All Questions
18 questions
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 ...
- 101
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
$$...
- 839
5
votes
0
answers
198
views
Heuristic and graphic representation of BV functions and their singularities
This question is about some heuristics and graphs of BV functions.
In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are
the Heaviside function, whose ...
- 839
4
votes
1
answer
365
views
Lusin Lipschitz approximation in BV and Sobolev space
Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that
Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...
- 839
5
votes
1
answer
220
views
Alberti rank one theorem and a blow-up argument
In this paper, it is written that Alberti’s rank
says that the singular part $D^s u$ with respect to $\mathcal L^d$ of the distributional derivative $Du$ of a function $u \in BV_{loc}(\mathbb R^d; \...
user123457
4
votes
1
answer
129
views
Meaningful generalization of viscosity solutions to higher order equations
Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?
user75426
2
votes
1
answer
713
views
Reference request: numerical analysis of PDEs and integro partial differential equations
I'm very new to the field of numerical analysis of PDE and integro partial differential equations.
My advisor (who is not a specialist in this area) highly recommended to read
Randall J. LeVeque'...
user102027
2
votes
2
answers
304
views
Why is the ellipticity condition important in the theory of viscosity solutions?
A fundamental assumption in User's guide (p. 2) is that the operator $F$ should be proper.
However, the role of this monotonicity condition (and especially of the "ellipticity condition" part) is ...
user102027
5
votes
1
answer
379
views
References on the obstacle problem for the heat equation
Can you point out some references that deal with the obstacle problem for the heat equation?
$$(OP) \quad\begin{cases}
\max\{\Delta u -\partial_t u, \varphi - u \} = 0 & \text{ in } (0,T)\times \...
user108310
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 ...
user60665
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 (...
- 303
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 ...
- 303
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....
user89890
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 ...
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 ...
- 16.6k
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)?
user60665
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 ...
- 11
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 ...
- 1,533