All Questions
Tagged with real-analysis ap.analysis-of-pdes
569 questions
11
votes
2
answers
1k
views
Harmonic oscillator in spherical coordinates
It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be ...
- 113
2
votes
0
answers
297
views
Examples of harmonic functions
I am looking for non-trivial examples (in the sense to be described below) of harmonic functions, which can be represented as cubes of smooth functions ($C^1$ would be also OK if this is important).
...
- 246
0
votes
1
answer
142
views
Showing that a Gaussian achieves equality in a logarithmic Sobolev inequality
Consider the following Logarithmic Sobolev inequality on page 210 of Analysis by by Elliott H. Lieb, Michael Loss (GSM 14): for $f\in H^1(\mathbb R^n),$ and $a>0$ any positive number,
$$
\frac{a^2}{...
- 1,755
0
votes
2
answers
241
views
Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$
Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions).
Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.
Link to the problem (paper "...
- 698
1
vote
1
answer
672
views
The first eigenfunction of fractional laplacian
Let $\Omega$ be bounded and smooth domain in $\mathbb{R}^n$, $s\in(0,1)$, $e_1\in \mathbb{H}^s(\Omega)$ the first eigenfunction of fractional laplacian $(-\Delta)^s$ with eigenvalue $\lambda_1>0$, ...
- 339
2
votes
0
answers
153
views
unique continuation in a disk
Let $D$ be the unit disc in $\mathbb R^2$ centered at the origin. Let $w \in C^{\infty}_c(D)$ satisfy
$$ (1-r^2)^2\Delta w +w =0.$$
Prove that $w \equiv 0$.
- 4,143
0
votes
0
answers
981
views
Weak $H^1$ convergence implies strong $L^p_{\text{loc}}$ convergence
On page 210 of Analysis by by Elliott H. Lieb, Michael Loss (GSM 14), there is a theorem stating that if $f_j \to f$ weakly in $H^1(\mathbb R^n)$ then $\mathbf 1_A f_j \to \mathbf 1_A f$ strongly in $...
- 1,755
3
votes
0
answers
96
views
A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$
Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves
$$\begin{cases} -\Delta ...
- 969
4
votes
1
answer
213
views
Mapping properties of backward and forward heat equation
In a previous question on mathoverflow, I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The ...
- 536
6
votes
2
answers
1k
views
Properties of heat equation
** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...
user121558
8
votes
1
answer
380
views
Lavrentiev phenomenon between $C^1$ and Lipschitz
Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere)
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that
$$
\inf_{y\in Lip([a,b])}F(y)<\inf_{...
- 1,309
2
votes
0
answers
253
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
- 197
5
votes
1
answer
260
views
Approximate Sobolev embedding
It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$
We then define ...
- 124
-3
votes
1
answer
200
views
Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$
The Euler equations are given as $$ \pmb{u}_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$
Where $$u = [u_1,u_2,\ldots u_n]^T$$
Now I want to rewrite these same equations but with a new ...
- 59
11
votes
0
answers
2k
views
A question on trig series
Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$
$$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
- 903
3
votes
1
answer
846
views
Aleksandrov maximum principle for semi-convex function
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
1
vote
0
answers
59
views
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
4
votes
0
answers
135
views
Zygmund class, Schwartz class and Littlewood-Paley projection operators
I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions:
Consider the Zygmund class of functions defined as ...
- 349
-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
6
votes
0
answers
151
views
Gap between consecutive Dirichlet eigenvalues
Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say
$$ -\Delta \phi_k = ...
- 4,143
2
votes
0
answers
144
views
Does this geometric PDE have a solution?
Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes.
Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
- 6,741
7
votes
1
answer
735
views
Conserved positive charge for a PDE
Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$:
\begin{equation}
\frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
- 642
1
vote
0
answers
70
views
Examples of reaction-diffusion systems with analytical solutions
I want to study how some numerical schemes work on $2$-dimensional reaction-diffusion systems on rectangles with Neumann Boundary conditions and I search for a while for a problem of the form:
$$\...
- 1,759
0
votes
1
answer
185
views
"Arc" length parametrization for surfaces
If we have a function $\phi:\Omega\subseteq\mathbb{R}^2\to\mathbb{R}$, $\phi\in C^2(\Omega)$ is there a way to find a function $d:\Omega\to\mathbb{R}, d\in C^2(\Omega)$ so that:
$|\nabla d(x,y)|=1,\ \...
