# Reference request to have a deep understanding of solutions to PDE

I come from an engineering background that heavily involves modelling in many fields especially data science and control theory. I have dealt with PDEs many times while modelling data features such as in control systems in analyzing natural phenomena around the system. Many of these PDEs I have encountered are elliptical by class (such as the Laplacian) and while it all ends by just finding the solution to this PDE, I am looking to learn more on how to apply "hardcore analysis" on the solution obtained in solving PDEs to have a better understanding of what I have and what special properties this solution belonging to this type of class has.

I have specifically highlighted the word solution because a quick search on this topic gave me a whole different meaning about this term which appears to be generalized to spaces that allows non-smooth solutions. In fact I have yet to see a natural phenomena modelled to have solution that is "smooth".

I wish to know "what key ideas need to be known" to have a rigorous understanding in this topic as I am not looking on functional analysis on one side and PDEs on the other side. I am tending to look for the intersection between these two. Furthermore, what references are recommended? I have a good knowledge in advanced calculus and measure theory.

I firmly believe that performing analysis on solutions to PDEs using advanced mathematical concepts can improve my perspective on modelling.

• Maybe you can start by reading a book on Sobolev spaces Sep 29 '21 at 15:38
• Not trying to be snarky, but Ive often been amazed at the depth and simplicity (and learning speed) of videos, eg youtube. Good luck Oct 1 '21 at 21:32

A few good starting points where you can't go wrong:

• Yu. V. Egorov and M. A. Shubin: Partial Differential Equations I.
• Sergei L. Sobolev: Partial Differential Equations of Mathematical Physics.

And an MSE favorite seems to be:

• Lawrence C. Evans: Partial Differential Equations.

These'll help you understand classes of partial differential equations, ideas of well-posedness, function spaces and weak solutions, etc.

• In my biased opinion, the book by Evans is one of the best math books ever written. Go through the chapters slowly and do the exercises at the end. It helps to have $5-10$ similar students who can do the exercises (so that you can compare answers). Oct 1 '21 at 19:48
• @axion004: It’s a matter of taste and personal preparation, but I concur with you completely on the quality of Evans’ book. Oct 1 '21 at 19:52

Since you have a good background in measure theory and advanced calculus, I would start by reading the classical textbook

1. Partial Differential Equations by Lawrence Evans.

and then (if you are interested), read more advanced material on PDE theory

1. Elliptic Partial Differential Equations by Han and Lin.
2. Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger. This is aimed at researchers in partial differential equations and covers more advanced topics.

In combination with these you should also have a good understanding of functional analysis, Sobolev spaces, and common techniques used in applied mathematics. Some books on these include

1. Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis.
2. Methods of Applied Mathematics by Todd Arbogast and Jerry Bona. This is tailored towards students in science in engineering and is freely available through Todd's website.
3. Sobolev Spaces by Adams and Fournier.

Evans PDE book is a comprehensive survey of modern techniques in the theoretical study of PDE. The book covers the following material

Chapter 1: This chapter surveys the principal theoretical issues concerning the solving of partial differential equations. Evans introduces some common notation (such as partial differentiation) and discusses examples of commonly studied partial differential equations. He then discusses strategies of studying different PDEs and introduces functional analysis as a method of studying existence and uniqueness results for partial differential equations.

Chapter 2: Evans discusses the four fundamental linear partial differential equations which have explicit solutions. These are

$$\text{The transport equation: }\quad u_t + b\cdot Du=0$$ $$\text{The Laplace equation: }\quad \Delta u=0$$ $$\text{The heat equation: }\quad u_t - \Delta u=0$$ $$\text{The wave equation: }\quad u_{tt}-\Delta u=0$$

Both inhomogenuous and homogenuous solutions are discussed and Evans emphasizes the importance of finding explicit solutions through inequalities, integration by parts, Green's formula, convolution, energy methods, maximum principles, etc. (which are the backbone of finding explicit solutions). This chapter is very useful in that it is one of the rare occasions where one can find explicit formulas for PDEs.

Chapter 3: This chapter focuses on investigating nonlinear first-order partial differential equations of the form

$$F(Du,u,x)=0,$$ where $$x\in U$$ and $$U$$ is an open subset of $$\mathbb R^n$$. Here $$F:\mathbb R^n \times \mathbb R\times \overline{U}\to\mathbb R$$ is given, where $$u:\overline{U}\to\mathbb R$$ is the unknown and $$u=u(x).$$ A subsection discussing the method of characteristics is discussed (however, I prefer the introduction in the PDE book by Robert McOwen). The later sections give a good introduction to Hamilton-Jacobi equations, conservation laws, and representation formulas for weak solutions.

Chapter 4: This chapter is a standalone chapter that contains a wide variety of techniques that are sometimes useful for finding explicit solutions to partial differential equations. Evans goes through detailed examples using separation of variables, similarity solutions, transformational methods, converting nonlinear PDE to linear PDE, and both asymptotics and power series. It is nice that Evans shows how to solve some nonlinear PDEs with techniques from linear PDEs. This chapter could be skipped on a first read-through.

