Questions tagged [fractional-calculus]
Questions regarding derivatives and integrals of non-integer order.
72 questions with no upvoted or accepted answers
7
votes
0
answers
351
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 ...
- 161
7
votes
0
answers
3k
views
Definition of homogeneous Sobolev spaces
As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...
- 633
6
votes
0
answers
271
views
Existence of a limit of alpha-difference quotient for Hölder functions
Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that
\begin{equation}
\sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
- 61
5
votes
0
answers
186
views
Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality
It is known that the fractional $L^p$ Sobolev inequality
$$
\|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
$$
...
5
votes
0
answers
163
views
Commutator of pseudodifferential operator and multiplication operator
Cross-post from math.sx.
Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the ...
- 245
4
votes
0
answers
92
views
Well-posedness for linear transport equations with fractional diffusion term
I have a rather applied problem where I consider an equation of the form
$$ \partial_{t} u + V\cdot \nabla u = -(-\Delta)^s u, \quad (t,x) \in (0,\infty)\times \mathbf{R}^d, \quad u(0,x) = u_0. {}$$
...
4
votes
0
answers
140
views
Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
- 1,075
4
votes
0
answers
176
views
If $u \in W^{\alpha,p}(0,T;X)$ for $\alpha \in (0,1)$, then $f(u) \in W^{\alpha,q}(0,T;Y)$ for some good $f:X \to Y$
Let $u \in W^{\alpha,p}(0,T;X)$ for some reflexive Banach space $X$ (you can also take a Hilbert space if it helps) for $\alpha \in (0,1)$ and $p \geq 2$. I am fine with both the Sobolev-Slobodeckij ...
- 51
4
votes
0
answers
154
views
Covergence of fractional Taylor series
Let $f(x)$ be a function that is continuous and infinitely smooth on entire $\mathbb R$. Let's consider Taylor-Maclaurin series for this function:
$$f(x) = \sum_{0}^{∞}\frac{f^n(x_0)(x-x_0)^n}{n!}$$
...
4
votes
0
answers
103
views
Normalisation in fractional integration and Brownian motion
Fractional Brownian motion comes in two forms (following Marinucci and Robinson 1998) for fraction $\alpha$ and Brownian motion $W_s$:
Type II (Levy, Volterra, Riemann)
$$ \tilde W^\alpha_t = \int_0^...
- 41
3
votes
0
answers
231
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:
$$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$
...
- 4,169
3
votes
0
answers
204
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 ...
- 3,098
3
votes
0
answers
219
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 ...
- 169
3
votes
0
answers
110
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 ...
- 459
3
votes
0
answers
119
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 ...
- 31
2
votes
0
answers
56
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 ...
- 93
2
votes
0
answers
59
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
$$ \sideset{_0^R}{}{D^{\alpha}f(t)}
=\frac{1}{\Gamma{(1-\alpha)}}
\frac{d}{dt}\left(\int_{0}^{t}
\...
- 852
2
votes
0
answers
44
views
Fractional Laplacian in higher order case
Let $n\geq 2$ and $\sigma \in (0,\frac{n}{2})$. Denote the critical Sobolev exponent $2_{\sigma}^*:=\frac{2n}{n-2\sigma}$, consider Sobolev space $E$ which is the space of real-valued functions $u\...
- 471
2
votes
0
answers
67
views
A mapping property for fractional Laplace--Beltrami operator
Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
- 4,135
2
votes
0
answers
47
views
Nonlocal perimeter of level sets
Let $u \in W^{s,1}(B)$ be given and $k < l$ be two numbers, then I am looking for a way to bound the following term from above. Here $B$ is the euclidean ball.
$$
\int_{B: u < k} \int_{B:u>l} ...
- 455
2
votes
0
answers
140
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 $s\in (0,2)$?
- 537
2
votes
0
answers
175
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
$$
\int_\Omega v (-\Delta)^su \, d x = \int_\Omega (-\Delta)^{s/2}v(-\Delta)^{s/2}u \...
- 99
2
votes
0
answers
62
views
Expansion of the Riemann-Liouville integral
Riemann-Liouville integral operator of the $\alpha$-th order ($\alpha > 0$) is defined as
$$
I^{\alpha} x(t) := \frac{1}{\Gamma(\alpha)}\int\limits_{0}^{t} (t-\tau)^{\alpha - 1} x(\tau)d\tau
$$
I ...