- 1,759
2
votes
1
answer
291
views
An inequality involving fractional Laplacian
I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function):
$$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$
...
- 339
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
3
votes
1
answer
266
views
If subharmonic functions converge weakly to a subharmonic limit, why do their smoothings converge uniformly on compact sets?
Let $u_k$ be a sequence of subharmonic functions on an open set $X$ and $\psi_\delta$ a family of standard mollifiers with compact support. Hörmander claims in The Analysis of Linear Partial ...
- 1,141
3
votes
1
answer
1k
views
Continuation (extension) of harmonic functions
Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
- 4,143
4
votes
1
answer
1k
views
Convergence of semi convex functions
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
3
votes
1
answer
147
views
wave equation with vanishing trace at infinity
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
\Box u+qu=0\,\quad &\text{on $(0,\infty)\times \...
- 4,143
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
2
votes
1
answer
246
views
Eigenfunctions of the fractional Laplacian are smooth?
Let $\Omega\subset\mathbb{R}^n$ open, bounded with smooth boundary, let $s\in(0,1)$. I know that the fractional Laplacian has a sequence of eigenfunctions $\{e_k\}_{k\in\mathbb{N}}\subset H^s(\mathbb{...
- 339
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
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
1
vote
1
answer
123
views
Are there $f, g$ such that $\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$ where $f'=\frac{\partial f}{\partial \theta}$
Let $f,g$ are the functions of $S^{1}$. Are there $f, g$ such that
$$\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$$
where $f'=\frac{\partial f}{\partial \theta}$?
- 153
1
vote
0
answers
257
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
$$
|\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s}...
- 99
3
votes
1
answer
137
views
Variational formulation of an elliptic pde
Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, what is the variational formulation of the following problem:
$$ \text{div}(y^a\nabla_{x,y}V)=0,\quad\text{on }\mathbb{R}^n\times(0,\infty),$$
$$ V(x,...
- 339
4
votes
1
answer
216
views
Finding super(sub)-harmonic functions for an elliptic operator
I am looking for a super(sub) harmonic function for an elliptic operator.
Let $n$ be a positive integer. We denote by $(\cdot,\cdot)$ and $|\cdot|$ the standard inner product and norm on $\mathbb{R}^n$...
- 721
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
0
answers
136
views
Fractional Laplacian of smooth cut off functions
Suppose we have a smooth compactly supported function $\phi\in C^{\infty}_c(B_\epsilon(0))$ such that $0\leq \phi \leq 1$, $\phi\equiv 1$ on the unit ball and $\phi$ vanishes outside $B_\epsilon(0).$
...
- 537
1
vote
1
answer
196
views
Solving an equation with fractional laplacian [closed]
Let $s\in (0,1)$, how i can solve the equation:
$$ (-\Delta)^su=1,\quad\text{in}\quad(-1,1)?$$
I have no idea, any help would be appreciated.
- 339
2
votes
0
answers
58
views
Uniqueness for a certain semilinear equation
Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation
$$
\begin{aligned}
\begin{...
- 4,143
0
votes
0
answers
239
views
Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
2
votes
1
answer
165
views
Is Sommerfeld radiation condition invariant under translations?
A smooth function $U:\mathbb{R}^3\setminus B_{r_0}(0)\to\mathbb{C}$ (for some $r_0>0$) satisfies the Sommerfeld Radiation Condition with index $k$, denoted $U\in \texttt{SRC}$, whenever
$$
\lim_{r\...
- 146
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?
$$(-\Delta)^\alpha u(x) + M u'(x) + K u(x) + C = 0 \quad \text{ in } (0,\infty)$$
$$u(x) = a, \quad ...
- 99
10
votes
2
answers
371
views
Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?
Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
- 311
3
votes
2
answers
410
views
Is a bounded sequence of $H^1(\Omega)$ tight?
Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $ (u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$.
Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: ...
- 1,101
1
vote
1
answer
101
views
Poincaré-type Inequality
In Lieb's paper "On the lowest eigenvalue of the Laplacian
for the intersection of two domains" one finds the following remark:
Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\...
- 69
2
votes
0
answers
164
views
What are (the different aspects of) harmonic analysis good for?
Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have ...
- 2,071
4
votes
0
answers
151
views
Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$
Let $\Omega$ be a bounded domain in $\Bbb R^n$. Define
$$
\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \},
$$
i.e. it the ring of thickness $r$ at the boundary of $\Omega$. Intuitively, ...
- 1,245