All Questions
1,304 questions
1
vote
0
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59
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Identification of a limit point of a sequence of solution of ODE
Let $v^0$ and $v^1$ be the following vector fields over $\big(\mathbb{R}_+^*\big)^3$: for $x\in\big(\mathbb{R}_+^*\big)^3$ and $1\leq i\leq 3$,
\begin{align*}
& v^0_i(x)=x_i(x_{i-1}-x_{i+1}) \\
&...
- 449
7
votes
0
answers
351
views
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 ...
- 161
2
votes
1
answer
205
views
Estimates for an elliptic PDE
Let's say I have an equation of the form $\Delta A = J$ where $J=u\nabla u + A|u|^2$ (Clarification: We are on $\mathbb{R}^3$ and $u$ is assumed to be in $H^1(\mathbb{R}^3)$). Then I could simply ...
- 469
8
votes
1
answer
498
views
Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?
Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space:
$$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...
- 287
0
votes
0
answers
164
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Bound for the $\ell^3$ norm for the one-dimensional propagator
Problem: In Appendix (A.6) of Main paper is written
$$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
\frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
\...
- 101
5
votes
2
answers
939
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Second order differentiability of subharmonic function almost everywhere?
The following general definition of subharmonic function comes from the classical text book [elliptic partial differential equations of second order] by Gilbarg and Trudinger.
We call a function $u$ ...
- 1,350
0
votes
0
answers
70
views
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 $\...
- 99
14
votes
4
answers
1k
views
Is every continuous microlocal operator a pseudo-differential operator?
Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space.
Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal.
By being microlocal I mean that the wave ...
- 8,086
-1
votes
1
answer
114
views
Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]
Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
- 217
2
votes
1
answer
327
views
Fractional Laplacian and convolution $(-\Delta)^\alpha (u \ast \eta_\epsilon) = (-\Delta)^\alpha u \ast \eta_\epsilon$?
For $u \in L^\infty(\mathbb R)$ and $\eta_\epsilon$ mollifier, it is well-known that for the (distributional) derivative it holds that $(u \ast \eta_\epsilon)' = u'\ast \eta_\epsilon$.
Is it also true ...
- 839
4
votes
1
answer
614
views
Application of uniform boundedness (Banach-Steinhaus) principle
$\DeclareMathOperator{\loc}{\mathrm{loc}}$This is from Lemarié-Rieusset's book "The Navier-Stokes problem in the 21st century", from the proof of a result about stationary solutions to ...
- 287
3
votes
1
answer
192
views
Laplace eigenfunction on a polygonal domain symmetric about an axis
Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
- 33
2
votes
1
answer
278
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Weak-star approximation of smooth functions in weak $L^p$-space
It is well known that the weak space $L^{p,\infty}$ has less density property contrary to standard $L^p$ space. Related to this one, I'm struggling to prove the following statement which is given in ...
- 323
0
votes
1
answer
147
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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
0
answers
57
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Existence of measure-preserving Lagrange flow for inhomogeneous transport equation
I asked this question on stackexchange:
Let us consider the Cauchy problem for the transport equation
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
- 173
1
vote
1
answer
187
views
Fractional Laplacian equation on a ball and explicit solutions
Let us consider
\begin{align*}
(-\Delta)^s u &= 0 && x \in B_r(0) \\
u&=0 && x \in \mathbb R^N \setminus B_r(0),
\end{align*}
where $$
(-\Delta)^s u(x) = \int_{\mathbb{R}^N} \...
- 99
4
votes
0
answers
146
views
Fourier transform without characters (Eigenfunctions of an operator)
Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form
$$
V(x) = V_0 \mathbf 1_{[-a,a]}(x),
$$
where $\mathbf 1_{[-a,a]}$ denotes the ...
- 1,755
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 $\mathbb R^d$
$$
\begin{...
- 446
1
vote
1
answer
2k
views
Sobolev embedding in the space of continuous functions [duplicate]
Let $I = \mathbb{R}$ and let $W^{1,2}(I,\mathbb{R})$ be the Sobolev space of function from $I$ to $\mathbb{R}$ (one time weakly differentiable and contained in $L^{2}$) and $C^{0}(I,\mathbb{R})$ be ...
- 11
10
votes
1
answer
2k
views
Chain rule for distributional derivative
Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$).
Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
- 145
2
votes
0
answers
65
views
Regularity of solution to first order time dependent variational problem
Consider the following first order evolution problem over some regular bounded domain $\Omega\subset\Bbb R^d$
$$\frac{\partial\phi}{\partial t}(\mathbf{x},t) +\vec V(\mathbf{x},t)\cdot \nabla\phi(\...
- 153
5
votes
1
answer
462
views
Backward uniqueness for a heat equation with a drift
Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or ...
- 931
4
votes
0
answers
143
views
If theorem valid for compactly supported distribution then is it also valid for tempered distribution?
I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.
For instance,
Theorem: Any $A \in \Psi^{m}$ ...
- 309
2
votes
0
answers
84
views
Method of characteristics and explicit formula for an IBVP for the transport equation
Let us assume $c:(0,1) \to (\epsilon,+\infty)$ and consider the PDE
$$
\begin{cases}
u_t+c(x)u_x = 0, \\
u(0,x) = g(x) \\
u(t,0) = f(t)
\end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...
