential ideas in the clearest possible ways, and thus, in various places, the assumptions/conditions imposed are not sharp. In many cases, the readers can improve the assumptions/conditions imposed right away. I will refer to a list of research articles and monographs at the end of each chapter that provide more general pictures of the situations.

The homework problems given in this book are of various level of difﬁculties. Most of the times, the exercises in corresponding sections are helpful for further understandings

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of relevant methods, ideas and techniques. Few of the problems are open ended and are related to some active research directions. I would like to thank my Ph.D. student, Son Tu, who provided me the ﬁrst draft of some of these notes based on a graduate topic course (Math 821) that I taught in Fall 2016 at UW Madison. Solutions to some problems were provided by him as well. I have been sitting on the notes for a long time before putting some real effort to have this second draft. Besides, I have also used some parts of my lecture notes taught at a topic course at University of Tokyo, Tokyo, Japan (September 2014), two topic courses at University of Science, Ho Chi Minh city, Vietnam (July 2015, July 2017) to form parts of this book. I would like to thank Professors Yoshikazu Giga, Hiroyoshi Mitake (University of Tokyo), Huynh Quang Vu (University of Science, Ho Chi Minh city) for their hospitalities. I would like to thank my wife, Van Hai Van, and my daughter, An My Ngoc Tran, for their constant wonderful supports during the writing of this book. Besides, I am extremely grateful for the friendships and the supports from Wenjia Jing, Hiroyoshi Mitake, Yifeng Yu. In this second draft, Nam Le suggested me to include the characterization of the Legendre transform. I thank Nam for this very useful suggestion. I appreciate Yeon-Eung Kim, Yuchen Mao, Son Tu, Son Van, Lizhe Wan for pointing out various typos and giving some good suggestions in this draft. I am supported in part by NSF grant DMS-1664424 and NSF CAREER grant DMS-1843320 during the writing of this second draft.

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CHAPTER 1

Introduction to viscosity solutions for Hamilton–Jacobi equations

1 Introduction

Basic notions. Let u : n → following.

be a smooth function. We have some basic notions as

• Du(x) = ∇u(x) =

∂u ∂x

( x ),

.

.

.

,

∂u ∂x

(x)

.

1

n

• D2u(x) = Hessian of u at x =

∂

2

u

2

(

x

)

∂ x1

...

∂ x∂12∂ux2 (x ) . . . ∂ x∂12∂uxn (x ) ... ... ... .

∂ x∂n2∂ux1 ( x ) ∂ x∂n2∂ux2 ( x ) . . .

∂

2

u

2

(

x

)

∂ xn

• The Laplacian ∆u(x) = tr(D2u(x)) =

n

∂

2

u

2

(

x

)

is

the

trace

of

D2u( x ).

i=1 ∂ xi

For u : n × [0, ∞) → smooth, we write

• Du(x, t) = Dx u(x, t) and ut (x, t) = ∂∂ ut (x, t). • D2u(x, t) = Dx2u(x, t), and ∆u(x, t) = ∆x u(x, t).

The following equations are of interests.

Cauchy problem. We consider the initial value problem

ut(x, t) + F (x, Du(x, t), D2u(x, t)) = 0

in n × (0, ∞),

u(x, 0)

= u0(x) on n, (C)

where u : n × [0, ∞) → is the unknown. Here, the initial data u0 is given.

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Static (Stationary) problem. Given λ ≥ 0, we consider the equation:

λu + F (x, Du, D2u) = 0 in n.

(Sλ)

Here u : n → is the unknown. In both problems, F : n × n × n → is a given function, where n is the set of all symmetric matrices of size n. These problems come from a lot of

sources such as

• Hamilton–Jacobi equations (classical mechanics, n-body problems);

• Optimal control theory;

• Differential games (two players zero-sum differential games);

• Front propagation (level set method).

Next, we present few examples that lead to either a Cauchy problem or a static problem.

Example 1.1 (First-order front propagation). Consider a surface Γt ⊂ n under the law of motions at time t > 0 with the initial proﬁle Γ0. The goal is to study how {Γt}t≥0 evolves.

• The simplest example is Γ0 is the unit sphere, and every point is moving inward with constant (vector) speed 1, then Γt is remain a sphere for t ∈ [0, 1), and eventually shrinks into a point at t = 1, located at the center.

• If each point on the surface Γt is moving with variable velocity, then the situation becomes more complicated. Osher, Sethian [102] introduced the level set method (numerically) to study this problem. The rigorous treatment was developed later by Evans, Spruck [44] and Chen, Giga, Goto [25], independently.

Magically, we assume that Γt is the 0-level set of some function u(x, t), that is, Γt = {x ∈ n : u(x, t) = 0} .

We set u(x, t) > 0 in the region enclosed by Γt and u(x, t) < 0 elsewhere. Assume u and Γt are smooth, and the given velocity at x ∈ Γt is V (x) = a(x)n(x), where n(x) is the inward normal vector to Γt at x. Let us then try to ﬁnd a PDE for u(x, t) based on this given law of motions.

