All Questions
1,304 questions
3
votes
0
answers
223
views
Sobolev space under Mellin transform
The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
user127612
2
votes
3
answers
979
views
"Must read" papers in functional analysis and PDE (à la Trefethen) [closed]
The following question has been inspired by "Must read" papers in numerical analysis.
In 1993, Prof. Trefethen published a NA-net posting with a list of thirteen papers that he had used in ...
3
votes
1
answer
752
views
A distributional normal derivative for functions in $H^1(\Omega)$
Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.
For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...
- 266
2
votes
0
answers
683
views
Laplace problem with Robin boundary condition on a wedge
I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let
\begin{equation*}
\Omega = ...
6
votes
2
answers
2k
views
Equivalent Norms on Sobolev Spaces
When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ^{\...
- 233
4
votes
0
answers
169
views
Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”
I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
- 141
1
vote
0
answers
87
views
Global solution of nonlinear Schrödinger equation via blow-up argument
Set $u_0\in H^1 (\mathbb{R} ^N)$ and $1<\alpha < \frac{N+2}{N-2}$.
I want to show that there exists $\varepsilon > 0$ s.t. if $\Vert u _0 \Vert _ {H^1} < \varepsilon$,
then there is ...
- 383
8
votes
1
answer
2k
views
Definitions of Hilbert Bundles
I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...
1
vote
1
answer
285
views
Recover norm from integral
I am given the following expression where $f \in L^2(\mathbb{R}^2, \mathbb{R}^{2 \times 2})$
$$\int_{\mathbb{R}} \int_{\mathbb{R}} \langle g(x), f(x,y) h(y)\rangle dx dy.$$
The functions $g$ and $h$ ...
user121558
3
votes
1
answer
219
views
Positivity of generalised heat kernels
Let $K_\alpha(t,x)$ be the (generalised or fractional) heat kernel which corresponds to the fractional heat equation (I'm not sure that's the right name) in $\mathbb R^n$
$$
u_t=(-\Delta)^\alpha u, \...
- 2,933
1
vote
0
answers
93
views
Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case
By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user123457
1
vote
0
answers
103
views
Choosing the weight in a particular definition of Besov spaces
Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" ...
- 5,380
1
vote
0
answers
45
views
How can we show that a solution depends on more than one variable? [closed]
I have obtained theoretically a solution to a nonlinear Schrödinger-type equation in dimension two. I also proved that is not constant. Now, I wonder if it depends on two variables and not only in one,...
- 383
1
vote
3
answers
166
views
Let $u_n\in\mathcal{D}'(\mathbb{R}^n)$ have $u_n\to0$ where $u_n\in C_c^\infty$ have uniformly compact support. Does $u_n\to0$ in $C_c^\infty$? [closed]
Suppose we have functions $u_n\in C_c^\infty(\mathbb{R}^d)$ with support all lying in $B(0,R)$, and suppose $u_n\to 0 $ in $\mathcal{D}'(\mathbb{R}^n)$, i.e. for all $\eta\in C_c^\infty(\mathbb{R}^d)$,...
- 1,247
0
votes
0
answers
83
views
$ 0 $ locates in the continuous spectrum of Schrodinger operators?
This is question is motivated by Non-closed range space of Laplace operators?. We aim to determine what kind of potential will make corresponding schrodinger operators possess non-closed range.
For ...
- 269
0
votes
0
answers
117
views
Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1
2
votes
1
answer
181
views
Estimate of semigroup with dual norm?
Consider a semigroup $(T(t))_{t\in\mathbb{R}^+}$ generated by a densely defined strictly positive symmetric linear operator $A: D(A) \subset X \to X$, where $X$ is a Banach space with norm $\|\cdot\|$....
- 319
4
votes
0
answers
111
views
A reference for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$
Let $\Omega$ be an open domain with nice boundary and $u\in W^{1,p}(\Omega)$. I believe that $|u|^p\in W^{1,1}$ with
$$
\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u
$$
but couldn't find a good ...
- 1,245
2
votes
1
answer
288
views
order of the singularity of a Green's function to the fractional Laplacian
I was looking at a problem which involves the Green's function of a fractional Poisson equation.
To fix notation, let $D\subset \mathbb{R}^n$ very nice, i.e. a hypercube, and
\begin{equation}
\begin{...
- 161
2
votes
0
answers
62
views
Existence and uniqueness for semilinear problem
Consider the following problem:
$$-\Delta u + [(u)^+]^\alpha = 0,$$
where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...
user139844
2
votes
0
answers
190
views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
1
vote
1
answer
2k
views
Lipschitz functions and $W^{1,\infty}$
I am not sure my question is research type, but I am sure I can find here an answer.
So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295:
Theorem 4 (...
- 1,594
1
vote
1
answer
165
views
Morrey condition (integral condition) and (local) Holder condition
Let $x \in \mathbb{R}^n$ and $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$)
$$\limsup_{r \to 0} r^{-\alpha \beta}\frac{...
- 839
1
vote
1
answer
165
views
Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous?
Let $z(x):=\int_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$
Moreover, we know ...
- 536
2
votes
0
answers
77
views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
1
vote
0
answers
196
views
Compact embedding result
Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
- 395
3
votes
1
answer
247
views
Are there fundamental solutions of the laplacian that decay rapidly?
The question
I consider the Laplacian $\Delta = \partial_1^2 + \partial_2^2 + \partial_3^2$ in $\mathbb{R}^3$. By the "standard" fundamental solution of the Laplacian, I mean the function
$$ \...
