Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

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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 $1<p<\frac{N+2s}{N-2s}$. ...
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Fractional partial derivatives and integrals of De Bruijn's $H_t(z)$-function. Does a simpler form exist for the $z$ derivative/integral?

$\newcommand\KummerU{\text{KummerU}}$ $\newcommand\Hypergeom{\text{Hypergeom}}$ During 2018/2019, the polymath 15 project managed to successfully reduce the upper bound of the De Bruijn-Newman ...
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4 votes
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129 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}<...
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Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?

In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality: $$\...
3 votes
3 answers
173 views

Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives

So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
5 votes
2 answers
207 views

Morse index in PDEs

I have encountered with the term "Morse index" multiple times while reading papers in PDEs (e.g. [1] and [2]). However the definition differs for each context. As far as I know this came ...
5 votes
0 answers
100 views

Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality

It is known that the fractional $L^p$ Sobolev inequality $$ \|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy $$ ...
4 votes
1 answer
132 views

prove spectral equivalence bounds for inverse fractional power of matrices

The question is an extention to the answered question prove spectral equivalence bounds for fractional power of matrices. Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and ...
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3 votes
1 answer
56 views

prove spectral equivalence bounds for fractional power of matrices

Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds $$ c^- x^\top D x \le x^\top A ...
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-1 votes
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Is integration by parts allowed for the $J^s$ derivative, where $s \in \mathbb R$

I am having the following integral: $$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$ where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$, $...
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Difference between summation for "$\aleph$" terms and summation for "$\aleph_0$" terms

Addition: Could we say that the dimension of a space is "$\aleph_0$" or"$\aleph$"? I guess that every elementary functions can be uniquely expanded as integer order power series ...
1 vote
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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 ...
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Explicit solution for 1d fractional problems

Consider the fractional problems $$ \begin{cases}\partial_t u(t,x) + (-\Delta)^s u(t,x)= 0 \quad t >0, \ x \in (-1,1),\\ u(t,x) = 0\quad t \ge 0, \ x \in \mathbb{R} \setminus (-1,1), \\ u(0,x) =...
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4 votes
1 answer
79 views

How to validate the exponentiality of fractional calculus?

Is it true $$ \frac{1}{\Gamma(1-\nu)}\frac{1}{\Gamma(\nu)} \int_{0}^{x}(x-y)^{-\nu}dy\int_0^y (y-t)^{\nu-1}f(t)dt = \int_0^x f(u)du$$ for any continuous function $f(x)$ such that $f(0)=0$ and $0<\...
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1 vote
1 answer
185 views

Fourier transform of the fractional Poisson kernel

Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows: $$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+...
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2 votes
0 answers
28 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} ...
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Euclidean log Sobolev inequality for fractional Laplacian

I am interested in this work of Cotsilois and Tavoularis, which claims the following Euclidean log Sobolev inequality for the fractional Laplacian, generalizing the classical Gross-Stam log Sobolev ...
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64 views

Fractional Laplacian of smooth cut off functions

Suppose we have a smooth compactly supported function $\phi\in C^{\infty}_c(B_\epsilon(0))$ such that $0\leq \phi \leq 1$, $\phi\equiv 1$ on the unit ball and $\phi$ vanishes outside $B_\epsilon(0).$ ...
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1 vote
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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 $[m,n]$ : ...
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1 vote
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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^\...
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42 views

How to prove this estimate involving the Stein Derivative?

Recall the Stein Derivative, $$\mathcal{D}^{s} f(x)=\left(\int \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}} d y\right)^{1 / 2}.$$ I want to show that, $$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \...
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Normal vector to a level set and fractional Laplacian

Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
1 vote
1 answer
158 views

Why should we model infectious diseases with fractional differential equations?

With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I ...
0 votes
0 answers
26 views

Spectrum of a linear operator depending of the fractional laplacian operator

Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the periodic fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...
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Lack of conformal invariance for fractional Laplace operators on a closed Riemannian surface

Let $(\Sigma,g)$ be a compact two-dimensional Riemannian manifold with no boundary. Let us denote by $\{(\lambda_k,\phi_k)\}_{k=0}^{\infty}$ the spectral data for the Laplace--Beltrami operator on $(\...
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1 answer
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Inequality involving the fractional Laplacian

Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...
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5 votes
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119 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 ...
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Explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)$

Do we know an explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)?$
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Difference between two fractional Schrödinger equations

