All Questions
Tagged with ap.analysis-of-pdes integral-operators
12 questions
2
votes
0
answers
52
views
On distributions and kernels
Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
- 1,429
0
votes
0
answers
77
views
$ \int_{\mathbb{R}^n} f R_1 f=0$ if $f\in L^2$ Riesz transform
How can I see that? It seems that it has to do with the adjoint of the Riesz transform $R_1^*=-R_1$, but here we do not have the complex $L^2$ scalar product.
- 1
1
vote
0
answers
74
views
Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$
Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
- 63
4
votes
1
answer
145
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Boundedness of Riesz potential on Hardy space
I encounter the following claim in one paper:
If $(-\Delta)^{\frac14}u\in L^{2,\infty}(\mathbb{R})$, then $u\in BMO(\mathbb{R})$. Equivalently in its dual version, if $u\in \mathcal{H}^1(\mathbb{R})$,...
- 633
3
votes
0
answers
107
views
Boundedness of Calderon-Zygmund type operator
I am trying to prove the following fact.
Suppose $\varphi\in C_c^\infty(\mathbb{R}^1)$. Define
$$T(u)(x):=P.V.\int_{\mathbb{R}^1}\frac{\varphi(x)-\varphi(y)}{|x-y|^{\frac32}}u(y)dy$$
where P.V. means ...
- 633
2
votes
0
answers
69
views
Unique continuation for integral operator
I accidentally met such question. Let's start from easy ones.
Let $\Omega$ be an open convex domain in $\mathbb{R}^2$ and $u(x)$ satisfies that
$$u(x) = \int_{\partial\Omega} \nabla_y G(|x-y|)\cdot ...
- 151
2
votes
0
answers
286
views
Modified variational formulation of heat equation
The heat kernel $u:\mathbb{R}^n\times (0,\infty)$ is defined as the solution to
$$
u_t = \Delta u,
$$
subject to certain boundary conditions and can alternatively be described, in variational form, as ...
- 5,405
4
votes
0
answers
254
views
Lower semi-continuity of integration
I've found many papers characterizing the weak lower semi-continuity of
$$
\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,
$$
on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....
- 5,405
2
votes
3
answers
541
views
BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?
I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
- 159
0
votes
1
answer
268
views
Linear operator has one-dimensional kernel
Let $S_{\lambda}$ be a family of linear bounded operator on $L^2(\mathbb{R}^n)$ depending on some parameter $\lambda$, I have recently encountered several problems that dealt with the question whether ...
- 329
0
votes
0
answers
59
views
$L^{\infty}$ norm of Integral-Differntial equation's solution
Let $\phi(t,z)\in C^{1,2}$ is a function taking values in $\mathbb{R}^D$
, $\rho_{i,j}$ and $\mu_i$ be vector-valued functions and consider the non-linear PDE
$$
\partial_t\Phi(t,z) - \phi(0,z) - \...
- 5,405
22
votes
5
answers
10k
views
Can an integral equation always be rewritten as a differential equation?
Given an integral equation is there always a differential equation which has the same (say smooth) solutions?
It seems like not but can one prove this in some example?
Edit: Naively I'm hoping for ...
- 5,343