Questions tagged [fractional-calculus]
Questions regarding derivatives and integrals of non-integer order.
74
questions with no upvoted or accepted answers
7
votes
0
answers
293
views
Fractional Laplacian and chain rule
For the classical Laplacian, we have
$$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$
for smooth functions $h$ and $u$.
Does a similar chain rule hold (up to a reminder term) also for the ...
6
votes
0
answers
263
views
Existence of a limit of alpha-difference quotient for Hölder functions
Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that
\begin{equation}
\sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
5
votes
0
answers
164
views
Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality
It is known that the fractional $L^p$ Sobolev inequality
$$
\|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy
$$
...
5
votes
0
answers
142
views
Commutator of pseudodifferential operator and multiplication operator
Cross-post from math.sx.
Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the ...
5
votes
0
answers
3k
views
Definition of homogeneous Sobolev spaces
As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...
4
votes
0
answers
128
views
Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
4
votes
0
answers
149
views
If $u \in W^{\alpha,p}(0,T;X)$ for $\alpha \in (0,1)$, then $f(u) \in W^{\alpha,q}(0,T;Y)$ for some good $f:X \to Y$
Let $u \in W^{\alpha,p}(0,T;X)$ for some reflexive Banach space $X$ (you can also take a Hilbert space if it helps) for $\alpha \in (0,1)$ and $p \geq 2$. I am fine with both the Sobolev-Slobodeckij ...
4
votes
0
answers
142
views
Covergence of fractional Taylor series
Let $f(x)$ be a function that is continuous and infinitely smooth on entire $\mathbb R$. Let's consider Taylor-Maclaurin series for this function:
$$f(x) = \sum_{0}^{∞}\frac{f^n(x_0)(x-x_0)^n}{n!}$$
...
4
votes
0
answers
102
views
Normalisation in fractional integration and Brownian motion
Fractional Brownian motion comes in two forms (following Marinucci and Robinson 1998) for fraction $\alpha$ and Brownian motion $W_s$:
Type II (Levy, Volterra, Riemann)
$$ \tilde W^\alpha_t = \int_0^...
3
votes
0
answers
228
views
Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?
The well-known integral expression for the entire function:
$$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$
...
3
votes
0
answers
199
views
Infinite partial fraction expansions to compute fractional iterations and recurrences
Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how ...
3
votes
0
answers
213
views
About generalized binomial theorem and Grünwald-Letnikov fractional derivative
I have run into a problem while computing the fractional derivatives of order $\alpha$ for the Riemann zeta function. My Theorem states
Let $s\in\mathbb{C}$, $\mathfrak{Re}(s)>1$, then the ...
3
votes
0
answers
104
views
Rearrangement in Bessel function spaces
I consider, for $0<s<1$, the Bessel function space
$$
L^{s,2}(\mathbb{R}^d) = \left\{f\in L^2(\mathbb{R}^n) : (1+|\cdot|)^{s/2}\hat{f}(\cdot)\in L^2(\mathbb{R}^n)\right\}.
$$
The question I ...
3
votes
0
answers
118
views
A question about fractional integrals
I am starting from Theorem 3.4 in Struwe's book Variational methods. The authors proves an existence result for a problem with critical growth. I would like to replace the laplacian with the ...
2
votes
0
answers
50
views
Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in
H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of
the ...
2
votes
0
answers
88
views
Fractional power of a matrix with complex spectrum
I will use the notation I found here https://arxiv.org/abs/1812.01206. Please forgive me if this is a poorly stated question. I'm not sure of the things I wrote in parenthesis.
The paper makes a claim ...
2
votes
0
answers
46
views
Is there a fractional derivative that preserves the composition of the one-parameter Mittag-Leffler function with $x\mapsto x^{\alpha}$?
Let $\alpha\in (0,1)$. The Riemann-Liouville fractional derivative of order $\alpha$ is defined by
$$ \sideset{_0^R}{}{D^{\alpha}f(t)}
=\frac{1}{\Gamma{(1-\alpha)}}
\frac{d}{dt}\left(\int_{0}^{t}
\...