- 53
0
votes
1
answer
386
views
Functions satisfying Neumann boundary condition
I have a question about functions satisfying a condition.
Let $D \subset \mathbb{R}^d$ be a Lipschitz domain. That is, for each $x \in \partial D$, there exists an open neighborhood $U$ of $x$ in $\...
- 721
1
vote
1
answer
259
views
Heat equation with source term in $L^1$
To simplify, let us work on $Q_T:=[0,T]\times\mathbb{T}^N$ where $\mathbb{T}^N$ is the $N$-th dimensionnal torus.
Consider $(S_n)_n$ a sequence of $L^1(Q_T)$ and $(z_n)_n$ the sequence of solutions ...
- 3,425
3
votes
1
answer
218
views
Short time $L^1$ bounds for semigroups obtained from elliptic operators
Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given ...
- 31
4
votes
0
answers
145
views
An embedding question: Morrey spaces
Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?
- 43.2k
4
votes
3
answers
1k
views
A space of distributions vanishing on the boundary
The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...
- 5,407
1
vote
1
answer
103
views
Is $X = \{ B \in L^\infty(\mathbb R^n,\mathbb R^n): \nabla \cdot B \in L^\infty(\mathbb R^n,\mathbb R^n) \}$ a dense subspace?
The Sobolev space $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ is not dense in $L^\infty(\mathbb R^n,\mathbb R^n)$. In fact the functions in $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ are Lipshitz, and not ...
- 13
2
votes
1
answer
699
views
Schwartz kernel theorem
I would like to understand how the Schwartz kernel theorem works for some more difficult cases and therefore would like to discuss an example from scratch:
Let the Dirichlet Laplacian on the half-...
- 23
4
votes
0
answers
2k
views
Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity
Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
1
vote
1
answer
654
views
Properties of the trace term in the Itō formula
Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where
$U,H$ are separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ is nonnegative ...
- 167
3
votes
1
answer
415
views
Continuity of convolution on L¹
Question
Let $f,g\in L^1(\mathbb{R}^n)$ be nonnegative and assume $\int f\,g \; dx <\infty$. Let $\Phi_\varepsilon$ be the heat kernel at time $\varepsilon$. Denote with $f_\varepsilon = f \ast \...
- 1,081
3
votes
1
answer
670
views
A specific mollified functions in the Sobolev space H^1(R)
Let $u>0$ be in $H^{1}(\mathbb{R})=W^{1,2}(\mathbb{R})$, we know that the set of $C^{\infty}$ functions with compact support are dense in the Sobolev space $H^{1}(\mathbb{R})$. Hence, we have a ...
- 31
4
votes
0
answers
174
views
Constant in trace theorem for balls
Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
user127450
2
votes
1
answer
244
views
Smoothness of distributions defined by oscillation integrals
In M.A. Shubin's book Pseudodifferential Operators and Spectral Theory, we have the following statement.
Let $X\subset\mathbb{R}^n$ be an open set, and fix a symbol $a\in S_{\varrho,\delta}^m(X\...
- 1,247
1
vote
0
answers
299
views
How can I obtain this inequality (from Evan's PDEs)
I am self studying PDEs from Evans' "Partial Differential Equations" textbook. Currently, I am going through Theorem 1 from Section 5.7 (Rellich-Kondrachov compactness theorem) and am having ...
- 185
1
vote
0
answers
52
views
Asymptotically periodic potentials
Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials?
- 69
0
votes
1
answer
272
views
A condition for Laplacian
Let $u\in L^{2}(\mathbb{R}^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(\mathbb{R}^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(\mathbb{R}^{2})$?
Thank you in advance.
- 135
7
votes
1
answer
489
views
When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
2
votes
0
answers
207
views
Smoothing properties of analytic semigroups
Assume $A$ is a second order operator, generator of a positive analytic semigroup on the $L^p$-spaces with $p\in (1,\infty)$ and domain $D(A_p)=W^{2,p}$. Do we have regularity estimates
$\|T_p(t)f\|_{...
- 21
7
votes
2
answers
632
views
Inverse of partial differential operator as a smooth tame map
Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an ...
- 5,824
1
vote
1
answer
98
views
Sign-changing solutions for initial-boundary value problem for $\partial_t u + \partial^4_x u = 0$
Can you point out a reference for the fact that solutions for the initial-boundary value problem associated to $$\partial_t u + \partial^4_x u = 0$$ with $u(0,\cdot) >0$ can change sign (that is, ...
user123672
8
votes
0
answers
260
views
Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$
I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
- 503
2
votes
2
answers
148
views
Solution of hyperbolic equations with $V^*$ data
Let $V\subset H\subset V^*$ a Hilbert triple and consider a 2nd order evolution equation of the form $$u''(t)+Au(t) = f(t)\quad \text{ in }\ L^2(0,T;V^*),$$ where $f\in\ L^2(0,T;H)$.
Can we let $f\in ...
- 123
1
vote
0
answers
78
views
Inequality for fractional power norm (sectorial operators)
How could we prove following inequality:
$\int\limits_{0}^{l} u^{3}(x) dx \leq \sqrt{l} \cdot|| u||_{\frac{1}{2}}^{3}$
where
$ || u ||_{\frac{1}{2}} = ||A^{\frac{1}{2}}(u)||_{L^{2}} + || u ||_{L^{...
- 31