All Questions
12,935 questions
7
votes
2
answers
567
views
Intuition for Agmon-Douglis-Nirenberg ellipticity
First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.
I am trying to understand the definition of ellipticity of systems due to ...
- 1,429
3
votes
1
answer
493
views
A strange condition of convexity?
During my research, I come across this question.
Let $f \in C^2(\mathbb R, \mathbb R_+^*)$ with $\forall x \in\mathbb R, f'(x) \geq |f''(x)+f(x)|$.
Is it true that $\forall x \in \mathbb R, f''(x) \...
- 4,074
1
vote
0
answers
98
views
criticality for nonlinear wave equations on manifolds
On $\mathbb{R}^{1+n}$, the initial value problem for the homogeneous wave equation
$$ \Box \phi = \partial_t^2 \phi - \Delta_{\mathbb{R}^n} \phi = 0, \\ (\phi, \partial_t \phi)|_{t=0} = (\phi_0, \...
- 914
2
votes
2
answers
328
views
Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
- 1,043
2
votes
1
answer
184
views
Prove if the fractional Laplacian of a function is bounded
Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$.
Here $(-\Delta)^s$ is the ...
- 123
2
votes
1
answer
207
views
Elliptic PDEs in BSDEs and in optimal control
This soft/reference question is related to this MO post of a similar nature.
What are some examples of elliptic PDEs appearing in control and BSDEs?
- 5,405
3
votes
0
answers
87
views
Instances of c-concavity outside of optimal transport?
Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
- 232
2
votes
0
answers
98
views
Question on Cauchy problem on manifolds
Let $(M,g)$ be a globally-hyperbolic manifold, i.e. $M=\mathbb{R}\times\Sigma$ and $g=-\beta^2 \, dt^2+h_{t}$. Furthermore, let $E$ be a vector bundle over $M$ and $\square$ a normally-hyperbolic ...
- 1,171
0
votes
0
answers
67
views
Examples of symmetry-breaking solitons which retain a subgroup symmetry
There are many works on spontaneous symmetry breaking in the Nonlinear Schrödinger equation with asymmetric soliton solutions.
However, all symmetry breaking soliton examples I have seen go from the ...
- 49
2
votes
0
answers
318
views
What are alternative or equivalent definitions of a positive-definite function on a group?
The standard definition of a positive-definite function on a group goes as follows:
Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
- 63
4
votes
1
answer
237
views
Closed-form solution to hyperbolic PDE
Let $A\in C^{\infty}(\mathbb{R}^2)$ be Lebesgue integrable, and $c_1,c_2\in C^{\infty}(\mathbb{R})$ also be Lebesgue integrable. Consider the hyperbolic PDE
$$
\begin{cases}
\partial_{x,y}u & = A\...
- 5,405
2
votes
0
answers
193
views
If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
- 90
1
vote
0
answers
70
views
Pohozaev type obstruction for higher order elliptic operators
I was reading about Pohazaev type obstructions: precisely, I mean the following kind of results. Let $h\in C^1(\mathbb{R}^n)$ and consider the following Dirichlet problem
$$
\begin{cases}
\Delta u + ...
- 87
0
votes
0
answers
80
views
What is the PDE corresponding to this weak formulation?
Consider a flow $(\mu_t)_{t\ge 0}$ such that
every $\mu_t$ is a probability on $\mathbb R_+$;
$\mu_0(dx) = \rho(x) \, dx$ with $\rho$ being a probability density (as nice as possible) on $\mathbb R_+$...
- 1,334
6
votes
0
answers
98
views
Conditions for completely positive maps to act homomorphically across multiple subalgebras
For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
- 61
2
votes
0
answers
45
views
Topology of an orbit space constructed from a Fréchet space under the "local" action of some "smooth" group
Let $G$ be a nontrivial connected compact subgroup of the general linear group $\operatorname{GL}(\mathbb{R}^3)$. For example, we may take $G$ to be $\operatorname{SO}(3)$.
Next, let $\mathcal{S}(\...
- 3,477
3
votes
1
answer
161
views
Equivalent definition for Skorokhod metric
I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$:
$$
d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
- 177
10
votes
0
answers
422
views
Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
- 825
1
vote
1
answer
113
views
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
- 825
2
votes
2
answers
307
views
Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
- 113
5
votes
1
answer
437
views
Elliptic PDEs in Finance
In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
- 5,405
0
votes
0
answers
25
views
Reference request heat equation with moving interface
Let $T,\sigma_1,\sigma_2>0$, $\lambda:[0,T]\to\mathbb{R}$ a continuous function and consider the following Cauchy problem on $[0,T]\times \mathbb{R}$:
$$
\begin{cases}
u_t = \sigma_1^2u_{xx} ~~~~\...
- 393
2
votes
0
answers
71
views
Any solution of an evolution problem tends to a steady state in $L^2$?
