All Questions
12,935 questions
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184
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Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?
$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...
- 365
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votes
0
answers
136
views
Why is it impossible to reduce a linear PDE of the second order in more than two independent variables to canonical form globally
It is known that in the case of more than two independent variables, it is usually not possible (especially in the case of PDE with the variable coefficients) to reduce a linear partial differential ...
- 133
0
votes
0
answers
216
views
Bound on integral of elliptic theta function
I need to prove that the following bound is true. I thought this might follow from the inversion property of the theta function, as the infinite sum in the integrand is precisely $\theta_3(0,\mathrm{e}...
0
votes
0
answers
74
views
Weak convergence of 4-th degrees
Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, $...
- 243
0
votes
0
answers
65
views
Interpolation with time continuity
If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in L^2(0,T;H^{-1}(\...
- 103
0
votes
1
answer
552
views
Teaching profession:Differential Equations and Mean Value Theorems
Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th ...
- 1,422
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votes
0
answers
83
views
Comparison between operators
I have found the following two concepts:
$\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The
operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$
and for any $\varepsilon>0$,...
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votes
0
answers
210
views
Weak derivatives and dual of Hölder functions
Let $0<\alpha<1$ and $f \in C^{\alpha}$ be a Hölder function (either with compact support on $\mathbb R^n$ or on a closed Riemaniann manifold).
From what I understand, the derivative of $f$ in ...
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0
answers
428
views
Given an even function how to obtain the most close odd function and vise versa?
Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$?
By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference $|...
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0
answers
178
views
$2$-normed Spaces
Someone suggested today that $2$-normed spaces are actually equivalent to normed spaces. Can anyone who's familiar with the topic provide a counterexample? (I can't access Gähler's original paper ...
- 1
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votes
0
answers
223
views
Solvable PDEs and their Green's functions
I have a class of PDEs of the form
$$
-\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0
$$
with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
- 1,687
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votes
0
answers
474
views
Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)
Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$.
Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
- 85
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votes
2
answers
180
views
A basic question about JL Lions' transformation of a Stefan problem
In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The ...
- 23
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votes
0
answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
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0
answers
537
views
matrix Khintchine inequality
The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then
\begin{equation*}
\left( \...
0
votes
2
answers
168
views
Let f:J→R be an absolutely continuous and f'\in...?
Let $f:J\rightarrow \mathbb{R}$ be an absolutely continuous.
Under what kind of extra condition for $f'$, (not $C$) holds the following relation?
$$
\Big | \frac{1}{|I_{1}|}\int_{I_{1}}f'(x)dx- \...
- 111
0
votes
0
answers
435
views
Analytic solution of Poisson's equation
Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square.
Impose a von Neumann boundary condition $...
- 409
0
votes
0
answers
405
views
Dual of the space of vector valued Borel measures
What is the dual of the space of all vector valued Borel measures?
- 21
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0
answers
533
views
Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$
Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.
Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X \...
user24451
0
votes
0
answers
156
views
Can a function be constructed from the direction of its gradient?
Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation
$Du=J|Du|$, $u|_{\partial \Omega}=f$
has a solution in a ...
- 23
0
votes
1
answer
2k
views
Second proof of Jordan-Von Neumann theorem
I am looking for a second proof of Jordan-Von Neumann theorem that characterizes inner product in normed spaces. The book "Inner Product Structures: Theory and Applications" talks about a second proof ...
- 113
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votes
0
answers
362
views
Banach-Mazur distance estimate finite-dimensional $\ell_p$ spaces
Hey my fellow Banach space guys. Sorry for another elementary question, and yes I have looked for the past hour and a half to see if I can find it on Google or in my books at home.
Fix $n\in\mathbb{...
- 1,591
0
votes
0
answers
68
views
can we say fixed point existance of a set valued map over a compact set is homotopy invariant?
Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(...
- 383
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votes
0
answers
515
views
If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?
Let $\phi_n$ be a sequence of mollifier converging to the identity
$$
\phi_n(x) \to \delta_{0}(x), \text{pointwise},
$$
with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in C^\infty_{\...
user24451
0
votes
0
answers
100
views
Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$
Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal P}$...
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votes
0
answers
96
views
partial maximum principle for elliptic differential operators
Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a self-adjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest ...
- 1,701
0
votes
0
answers
100
views
Parabolic PDE; uniform bound on approximations $u'_n$ in $L^2(0,T;V^*)$ without using orthogonal basis?
Let $V \subset H \subset V^*$ be a Gelfand triple, all Hilbert and separable spaces.
I consider the PDE with weak form: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that
$$\langle u'(t), v(...
- 405
0
votes
0
answers
149
views
Does this sequence of H\"older functions have a limit?
Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
$$\lim_{n\...
- 91
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votes
0
answers
113
views
Reference Search for a Functional Minimization Problem
Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is
$$M \equiv\int_{x=-\infty}^\infty u(x) ~ ...
- 151
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votes
0
answers
146
views
How to bound Haar coefficients in terms of total variation?