2
votes
0
answers
40
views
Fractional Laplacian in higher order case
Let $n\geq 2$ and $\sigma \in (0,\frac{n}{2})$. Denote the critical Sobolev exponent $2_{\sigma}^*:=\frac{2n}{n-2\sigma}$, consider Sobolev space $E$ which is the space of real-valued functions $u\...
2
votes
0
answers
49
views
A mapping property for fractional Laplace--Beltrami operator
Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
2
votes
0
answers
41
views
Nonlocal perimeter of level sets
Let $u \in W^{s,1}(B)$ be given and $k < l$ be two numbers, then I am looking for a way to bound the following term from above. Here $B$ is the euclidean ball.
$$
\int_{B: u < k} \int_{B:u>l} ...
2
votes
0
answers
125
views
Computing the fractional laplacian of a logarithm function
Are there explicit formulas to compute the fractional laplacian $(-\Delta)^{s/2}\log |x|$ in $\mathbb{R}^n$ for $s\in (0,2)$?
2
votes
0
answers
164
views
Boundary terms in integration by parts for the fractional Laplacian
Let $u,v \in C^\infty(\Omega)$ and assume that $v$ is compactly supported inside a domain $\Omega$.
Is it true that
$$
\int_\Omega v (-\Delta)^su \, d x = \int_\Omega (-\Delta)^{s/2}v(-\Delta)^{s/2}u \...
2
votes
0
answers
61
views
Expansion of the Riemann-Liouville integral
Riemann-Liouville integral operator of the $\alpha$-th order ($\alpha > 0$) is defined as
$$
I^{\alpha} x(t) := \frac{1}{\Gamma(\alpha)}\int\limits_{0}^{t} (t-\tau)^{\alpha - 1} x(\tau)d\tau
$$
I ...
2
votes
0
answers
142
views
Approximation in fractional Sobolev space
Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$.
How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$?
Recall that,
$$|u|^p_{W^{s,p}(\Omega)}= ...
2
votes
0
answers
193
views
Eigenvalues and kernel of the the fractional laplacian in the $d$-dimensional torus
Let $\mathbb{T}^d$ be the $d$-dimensional torus. Consider the operator
$$
\Delta^{(\alpha/2)}u(x):= \int_{\mathbb{T}^d} \frac{u(x+y)+u(x-y)-2u(x)}{(d_{\mathbb{T}^d}(x,y))^{d+\alpha}}
dy$$
Where $u$ ...
2
votes
0
answers
141
views
Fractional derivative of the Wright function
It is mentioned in some papers (Appendix in this paper, for example) that the (formal) solution of the fractional drift (or transport) equation
$$
\partial_{t}^{\alpha}u(t,x)+\partial_{x}u(t,x)=0\quad\...
1
vote
0
answers
44
views
Extension operator maps which fractional Sobolev space to $W^{p,s}(R)$
Let us assume we have the following extension operator:
$$
\operatorname{ext}_R^\sigma u=
\begin{cases}
u(x) & \text{if }x \in (0,T)\\
u(0) & \text{if }x \in(0,T)^c.
\end{cases}
$$ We ...
1
vote
0
answers
59
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
1
vote
0
answers
61
views
Variation of the fractional derivatives
$\DeclareMathOperator\AC{AC}\DeclareMathOperator\Lip{Lip}$Suppose we have $f\in L^1(\mathbb{R})\cap \AC(\mathbb{R})\cap \Lip(\mathbb{R})$ and $f$ piecewise linear function, bounded and $|f|\leqslant \...
1
vote
0
answers
71
views
Domain where the fractional Laplacian operator is a closed operator
Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...
1
vote
0
answers
45
views
A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
1
vote
0
answers
39
views
Functional inequality for fractional Laplacian
Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...
1
vote
0
answers
55
views
Precise decay of solution fo fractional Schroedinger equations
Let us consider the time-independent fractional Schroedinger equation $$(-\Delta)^s u + u = \vert u \vert^{p-1}u$$ in $\mathbb{R}^N$, where $0<s<1$, $N>2s$ and $1<p<\frac{N+2s}{N-2s}$.
...
1
vote
0
answers
134
views
Poisson Kernel and solution formula for fractional elliptic problem
$$
k (-\Delta)^s u + u = 0, \qquad x \in U, \\
u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,
$$
with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
1
vote
0
answers
190
views
Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
1
vote
0
answers
125
views
Multivariable fractional calculus
I recently started to learn about nonlocal PDEs. Sorry if I am missing something really standard in this field.