Let us consider the fractional Schrödinger equation with periodic boundary conditions $$ \begin{cases} iu_t\mathbf{+}(-\Delta)^{\alpha}u= \pm |u|^2u,\; x \in \mathbb{T}, t \in \mathbb{R}_+\\ u(x,0)=...
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1 answer
203 views

Explicit expression for the fractional Laplacian of $1/(1+|x|^2)^s$

For $s\in (0,1)$, is there are an explicit expression for $$(-\Delta)^s \left(\frac{1}{\left(1+|x|^2\right)^s}\right)?$$ Edit: My goal is to show that for the function $u(x)=\frac{1}{\left(1+|x|^2\...
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2 votes
0 answers
80 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 $s\in (0,2)$?
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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 ...
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Relation between the eigenvalues of the weighted laplacian and fractional laplacian?

Consider the eigenvalue problem $-\Delta u = \lambda u\rho$ for $u\in \dot{H}^{1}(\mathbb{R}^n)$ with $n\geq 3$ and weight $\rho\in L^{n/2}(\mathbb{R}^n).$ Let $(\lambda_k, \psi_k)$ be the increasing ...
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Are there any known results for the spectrum of $(-\Delta)^s/V^{p-1}$?

I am interested in generalizing some results known for the $\frac{-\Delta}{U^{p-1}}$ where $U$ is a Talenti bubble to the non-local operator $\frac{(-\Delta)^s}{V^{p-1}}$ where $U$ and $V$ are bubbles ...
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1 vote
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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 $f_0(x)\ge 0$, $\forall x\ge 0$....
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6 votes
1 answer
146 views

Fractional derivative notation in wave turbulence

This is my first question in MathOverflow and I will do my best to format it correctly and make it clear. I am reading a paper on dispersive wave turbulence which introduces the following family of ...
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2 votes
1 answer
136 views

Property about the fractional Laplacian

Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...
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2 votes
1 answer
184 views

Limit of $e^{-t(-\Delta)^\alpha}$ when $\alpha \to 1$

Let us consider the fractional heat semigroup $\left(e^{-t(-\Delta)^\alpha}\right)_{t\ge 0}$ for $\alpha\in (0,1)$ (the fractional power is taken in whole $\mathbb R^d$). Is there any result about the ...
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2 votes
0 answers
106 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 \...
1 vote
0 answers
55 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 ...
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Spectral fractional Laplacian of power-function $(-\Delta)^s x^{\alpha}$ in $(0,1) \subset \mathbb R$

How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space....
1 vote
1 answer
325 views

A mysterious expression from a discriminant

I recently investigated $\textrm{Discriminant}_u(\omega P(u) +(z-u)P'(u))$, where $P(u) := u^3 + au + b$ and $\omega$ is a real parameter (with $\omega\in(0,1)\cup(1,3)$) associated with the order of ...
0 votes
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71 views

Inverse of the Riesz potential of a measure

Let $0<\alpha<d$ and let $I_{\alpha}(f)$ be the Riesz potential of a function $f$ on $\mathbb{R}^{d}$, $$ I_{\alpha}(f)(x)=\int_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}dy. $$ Assuming $f$ ...
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4 votes
0 answers
121 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 ...
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6 votes
3 answers
1k views

A question on fractional derivatives

I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange. I just wanted to ask if there is a notion of ...
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2 votes
1 answer
500 views

Sobolev embedding for fractional Sobolev spaces

Let $\Omega\subset\mathbb{R}^2$ be open and of class $C^1$. The Sobolev embedding theorem implies that if $u\in W^{k,2}(\Omega)$ and if $k\in\mathbb{N}: k\geq 2$, then $u$ is continuous. Question. ...
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6 votes
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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 ...
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definition of functions that "weakly vanishes as $y\to\infty$" and find a proof of Theorem 9 in the article

I'm reading "Extension problem and fractional operators: semigroups and wave equations" by P.R. Stinga, i have two questions: in Theorem 7 the author use the state "weakly vanishes as $...
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0 votes
1 answer
548 views

Euler-Lagrange equation for a functional

What does it mean that the equation: $$ \text{div}_{x,y}(y^a\nabla_{x,y}u)=0,\quad \text{in }\mathbb{R}^n\times(0,\infty),$$ is the Euler-Lagrange equation for the functional: $$ J(u)=\int_{\mathbb{R}^...
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Fractional laplacian on $H^s(\mathbb{R}^n)$ and symmetry

Let $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n),$ i define the fractional laplacian of $u$ in the following way: $$(-\Delta)^su(x)=C(n,s)P.V.\int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\quad\...
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