I'm trying to get the basic idea behind the proof of Theorem 8.1 of this paper, but I'm having difficulty. Specifically, it says:
We shall show that there is a set $\Lambda_n\subset\mathcal{D}$ such ...
- 7,633
0
votes
2
answers
812
views
Estimates for Green's function
Let $n$ - dimension $\geq 3$.
Consider a compact manifold (M,g). Let $\epsilon_0$ denote the injectivity radius of $(M,g)$. Let $B_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < \...
- 3
0
votes
1
answer
347
views
Dual space of Bochner space: is there an easier proof to show they're isometric?
It is known that $[L^p(0,T;H)]^* = L^q(0,T;H^*)$.
If $p=q=2$ and $H$ is a Hilbert space, is there an easier proof to show that the spaces are isometric? The proof that I know for the general case ...
0
votes
0
answers
270
views
Gradient estimates for subsolutions of elliptic equations
Let $M$ be a Riemannian manifold. Assume $u \in C^\infty(M)$ such that $u>0$ and
$\Delta u + \lambda u = 0,$
where $\lambda \geq 0$. There is a poinwise estimate for $|\nabla u|$ in Peter Li's ...
- 56
0
votes
0
answers
137
views
$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$
Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in L^2(...
- 113
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votes
0
answers
93
views
Infinite limit in all points
Do there exist a Banach space (possibly nonseparable) $X$ and a mapping $F: X\to X$ such that
$$
\lim_{x\to a} \|F(x)\| = +\infty \quad \forall a\in X\quad?
$$
0
votes
0
answers
152
views
Need help determining whether a certain map is a $C^\ast$ homomorphism
Hello, I need help determining whether the map I defined between two algebras is a well-defined homomorphism of $C^\ast$-algebras. I ran into this problem while trying to define a "rotation map" ...
- 365
0
votes
0
answers
166
views
Harnack's Inequality and (hypo)elliptic PDE
Background: I am aware of the Harnack's Inequality for linear elliptic equations.
My questions are:
(a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
- 1
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 125
0
votes
0
answers
223
views
functional equation, how to solve
Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.
$$F(x, y) = \frac{x\circ Ay}{x^TAy}$$
$$G(x, y) = \frac{x\circ By}{x^TBy}$$
where $A$ and $B$ are ...
- 13
0
votes
0
answers
160
views
Is this function in the weighted Sobolev space $H^{2,-s}$?
I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order $...
- 1
0
votes
1
answer
656
views
Gel'fand Yaglom functional determinant of non-diagonal operator?
Introduction:
As a quick reminder, the Gel'fand Yaglom theorem uses the generalized zeta-function approach to compute functional determinants of differential operators. Given a differential operator ...
0
votes
0
answers
241
views
Continuity of a function
Let $f\in L^2(\mathbb{R}^3)$ with compact suppport and $z\in\mathbb{C}$. Is the following function continuous for $z\in Q = \{ z : \Re z\in [a,b], \Im \sqrt{z} \in (0,1] \}$:
$$ F(z)=\bigg(\alpha-i\...
- 71
0
votes
0
answers
189
views
functional maximization
Define a functional space of functions of the form $F(t)=p_1 exp^{-\mu_1(\delta-t)}+p'_1 (1-exp^{-\mu_1(\delta-t)}))$. $p_1,p'_1,\delta,\mu$ are parameters in [0,1] and trivially, variation of these ...
- 221
0
votes
0
answers
214
views
Splitting the action of functionals in duals of Sobolev spaces
Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
- 101
0
votes
1
answer
302
views
An interpolation inequality.
For all $s>0$ define for $\epsilon\in(0,1)$ the function:
\begin{equation}
g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.
\end{equation}
Prove that $\exists C>0$ and $\phi(s)$ such ...
- 45
0
votes
0
answers
558
views
Eigenfunctions of the laplace on the 2-sphere with conformal metric induced by schwarzschild
Hi,
it's well known that the coordinates $x_1$, $x_2$, $x_3$ are the first three eigenfunctions with positiv eigenvalues ($=2/r^2$) of the negative laplace-beltrami operator ${-}\triangle$ on the 2-...
- 143
0
votes
0
answers
335
views
A basis of the space of continuous function of countable ordinals $C(\alpha) = C [0, \alpha]$
A basis of the space of continuous function of countable ordinals $C({\alpha}) = C [0, {\alpha}]$, which consist of characteristics functions of clopen subsets of $C({\alpha})$, in some order. But can ...
- 1
0
votes
0
answers
205
views
damped wave equation
For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation.
My question is...why is the ...
- 115
0
votes
0
answers
80
views
relationship between different function classes
I was wondering if there is a survey of relationship between several different well-studied function classes ?
ps - The question may be vague but I am looking for something along the lines of - http:/...
- 1
0
votes
0
answers
218
views
Series of linear maps: on a paper by Evans and Hanche-Olsen
I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.
Let $L$ be a bounded self-adjoint
...
- 421