I know Riesz derivative is defined through Fourier transform:
$$\mathcal{F}(\partial^\...
1
vote
0
answers
199
views
Is the spectral fractional Sobolev norm equivalent to other norms (e.g. Gagliardo...)?
Let $s \in (0, 1)$ and $\Omega$ be a bounded subdomain of $\mathbb R^n$ with polygonal/polyhedral boundary. Let $\Delta$ be the Laplace-Dirichlet operator on $\Omega$ (i.e. the Laplace operator with ...
1
vote
0
answers
65
views
How to check that $f(t,x)\ge 0$ for any $x\ge 0$?
Suppose $f(t,x)$ satisfices $\partial_t f=\Lambda^{-\alpha}f\partial_xf$, for $0<\alpha<1$ and where $\Lambda=(-\partial_{xx})^{1/2}$, $f_0(x)=f(0,x)$ is odd, and $f_0(x)\ge 0$, $\forall x\ge 0$....
1
vote
0
answers
63
views
Fractional power of the operator $\mathcal{L}_t[t f(t)](x)$ and equivalence of divergent integrals
I wonder whether an expression for fractional power of operator $\mathcal{L}_t[t f(t)](x)$ that involves Laplace transform can be derived?
I am asking this because this operator preserves the area ...
1
vote
0
answers
45
views
Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder
Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
1
vote
0
answers
50
views
Functions that vanish weakly to $\infty$ and a uniqueness problem
I am reading the article "User’s guide to the fractional Laplacian and the method of semigroups" by P.R. Stinga, there is a link. At page 17, in theorem 7, the author state that, for a given ...
1
vote
0
answers
40
views
Extension problem of fractional laplacian and Fourier transform of $L^1_\text{loc}$ function?
I have understand the proof oh the lemma 4.1.9 "SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY" by C.Bocur, there is a link. In this paper, we define, for $u\in L^1_\text{loc}(\...
1
vote
0
answers
200
views
Fractional Laplacian extension problem and uniqueness question
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. Consider the following problem:
$$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...
1
vote
0
answers
68
views
A question about extension problem related to fractional laplacian
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. link. At page 2, for a function $f\colon\mathbb{R}^n\to\mathbb{R}$, we ...
1
vote
0
answers
377
views
Stein's extension operator for fractional Sobolev spaces
In his book Singular Integrals and Differentiability Properties of Functions,
Stein constructs an extension operator $\mathcal{E}:W^{m,p}(\Omega)\rightarrow
W^{m,p}(\mathbb{R}^{N})$, $m\in\mathbb{N}$, ...
1
vote
0
answers
91
views
Equivalence of Cauchy Type Problems and Volterra Integral Equations
The following Theorem is from the book Theory and Applications of Fractional Differential Equations by Kilbas, Srivastava and Trujillo. It's Theorem 3.10 from page 163.
Theorem.
Let $ \alpha \in \...
1
vote
0
answers
51
views
Limit contration rates and expansion rate solenoid map
Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
1
vote
0
answers
105
views
formula of spectral fractional laplacian
In fact, if $U$ is a solution of
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s U &=f &&\text{in } \Omega \\
U &= g &&\text{on } \partial \Omega \\
\end{...
1
vote
0
answers
56
views
Solution of two coupled fractional differential equations
The following arose in a physical problem:
Solve the two equations
\begin{eqnarray}
p(t)+D^\alpha x(t)+\omega x(t)=0\\
_T^- D^\alpha p(t)-\omega p(t)+k x(t)=0
\end{eqnarray}
subject to the ...
1
vote
0
answers
61
views
Fractional Leibniz rule with Lorentz spaces
The "fractional Leibniz rule" asserts that
$$\Vert D^s(fg)\Vert_{L^r}\lesssim\Vert D^sf\Vert_{L^{p_1}}\Vert g\Vert_{L^{q_1}}+\Vert f\Vert_{L^{p_2}}\Vert D^sg\Vert_{L^{q_2}}$$
where
$$\frac{1}{p_i}+\...