Chapter 5: In my opinion, this is one of the most important chapters in the book. Evans dives into the more theoretical aspects of linear partial differential equations. He starts by introducing both Hölder and Sobolev spaces and then goes through local and global approximation theory through smooth functions (through the use of mollifiers). Next, he discussions the extension and trace theorems. In section $$5.6$$, Evans lays out crucial information about the embeddings of Sobolev spaces into others (what is known as Sobolev inequalities). He later discusses the Poincaré inequality. I would spend some time reading this chapter slowly (and do the exercises at the end of the chapter).

Chapter 6: This chapter is a generalization of Chapter $$2$$ (and the Laplace equation) with the new tools introduced in Chapter $$5$$. Evans discusses (and motivates) the existence weak solutions by the Lax-Milgram theorem, energy estimates, the Fredholm alternative, and applications of the Rellich-Kondrachov compactness theorem. In section $$6.3$$, Evans presents the regularity problem for weak solutions - Is the weak solution $$u$$ of the PDE

$$Lu=f \quad \text{in U}$$ smooth? He provides the proper motivation for the problem and then discusses the interior regularity theorems (more information about this material is given in the book by Han and Lin.) In section $$6.4$$, he discussions several nice applications of the maximum principle and the difference between the weak and strong maximum principles. The final section is devoted to eigenvalues and eigenfunctions of the Laplacian.

Chapter 7: This chapter is a natural extension of Chapter $$6$$ and covers linear partial differential equations which involve time. These are known as evolution equations where the solution evolves in time from a given initial configuration. Here, Evans gives a nice description of how to apply energy methods to general second-order parabolic and hyperbolic equations. The final subsection gives an outstanding introduction to the abstract theory of semigroups and describes contraction semigroups, the resolvent, and the Hille-Yosida theorem. There is also a nice application of semigroup theory to second-order parabolic PDEs.

Chapter 8: This chapter gives a wonderful introduction to the calculus of variations (you may also want to look at the book by Gelfand and Fomin). These are an important class of nonlinear problems which can be solved through using straightforward techniques from nonlinear functional analysis. Evans discusses the Euler-Lagrange equations, coercivity, lower semicontinuity, and the existence/uniqueness of minimizers. He then motivates the definition of the weak solution and shows how to find weak solutions to the Euler-Lagrange equations. In section $$8.3$$, the regularity theorems are discussed (which follow similarly from the analysis done in chapter $$6$$). The final subsections given an overview of various constraints, incomprehensibility, and critical points. The famous Moutain Pass theorem is discussed in section $$8.5.1$$ which is followed by Noether's theorem in section $$8.6$$.

Chapter 9: In the same spirit as chapter $$4$$, this is a collection of various techniques for proving the existence, nonexistence, uniqueness, and other properties of solutions to nonlinear elliptic and parabolic differential equations. The familiar fixed point theorems are discussed and Evans discusses direct applications of Banach's Fixed Point Theorem. Supersolutions, subsolutions, and the nonexistence of solutions are discussed in the context of nonlinear partial differential equations. The final subsection gives a nice overview of gradient flows as an extension of the abstract semigroup theory developed in section $$7.4$$. As with chapter $$4$$, this chapter could be skipped on a first reading.

Chapter 10: This chapter is devoted to the study of Hamilton-Jacobi equations (which are a modern application of PDE theory to the study of optimal control problems). As usual, Evans gives a good motivation for the definition of a viscosity solution and then discusses the consistency of solutions. Section $$10.2$$ shows uniqueness of viscosity solutions to Halmilton-Jacobi PDEs with initial values. The final section gives a good introduction to dynamic programming and control theory.

Chapter 11: This chapter is an abstract generalization of the later half of chapter $$3$$. Evans discusses systems of nonlinear, divergence structure first-order hyperbolic PDE, which arise as models of conservation laws. He goes through the familiar conservation and mass, momentum, and energy and then later demonstrates how to obtain weak solutions. Properties of traveling waves, rarefaction waves, and shock waves are discussed (with some good pictures modeling the physical parameters). The later sections include theory about how to attack systems of $$2$$ or more conservation laws and information about entropy conditions.

Chapter 12: This is a fun chapter about nonlinear wave equations. Evans discusses the semilinear wave equation, quasilinear wave equation, and how to think about the conservation of energy (in the spirit of the energy functional described in section 8.6.2). A bunch of generalizations of linear wave equations are discussed: existence of solutions, Sobolev inequalities, energy estimates, and nonexistence of solutions.

In my opinion, the book is structured as

Chapters 1-4: These show how to find explicit solutions to linear PDEs. This material is covered in most standard PDE I/II courses. It is useful to understand how to apply Green's formula, integration by parts, inequalities, etc. as you will be doing a lot of this in the later chapters (and while reading PDE papers).

Chapters 5-8: These are the "bread and butter" of the book. Chapter $$5$$ demonstrates how use the Sobolev inequalities and chapter $$8$$ gives a bunch of different applications of the Euler-Lagrange equations. I would read these slowly and do the exercises at the end (I would suspect that these would be a "PDE 2" course in many universities).

Chapters 9-12: These are more advanced topics and Chapter $$10$$ gives a nice overview of some of optimal control theory. I like the presentation in chapter $$12$$ as it generalizes what is discussed in chapters $$3$$ and $$6$$.

As you are interested in the smoothness of solutions, I would review the regularity theorems presented in chapters $$5$$-$$7$$. You could then turn to other sources once you have mastered most of the exercises.

• Thank you very very much for your lengthy answer. I will enjoy reading all the content in it! Oct 5 '21 at 21:40