Intereting Posts

Can an observed event in fact be of zero probability?
Every ideal of $K$ has $\leq n$ generators?
Does “triangle” in English exclude degenerate triangles?
Converting from NFA to a regular expression.
Prove that $n\choose0$+$n+1\choose1$+$n+2\choose2$+…+$n+m\choose{m}$=$n+m+1\choose{m}$ combinatorically
Determinant value of a square matrix whose each entry is the g.c.d. of row and column position
Fraction field of $F(f)$ isomorphic to $F(X)/(f)$
Properties of $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$ where $\{a_n\}$ is unbounded, strictly increasing sequence of positive reals
Continuous extension of a real function
Zeta function zeros and analytic continuation
Lie bracket of a semidirect product
How can I show that $ab \sim \gcd (a,b) {\operatorname{lcm} (a,b)}$ for any $a,b \in R \setminus \{0\}$?
How do we approach this summation question?
Is there an elegant bijective proof of $\binom{15}{5}=\binom{14}{6}$?
I want to get good at math, any good book suggestions?

I am now a graduate student in mathematics, and I really want to learn more about PDE. I would say I have a very solid foundation in soft analysis, including functional analysis and harmonic analysis, but I did not receive a good training in hard analysis.

So I do not mind if the book requires a lot functional analysis, but I would like one that provides more intuition and motivation rather than just hard techniques and estimates.

I know Evan’s is the best, and I read most materials from the second part when I was doing a project on theory of distributions. But I cannot say I enjoy his style very much.

- estimating a particular analytic function on a bounded sector.
- A surjective map which is not a submersion
- $ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$
- Second Countability of Euclidean Spaces
- Proof Verification : Prove -(-a)=a using only ordered field axioms
- Integer parts of multiples of irrationals

So can someone introduce a gentler book? Thanks so much!

- To define a measure, is it sufficient to define how to integrate continuous function?
- Archimedean Property and Real Numbers
- When is the image of a null set also null?
- Find the Green's Function and solution of a heat equation on the half line
- Quasi-linear PDE with Cauchy conditions:
- Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
- If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_n $, prove that $\sum \frac{a_n}{b_n}$ diverges
- Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?
- Question about statement of Rank Theorem in Rudin
- Closed subset of compact set is compact

I really like *Functional Analysis, Sobolev Spaces and Partial Differential Equations* by Haim Brezis. It has more of a functional analysis flavor than many other books.

Aside from the standard PDE textbooks for beginner (Evans) and for researcher (GT), I have recently run into a great book, *Linear and Nonlinear Functional Analysis with Applications*, by Philippe G. Ciarlet.

Here I just need to quote the preface of it:

* (Begin quote)* Why write another textbook on functional analysis and its applications, since there are

already many excellent textbooks around?

Apart from the personal pleasure that such an exercise provides to an author, there are

other reasons: One, which perhaps constitutes the main originality of this text, was to assemble in a single volume the most basic theorems of linear and of nonlinear functional analysis; another reason was to simultaneously illustrate the wide applicability of these theorems by treating an abundance of applications.

Applications to linear and nonlinear partial differential equations treated here include

Korn’s inequality and existence theorems in linear elasticity, obstacle problems, the Babuska–Brezzi inf-sup condition, existence theorems for the Stokes and Navier–Stokes equations of ﬂuid mechanics, existence theorems for the von Karman equations of a nonlinearly elastic plate, and John Ball’s existence theorem in nonlinear elasticity. A variety of other applications deals with selected topics from numerical analysis and optimization theory, such as approximation theory, error estimates for polynomial interpolation, numerical linear algebra, basic algorithms of optimization, Newton’s method, or ﬁnite difference methods.

A special effort has been made to enhance the pedagogical appeal of the book. After Chapter 1, which is essentially a review of results from real analysis and the theory of functions that will be used in the text, self-contained and complete proofs of most of the theorems are provided. These include proofs that are not always easy to locate in the literature, or difficult to reconstitute without an extended knowledge of collateral topics; for instance, self-contained proofs are given of the Poincare lemma, of the hypoellipticity of the Laplacian, of the existence theorem for Pfaff systems, or of the fundamental theorem of surface theory.

Numerous ﬁgures and problems (almost 400) have also been included. Historical notes and

original references (at least those that I have been able to trace with a reasonable assurance of veracity) have also been included (mostly as footnotes), so as to provide an idea of the genesis of some important results.

It is my belief that this book contains most of the core topics from functional analysis

that any analyst interested in linear and nonlinear applications should have encountered at least once in his or her life. More speciﬁcally, linear functional analysis and its applications are the subjects of Chapters 2–6, while nonlinear functional analysis and its applications are the subjects of Chapters 7–9. *(End quote)*

Here is the Table of Contents, if you will be a researcher in a field relevant to PDE or applied math, you will like this book and appreciate its way of presentation. At least I already fell in love with this book.

You might check out Garabedian’s *Partial Differential Equations*. One of Evan’s students told me that he thought it was “dated”, and I guess it is a tad on the old side, but it is *very* readable, not too hard on the analysis, and covers a *lot* of territory.

As you said Evan’s *Partial Differential Equations* is a very good book. Gilbarg and Trudinger *Elliptic Partial Differential Equations of Second Order* is a masterpiece of the subject, but it is a very heavy book and sometimes notation is a nightmare (Schauder’s estimates made me cry 🙁 ). I also would recommend *An Introduction to Partial Differential Equations* by Renardy and Rogers: as Evan’s book it covers a lot of topics and there you can find chapters dedicated to the functional analysis that is needed.

Books I barely touched but that you could find interesting are *Partial Differential Equations: An Introduction* by Strauss and *Introduction to Partial Differential Equations with Applications* by Zachmanoglou and Thoe (the latter is a Dover publication and hence it should be the cheapest)

The book by F. Lin and Q. Han on elliptic PDE’s is a great reference, covering a large variety of topics.

- coproducts of structures
- Ring germs of $C^{\infty}$ functions on the real line
- Does a negative number really exist?
- Repeated Factorials and Repeated Square Rooting
- How many of the 9000 four digit integers have four digits that are increasing?
- What are relative open sets?
- Continuity of $L^1$ functions with respect to translation
- Proving that $\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$
- Complex analysis prerequisites
- $\mathbb Z_p^*$ is a group.
- Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?
- Probability that A meets B in a specific time frame
- Why is $\mathbb{Z}, n\ge 3$ not a UFD?
- A category of relations – or two different?
- monic, but not pointwise monic