- 839
7
votes
2
answers
508
views
Making the Fourier transform quantitative
I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand ...
- 225
22
votes
4
answers
3k
views
When to use more exciting function spaces than ordinary Sobolev spaces?
In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions.
For example, Sobolev spaces $L^2(0,T;H^...
- 405
1
vote
0
answers
103
views
Regularity results for non uniform elliptic equation
I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance,
$$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...
- 309
5
votes
3
answers
1k
views
Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$
Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$
with initial data $u(0,\cdot) = u_0$.
Fix $\alpha >0$. How can I estimate (without using explicitly ...
- 839
5
votes
1
answer
153
views
Why is density and separability needed for uniqueness of weak (time) derivatives?
Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
2
votes
0
answers
162
views
Explicit computation of a norm in context of operator-semigroups and differential equations
I am interested in the explicit calculation of the following norm $\vert \cdot \vert$.
Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter ...
- 185
5
votes
1
answer
166
views
Strong maximum principle for a PDE with coefficient in $L^1$
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$. Set $N = \frac{2n}{n-2}$. I am interested in the following equation:
$$
-\Delta \phi + R \phi + \phi^{N-1} = 0
$$
...
- 1,263
1
vote
0
answers
144
views
Liouville theorem for elliptic equation with advection term
How can one prove that any $L^2$ solution of
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$
is zero if $a(x)$ is a divergence-free vector field such that
$\int |\...
- 99
0
votes
1
answer
328
views
Domain of the fractional Laplacian operator
If $u:\mathbb R^n \to \mathbb R$ satisfies $$\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dxdy < \infty,$$
but $u$ is not in $L^2(\mathbb R^n)$, is $(-\Delta)^su$ well-...
- 109
3
votes
0
answers
467
views
Opposite of the curl operator and Biot-Savart kernel
Note: I just realized that using $\omega$ and $w$ might not have been the smartest choice of notation -- Sorry about that.
Let $\renewcommand{\div}{\operatorname{\div}}Q_0, Q_1$ be two real numbers, $...
- 1,326
12
votes
2
answers
5k
views
Where was/is Compensated Compactness used?
This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
- 261
1
vote
0
answers
259
views
Cut-off function and fractional Laplacian
Is there a smooth function $u$ such that $u = 1$ in $B_r(0)$, $u=0$ in $\mathbb R^N \setminus B_{2r}(0)$, and
$$
|\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s}...
- 99
1
vote
1
answer
134
views
Integrability of fractional heat kernel
In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (...
- 109
2
votes
0
answers
658
views
Convergence of operator in norm resolvent sense and their eigenvectors
Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
- 21
1
vote
1
answer
137
views
Adjoint operator of OU generator
The generator an OU process is given by
$$A = \operatorname{tr}(QD^2)+\langle Bx,D\rangle.$$
This one possesses an invariant measure given by
$$d\mu(x) = b(x) \ dx \text{ with } b(x) = \frac{1}{(4\pi)^...
- 192
0
votes
1
answer
120
views
Density property fractional heat kernel
Let us consider $$p_t^{(n+2)}(\tilde x) , $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (x_1, \ldots, x_n, 0, 0) \in \mathbb R^{n+2}$, and $p_t^{(n)}(x)$ is the heat kernel for $(-\...
- 109
3
votes
1
answer
763
views
Approximate Fréchet derivative of PDE solution operator
Let's assume that $F$ is the solution operator of some PDE problem, i.e. $F$ maps some parameter or boundary condition function to the solution of a given PDE. Let also be $\Pi$ be the evaluation ...
- 375
3
votes
2
answers
620
views
Interior smooth regularity
I recently read the PDE book of L. Evans, and in its chapter 6 some kinds of regularities of second-order elliptic equations were discussed. My question is about its proof of interior smooth ...
- 402
2
votes
1
answer
141
views
A quadratic estimate for the Schrödinger equation
It is known that the bilinear Strichartz estimate
$$\Vert e^{it\Delta}u_0\cdot \overline{e^{it\Delta}v_0}\Vert_{L_t^2L^2(\mathbb{R}^d)}\lesssim_{\delta} \Vert u_0\Vert_{H^{-1/2+\delta}(\mathbb{R}^d)}\...
- 943
0
votes
0
answers
239
views
Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
6
votes
0
answers
113
views
A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
- 163
1
vote
1
answer
131
views
Fractional Laplacian problem on half-line
Is it possible to obtain an explicit solution for the following fractional problem on the half-line?
$$(-\Delta)^\alpha u(x) + M u'(x) + K u(x) + C = 0 \quad \text{ in } (0,\infty)$$
$$u(x) = a, \quad ...
- 99
3
votes
0
answers
382
views
Green's function for Robin boundary condition
Let $\Omega$ be a bounded subset in $\mathbb{R}^n$, $n \ge 3$, with smooth boundary $\partial \Omega$. Assume given $a^{ij} \in W^{2, p}(\Omega)$ with $a^{ij} = a^{ji}$, $f \in L^p(\Omega)$ and $g \in ...
- 1,263
2
votes
1
answer
225
views
Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)
Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ ...
- 839
4
votes
1
answer
729
views
Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds
On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
- 43
9
votes
1
answer
2k
views
Density of smooth functions on Hölder spaces
The following result is often cited without reference in the context of PDEs:
Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
- 727