- 21
2
votes
0
answers
160
views
Approximation in fractional Sobolev space
Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$.
How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$?
Recall that,
$$|u|^p_{W^{s,p}(\Omega)}= ...
- 1,101
2
votes
0
answers
204
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 $u$ ...
- 446
2
votes
0
answers
143
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
$$
\partial_{t}^{\alpha}u(t,x)+\partial_{x}u(t,x)=0\quad\...
- 201
1
vote
0
answers
52
views
Extension operator maps which fractional Sobolev space to $W^{p,s}(R)$
Let us assume we have the following extension operator:
$$
\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}
$$ We ...
1
vote
0
answers
70
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
1
vote
0
answers
64
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 \...
- 89
1
vote
0
answers
89
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
$$D((-\Delta)^s) = \{u \in H^s(\...
- 111
1
vote
0
answers
52
views
A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
- 287
1
vote
0
answers
47
views
Functional inequality for fractional Laplacian
Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...
- 873
1
vote
0
answers
57
views
Precise decay of solution fo fractional Schroedinger equations
Let us consider the time-independent fractional Schroedinger equation $$(-\Delta)^s u + u = \vert u \vert^{p-1}u$$ in $\mathbb{R}^N$, where $0<s<1$, $N>2s$ and $1<p<\frac{N+2s}{N-2s}$.
...
- 459
1
vote
0
answers
152
views
Poisson Kernel and solution formula for fractional elliptic problem
$$
k (-\Delta)^s u + u = 0, \qquad x \in U, \\
u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,
$$
with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
- 839
1
vote
0
answers
210
views
Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
- 4,829
1
vote
0
answers
144
views
Multivariable fractional calculus
I recently started to learn about nonlocal PDEs. Sorry if I am missing something really standard in this field.
I know Riesz derivative is defined through Fourier transform:
$$\mathcal{F}(\partial^\...
1
vote
0
answers
248
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
68
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 $f_0(x)\ge 0$, $\forall x\ge 0$....
- 171
1
vote
0
answers
67
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 ...
- 10.1k
1
vote
0
answers
47
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 ...
- 339
1
vote
0
answers
52
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 ...
- 339
1
vote
0
answers
42
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}(\...
- 339
1
vote
0
answers
213
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:
$$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...
- 339
1
vote
0
answers
78
views
A question about extension problem related to fractional laplacian
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. link. At page 2, for a function $f\colon\mathbb{R}^n\to\mathbb{R}$, we ...
- 339
1
vote
0
answers
441
views
Stein's extension operator for fractional Sobolev spaces
In his book Singular Integrals and Differentiability Properties of Functions,
Stein constructs an extension operator $\mathcal{E}:W^{m,p}(\Omega)\rightarrow
W^{m,p}(\mathbb{R}^{N})$, $m\in\mathbb{N}$, ...
- 411
1
vote
0
answers
91
views
Equivalence of Cauchy Type Problems and Volterra Integral Equations
The following Theorem is from the book Theory and Applications of Fractional Differential Equations by Kilbas, Srivastava and Trujillo. It's Theorem 3.10 from page 163.
Theorem.
Let $ \alpha \in \...
- 11
1
vote
0
answers
53
views
Limit contration rates and expansion rate solenoid map
Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
- 1,043
1
vote
0
answers
105
views
formula of spectral fractional laplacian
In fact, if $U$ is a solution of
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s U &=f &&\text{in } \Omega \\
U &= g &&\text{on } \partial \Omega \\
\end{...
- 402
1
vote
0
answers
57
views
Solution of two coupled fractional differential equations
The following arose in a physical problem:
Solve the two equations
\begin{eqnarray}
p(t)+D^\alpha x(t)+\omega x(t)=0\\
_T^- D^\alpha p(t)-\omega p(t)+k x(t)=0
\end{eqnarray}
subject to the ...
1
vote
0
answers
66
views
Fractional Leibniz rule with Lorentz spaces
The "fractional Leibniz rule" asserts that
$$\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}+\...
- 943