Figure 1.1: Front propagation of {Γt}t≥0.

For a particle x(0) ∈ Γ0, we keep track with its position x(t) ∈ Γt for t ≥ 0 under this front propagation problem. First of all, we have

x (t) = a(x(t)) n(x(t)) = a(x(t)) Du(x(t), t) . |Du(x(t), t)|

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Moreover, in light of the fact that u(x(t), t) = 0,

d u(x(t), t) = ut (x(t), t) + Du(x(t), t) · x (t) = 0,

dt

which implies Thus, we obtain a PDE

ut (x(t), t) + a(x(t)) |Du(x(t), t)| = 0. ut + a(x)|Du| = 0 in n × (0, ∞),

which is a ﬁrst-order Hamilton–Jacobi equation.

Example 1.2 (Level set mean curvature ﬂow). Let κ(x) be the mean curvature at x ∈ Γt of the surface Γt. For example, if Γt is a sphere of radius R(t), then for x ∈ Γt, κ(x) = Rn−(t1) . Again, we assume that Γt is the 0-level set of some function u(x, t), that is,

Γt = {x ∈ n : u(x, t) = 0} .

Set u(x, t) > 0 in the region enclosed by Γt and u(x, t) < 0 elsewhere. Assume u and Γt are smooth, and the given velocity at x ∈ Γt is V (x) = κ(x)n(x), where n(x) is the inward normal vector to Γt at x. As above, for a particle x(0) ∈ Γ0, we keep track with its position x(t) ∈ Γt for t ≥ 0 under this mean curvature ﬂow motion. It is clear that

ut(x(t), t) + Du(x(t), t) · x (t) = 0,

where

x (t) = κ(x(t))n(x(t)) = −div Du(x(t), t) |Du(x(t), t)|

Thus the level set mean curvature ﬂow equation of interest is

Du(x(t), t) |Du(x(t), t)| .

Du ut = |Du|div

|Du|

in n × (0, ∞).

Of course, the Cauchy problem (C) is a general form of both above examples. From the PDE viewpoints, we focus on the following main issues

1. Well-posedness theory: Existence, uniqueness and stability of solutions;

2. Study ﬁne properties of solutions such as large time behavior, homogenization, dynamical properties.

Example 1.3 (one dimensional eikonal equation).

|u (x)| = 1 u(−1) = u(1) = 0.

in (−1, 1),

It is not hard to see that there are inﬁnitely many almost everywhere solutions to this equation. To design such a solution, one just need to draw its graph which is zero at the two endpoints ±1, and always has slope ±1 in between. Here are some simple but important observations.

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1. This eikonal equation has no classical solution (C1 solution).

2. If u is an a.e. solution, then so is −u. In a sense, if we want to select only one solution (well-posedness goal), then we have to breakdown the symmetry. Besides, we might need to be careful with stability then.

3. Clearly, we need to impose a bit more in order to get less solutions. This is typically the case in the theories of viscosity solution, renormalized solutions, etc.

2 Vanishing viscosity method for ﬁrst-order Hamilton–Jacobi equations

Let us look at the following simple Cauchy problem for Hamilton–Jacobi equation

ut + H(Du) = 0

u(x, 0)

= u0(x)

in n × (0, ∞), on n,

(1.1)

where H : n → is the given Hamiltonian, and u0 is the given initial data. Assume that H and u0 are smooth enough. One way to study the solution of (1.1) is using the idea of vanishing viscosity procedure. For each > 0, we consider

ut + H(Du ) = ∆u

u (x, 0)

= u0(x)

in n × (0, ∞), on n.

(1.2)

Under some appropriate assumptions on H and u0, (1.2) is a parabolic equation, which has a unique smooth solution u . The question is what happens as → 0. Do we have u → u for some function u and in some sense? If it is the case, do we have that u solves (1.1) in some sense? This is the idea of a selection principle, which often appears when one introduces some approximation processes to a nonlinear PDE.

Evans [36] ﬁrst showed that this procedure leads to u → u locally uniformly on n×[0, ∞), and u solves (1.1) in the viscosity sense, which will be deﬁned later. Later on, Crandall and Lions [32] proved the uniqueness of viscosity solutions to (1.1), thus, established the ﬁrm foundation for the theory of viscosity solutions to ﬁrst-order equations. Roughly speaking, the procedure is carried out as following.

• Equation (1.2) is a parabolic equation, and thus, it has maximum principle;

• Hamiltonian H(p) is nonlinear in p in general (e.g., H(p) = |p|2), so there is no way to use integration by parts technique to deﬁne weak solutions;

• There is a priori estimate for {u } >0: There exists a constant C > 0 independent of such that ut L∞( n×[0,∞)) + Du L∞( n×[0,∞)) ≤ C .

We will supply a proof of this later. Thus, {u (x, t)} >0 is equi-continuous and thus by the Arzelà-Ascoli theorem, there exists j 0 such that u j → u locally uniformly on

n × [0, ∞). We hence hope that u solves (1.1) naturally in some sense that ﬁts well

with the context of maximum principle.

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