All Questions
1,304 questions
0
votes
2
answers
218
views
Friedrichs/Poincare inequality on $S_n \times (0,\infty)$?
Should I expect the following Friedrichs/Poincare inequality to hold for $u \in C^\infty(S_n \times (0,\infty))$ with $u(x,0) = 0$:
$$\int_{S_n \times (0,\infty)}|u|^2 \leq C\int_{S_n \times (0,\infty)...
- 17
0
votes
1
answer
117
views
How to understand the unique continuation result
Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm
$$
\|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}.
$$
Suppose $K(x) \in C^1\left(\mathbf{R}^...
- 471
0
votes
1
answer
121
views
A simple bilinear estimate
Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that
$\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$.
Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$.
What is the optimal value of $t=t(\...
- 852
0
votes
1
answer
171
views
Looking for English version of a paper of Jean Ginibre
I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...
- 159
0
votes
1
answer
158
views
Oleinik inequality (one-sided Lipschitz condition) implies $BV_{\mathrm{loc}}$ for solution of conservation law
Consider the scalar conservation law
$$u_t+f(u)_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$
where $f \in C^{2}(\mathbb{R})$ is a strictly convex function ($f''>...
user139844
0
votes
1
answer
344
views
Is this PDE solvable?
Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is unit ball. I am trying to solve the following PDE for $f$:
$$\Delta f -\frac{ f }{r^2}+ \frac{ \left. f \right|_{\partial M}}{r^2} = 0, \qquad \text{...
- 969
0
votes
4
answers
1k
views
Does the Leibniz (product) rule hold for the spectral fractional Laplacian?
Does the Leibniz (product) rule hold in some sense for the spectral fractional Laplacian (at least in 1 dimension)?
user139845
0
votes
1
answer
81
views
Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?
Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:
$$u_t = grad[V(u)]$$
For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-...
- 7,789
0
votes
3
answers
320
views
Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]
Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1,...
user81900
0
votes
1
answer
254
views
Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases}
-\...
- 266
0
votes
1
answer
272
views
Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space,
$$A(\mathbb T):= \{f\in L^...
- 1,051
0
votes
1
answer
612
views
Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
- 213
0
votes
1
answer
126
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,143
0
votes
1
answer
140
views
Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
0
votes
1
answer
148
views
Proof of vanishing viscosity error rate
Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$.
What is a ...
user139844
0
votes
1
answer
326
views
Domain of the fractional Laplacian operator
If $u:\mathbb R^n \to \mathbb R$ satisfies $$\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dxdy < \infty,$$
but $u$ is not in $L^2(\mathbb R^n)$, is $(-\Delta)^su$ well-...
- 109
0
votes
1
answer
419
views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
0
votes
1
answer
104
views
Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
0
votes
1
answer
843
views
$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
- 35
0
votes
1
answer
332
views
Sobolev's lemma on manifolds
Let $M$ be a n-dimensional closed submanifold in $\mathbb{R}^m.$ I was looking for a version of Sobolev's lemma saying that for $f \in {W}^{k,2}$ we find a representative of $f \in C^{r}$ satisfying $...
- 3
0
votes
1
answer
130
views
Positive inuequality of Laplacian with a variable coefficient [closed]
Let $0 < a_0 \leq a(x)$ be a smooth function on $\mathbb{T}=[0,2\pi]$, $a(0)=a(2\pi)$, whether it holds that
$$
\int_\mathbb{T} a(x)|\partial_x \varphi|^2 \geq \int_\mathbb{T}\frac{\partial^2_x a }...
- 425
0
votes
1
answer
475
views
uniqueness for Poisson equation in R^d with mildly regular data
I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...
- 5,380
0
votes
1
answer
643
views
Is this (interpolation) inequality right?
Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap (\cap_{k=1}^\infty L^{5/3}(0,T;C^k(\bar{\Omega})))$.
Question: Can we say that $F$ ...
- 61
0
votes
2
answers
2k
views
fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
- 1
0
votes
1
answer
102
views
Limit of minimizers of a class of functionals
Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional
$$
\mathcal{F}(u)=\int_\Omega(|\nabla u|^2+h^{-1}|u-u_0|^2) \, dx
$$
where $ h>0 $ is a parameter and $ ...
- 1,219
0
votes
1
answer
154
views
Finite dimensionality of a subspace
Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...
- 4,143
0
votes
1
answer
130
views
Riesz transform after linear transformation
I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation
$$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$
I ended up with ...
