Skip to main content

All Questions

Filter by
Sorted by
Tagged with
7 votes
3 answers
2k views

Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs

What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in real analysis, complex analysis, functional analysis, ODEs, PDEs? The ...
0 votes
0 answers
96 views

Books on limiting properties of matrices with growing size

This question has been posted on Math-Se previously. I am studying asymptotic properties of the Projection Matrix $$ H_n=X'(X'X)^{-1}X $$ By the Gerschgorin disc theorem, the bounds on the ...
4 votes
1 answer
129 views

Meaningful generalization of viscosity solutions to higher order equations

Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?
2 votes
2 answers
317 views

Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative? More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of a function $$...
5 votes
1 answer
220 views

Alberti rank one theorem and a blow-up argument

In this paper, it is written that Alberti’s rank says that the singular part $D^s u$ with respect to $\mathcal L^d$ of the distributional derivative $Du$ of a function $u \in BV_{loc}(\mathbb R^d; \...
5 votes
0 answers
198 views

Heuristic and graphic representation of BV functions and their singularities

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...
4 votes
1 answer
365 views

Lusin Lipschitz approximation in BV and Sobolev space

Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...
12 votes
3 answers
881 views

Bibliographic request concerning an article by Bernstein and Robinson

Concerning the article "Bernstein, Allen R.; Robinson, Abraham. Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in finding ...
2 votes
1 answer
713 views

Reference request: numerical analysis of PDEs and integro partial differential equations

I'm very new to the field of numerical analysis of PDE and integro partial differential equations. My advisor (who is not a specialist in this area) highly recommended to read Randall J. LeVeque'...
1 vote
1 answer
151 views

Global wellposedness of the Cauchy problem for a third order PDE

Consider $$u_t-\alpha u_{xx} - \beta u_{xxx} = f(u_x)$$ with initial condition $u(0,x) = u_0(x)$, where $\alpha>0$, $\beta \in \mathbb{R}$, $u_0 \in C^\infty(\mathbb{R})$, and $f$ is Lipschitz (...
2 votes
2 answers
304 views

Why is the ellipticity condition important in the theory of viscosity solutions?

A fundamental assumption in User's guide (p. 2) is that the operator $F$ should be proper. However, the role of this monotonicity condition (and especially of the "ellipticity condition" part) is ...
5 votes
1 answer
379 views

References on the obstacle problem for the heat equation

Can you point out some references that deal with the obstacle problem for the heat equation? $$(OP) \quad\begin{cases} \max\{\Delta u -\partial_t u, \varphi - u \} = 0 & \text{ in } (0,T)\times \...
10 votes
3 answers
1k views

References: spectral analysis of the Laplacian operator

I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature....
2 votes
0 answers
90 views

Boundary regularity of solutions to semilinear heat equation

Consider the Cauchy IVP problem $$u_t - \Delta u + f(t,x,u) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$ with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$ Can you point out a ...
2 votes
3 answers
978 views

"Must read" papers in functional analysis and PDE (à la Trefethen) [closed]

The following question has been inspired by "Must read" papers in numerical analysis. In 1993, Prof. Trefethen published a NA-net posting with a list of thirteen papers that he had used in ...
3 votes
1 answer
329 views

Survey on functional equations and inequalities

Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)?
1 vote
1 answer
2k views

PhD in operator algebras and non-commutative geometry [closed]

I do not know whether it is a good place to ask this question or not. I want to PhD in operator algebras and non-commutative geometry. What are the best places in the world for that? I want a good ...
4 votes
2 answers
580 views

An analogue of Hilbert-Schmidt theorem for multilinear forms

Let $H$ be a (the) real separable Hilbert space. The Hilbert--Schmidt theorem says that a compact self-adjoint operator $A$ has an eigenfunction expansion. Instead of operator, we can think of a ...