Questions regarding derivatives and integrals of non-integer order.

**3**

votes

**1**answer

155 views

### On nonlinear fractional field equations

In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are ...

**3**

votes

**0**answers

91 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

54 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 ...

**1**

vote

**0**answers

40 views

### Another proof of the comparison principle for PDEs by the theory of viscosity solutions

In my research I study (fully) nonlinear PDEs with fractional derivatives. For an analysis of these, I use the theory of viscosity solutions.
So I am now at the end of my tether becasuse I can not ...

**1**

vote

**0**answers

295 views

### Green's function for fractional Laplacian

Consider the fractional differential equation
\begin{align}
D_{|x|}^\alpha u(x) +bu(x)=f(x)
\end{align}
with $0<\alpha<2$ on an unbounded domain. Instead of $D_{|x|}^\alpha$ one also often sees ...

**1**

vote

**0**answers

175 views

### Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...

**1**

vote

**0**answers

222 views

### A fractional calculus eigenvalue problem

One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$,
$$\int_z^\infty ...

**1**

vote

**0**answers

395 views

### Relation between interpolation spaces and besov spaces

Consider the following two norms:
The interpolation norm:
1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x ...