- 159
0
votes
1
answer
166
views
Relationship between elliptic and parabolic problems and their discretizations
Let us consider the fully nonlinear problem
$$
\begin{cases}
F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega \\
u=0 & \text{ in } \partial \Omega
\end{cases}
$$
Suppose that we know that the ...
0
votes
1
answer
125
views
Bounding integral expression with Sobolev norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
0
votes
1
answer
417
views
Application of Green function for non linear PDE [closed]
In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu_f=f$ where $u_f=G*f$.
Is the same thing hold for ...
- 309
0
votes
1
answer
98
views
Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?
Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$
where the coefficient $a$ are smooth and bounded and $D$ is a bounded
and smooth domain of $\mathbb R^d$
$$
\begin{...
- 446
0
votes
1
answer
147
views
The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]
Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
- 47
0
votes
1
answer
116
views
Fractional Laplacian and support
Let $u:\mathbb [-1,1] \to \mathbb R$ such that $\mathrm{supp}(u) \subset B_{1/2}(0)$. Under what assumptions on $u$ does it hold $$\mathrm{supp}\Big((-\Delta)^s u\Big) \subset B_{1/2}(0),$$
where $(-\...
- 217
0
votes
1
answer
76
views
Reverse Hölder type inequality for the Laplacian raised to a power
I am studying integrals of the form $\int (\Delta \rho)^{\alpha} f^{\beta}$ where $0<\alpha < 1, \beta \geq 0$ and $\rho, f \in C_c^{\infty}(\mathbb{R}^n).$ My goal is to obtain lower bound on ...
- 537
0
votes
2
answers
238
views
Fractional Laplacian of $(a-x)_+^\alpha$ in $(0,1)$
How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$?
Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
- 109
0
votes
1
answer
120
views
Density property fractional heat kernel
Let us consider $$p_t^{(n+2)}(\tilde x) , $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (x_1, \ldots, x_n, 0, 0) \in \mathbb R^{n+2}$, and $p_t^{(n)}(x)$ is the heat kernel for $(-\...
- 109
0
votes
1
answer
89
views
Does $\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le C$ imply $\Vert u_x (t,\cdot) \Vert_{L^2(\mathbb R)} \le C$ in the heat equation?
For the parabolic equation
$$u_t + f(u)_x - u_{xx} = 0$$
one has
$$\Vert u(t,\cdot) \Vert_{L^2(\mathbb R)} + 2\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le \Vert u(0,\cdot) \Vert_{L^2(\mathbb R)}.$$...
user140746
0
votes
1
answer
380
views
How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?
Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
- 550
0
votes
1
answer
233
views
Existence of subsequences convergence with weak topology
Let $\left\{ {{\varphi _n}} \right\}$ is the sequence bounded in ${L^\infty }\left( {0,\infty ;H_0^1\left( {0,1} \right)} \right)$. Is there exists $\varphi \in {L^\infty }\left( {0,\infty ;H_0^1\...
- 183
0
votes
1
answer
582
views
$L^2$ bound and interpolation of Hölder norm
Consider the function
$$F(x):=\int_{\mathbb R} f(t+x)f(t-x) \ dt .$$
Clearly, we have by Cauchy-Schwarz
$$\vert F(x) \vert\le \Vert f \Vert^2_{L^2} $$
$$\vert F'(x)\vert\le 2\Vert f' \Vert_{L^2} \...
user144765
0
votes
1
answer
175
views
Accessible reference for (scattering) $\Psi DO$'s on manifolds
I am currently trying to understand Hassell, Tao, and Wunsch's paper on Strichartz estimates on non-trapping asymptotically conic manifolds, however, my understanding of pseudodifferential operators ...
- 1,247
0
votes
1
answer
385
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
0
votes
1
answer
104
views
Poisson Equation across a Hypersurface [closed]
Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem
$ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
- 4,143
0
votes
1
answer
218
views
Heat semigroup dissipative
Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition.
On $L^2$ it would be completely trivial, but ...
- 167
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
1
answer
247
views
Gradient bounds on Newtonian potentials
Suppose $N \ge 3$ and let $\Phi(x):= C_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$.
Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \...
- 1,385
0
votes
1
answer
445
views
Domain of the Stokes operator
Let
$\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$)
$\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$
$\mathcal H:=\overline{\mathfrak ...
- 167
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
0
votes
1
answer
491
views
Is this set of function belongs to $L^\infty$?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal H^{N-...
- 679
0
votes
1
answer
781
views
How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?
Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
$\...
- 679