I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
- 1,759
0
votes
0
answers
108
views
Mixture of Ornstein-Uhlenbeck operators
Consider a set of isotropic, multivariate Gaussian densities with different centers $\mu_i\in \mathbb{R}^d $, $i\in\{1, \ldots, K\}$, which are denoted $\phi_i(x)$. They all have the same variance ...
- 555
3
votes
0
answers
173
views
$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature
This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ .
The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
- 155
1
vote
0
answers
64
views
The operator $D^{p}\colon \mathcal{S}\subset L^{1}(\gamma)\to L^{1}(\gamma)$ is closable for every integer $p =1,2,\dots$
I am reading Nourdin and Peccati’s textbook (Normal Approximations with Malliavin Calculus From Stein’s Method to Universality). My question is about Lemma 1.1.6. Which says
Lemma 1.1.6:
The operator $...
4
votes
1
answer
214
views
Equivalent Littlewood-Paley-type decompositions
The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that
\begin{...
- 989
0
votes
1
answer
91
views
Construct next polynomial from predecessor and resulting GCD
I have a sequence of polynomials built from an interpolation derived in a combinatorial problem. For each integer value of a parameter $n$ there is a different polynomial.
After trying to find a way ...
- 47
0
votes
0
answers
53
views
Non-linearity of viscosity solutions
I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 393
4
votes
0
answers
148
views
Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$
In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
- 91
8
votes
1
answer
357
views
Estimates of $\Delta|\nabla u|$ for harmonic function $u$
The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$,
$$
\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
- 155
0
votes
0
answers
56
views
Godunov splitting convergence research
The approximation of Godunov splitting on certain differential equations is known to be first order accurate. In 2011, a paper has also shown that it is first order accurate for nonlinear ordinary ...
- 101
7
votes
2
answers
592
views
Prove that the following function is positive
Consider the following function:
$$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right)
$$
This is Mehler's kernel, and can be ...
- 90
2
votes
1
answer
223
views
Sobolev regularity via Laplace spectrum
Fix a positive integer $n$ and let $\mu$ be the uniform measure on the sphere $\mathbb{S}^n$, with respect to its usual Riemannian metric $g$. Let $\nabla$ be the Laplacian on $(\mathbb{S}^n,g)$ and ...
- 5,405
2
votes
0
answers
126
views
On improving the regularity of solutions to nonlinear parabolic pde
There's a common reasoning i have seen in many papers/books that work with parabolic pde's (most specifically in the context of geometric flows) but I can't fully grasp it. This reasoning is that to ...
5
votes
1
answer
221
views
In what sense does the Laplacian on compact intervals converge to one on all of $\mathbb{R}$?
I guess this topic may have been addressed somewhere but I cannot really find a reference myself, so I ask here.
For each $N \in \mathbb{N}$, consider the Laplacian $\Delta$ on the interval $[-N,N]$ ...
- 3,477
0
votes
0
answers
163
views
Generalization of polynomial coefficients
I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
- 47
2
votes
0
answers
77
views
Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem
Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...
- 809
3
votes
0
answers
41
views
Functional of fully nonlinear equations
Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
- 471
1
vote
0
answers
147
views
Gelfand's representation on matrices: construct maximal ideal in matrix algebra
I would like to see a constructive proof (some algorithm?) of the following statement:
Let $A_1, A_2, \dotsc ,A_k \in M_n(\mathbb C)$ be some commuting matrices, let $B$ be the commutative algebra (...
- 807
2
votes
1
answer
152
views
Growth rate of elementary sequences
We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
0
votes
0
answers
81
views
Is a bounded measurable convex function above its interior lower semi-continuous convex envelope?
Let $E$ be a locally convex topological vector space, let $C$ be a convex set which matches the closure of its relative interior $\mathring C=\{ x\in C : \forall y\in C,\exists z\in C,~x\in\mathopen]y,...
- 204
0
votes
0
answers
37
views
Finding an element of Gelfand triple with a designated time derivative
Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
where $V'$ is the dual of $V$ and the inclusions are ...
- 3,477
0
votes
0
answers
33
views
On the I-method's energy increment calculation in a paper of Dodson
I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem ...
- 870
1
vote
0
answers
91
views
Parabolic regularity for weak solution with $L^2$ data
I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions:
$$\begin{cases}\...
- 1,759
1
vote
0
answers
115
views
Looking for examples of kernels with scalar Pick property but not the complete Pick property
I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy.
A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...
- 151
1
vote
1
answer
309
views
Eigenvalues of a Schrödinger operator
I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator
$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$
$$\varphi(0) = \...
- 337
9
votes
1
answer
639
views
Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
- 807
1
vote
0
answers
70
views
On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$
Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...
- 90
3
votes
0
answers
76
views
Absolute continuity of $t \to \lVert u(t) \rVert^2_{H}$ and Gelfand triple : are they equivalent?
Let $V$ be a separable Banach space and $H$ be a separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
and the inclusion maps are continuous with dense images. Here $...
- 3,477