All Questions
Tagged with real-analysis ap.analysis-of-pdes
569 questions
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Continuation of a smooth function
Setting
Suppose I have two bounded open domains $\Omega' \subset \Omega \subset \mathbb{R}^n$ (I'm particularly interested in case n = 2 or n = 3). We assume that all boundaries of domains are $C^\...
- 4,713
2
votes
1
answer
128
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On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 28k
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votes
1
answer
217
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About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain
I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$:
Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
- 111
4
votes
2
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364
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Nontrivial invariant transformations for heat equations
It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by
$$ v(r,\theta) = u(\frac{1}{r},\theta)$$
is also harmonic for $r>0$. Note that the Kelvin ...
- 4,143
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0
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28
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Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
- 21
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84
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About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
7
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2
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627
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Elliptic regularity on manifolds: Is this true?
Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
- 28k
4
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1
answer
279
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Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
- 24.2k
0
votes
1
answer
140
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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 ...
- 128k
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220
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Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
- 517
7
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2
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567
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Intuition for Agmon-Douglis-Nirenberg ellipticity
First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.
I am trying to understand the definition of ellipticity of systems due to ...
- 21.5k
1
vote
1
answer
113
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An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
- 6,035
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0
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53
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Non-linearity of viscosity solutions
I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 6,367
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52
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Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 3,065
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1
answer
110
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Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$
Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying
$$
0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2
$$
where $\Delta$ denotes the Laplace ...
- 109k
0
votes
1
answer
185
views
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
- 10.3k
3
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83
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Embedding theorems for Dini continuous functions
Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
- 3,388
8
votes
1
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496
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A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
- 6,035
2
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0
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116
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Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives
If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$?
It is clear that $(\nabla_x^2 + \nabla_y^2) f \...
- 6,367
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votes
1
answer
507
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Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 684
23
votes
5
answers
2k
views
PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
- 2,689
5
votes
0
answers
213
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,429
2
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0
answers
120
views
Closure of Laplacian
Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator
$$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$
There are two ...
- 1,171
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0
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120
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Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$
I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$:
$$\frac{1}...
- 1
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0
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180
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Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
1
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0
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113
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Computing a limit for the Weierstrass function
Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
- 4,143
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votes
1
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143
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An estimate of the integral of the higher order derivative of a bump function
Let $\kappa_1>0$, $\beta\in [0, 1]$ and $b: \mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ such that for all $t\ge0$ and $x,y \in \mathbb R^d$ we have $|b(t, 0)| \le \kappa_1$ and $|b(t, x) - b(t, ...
- 128k
0
votes
2
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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 $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
CommunityBot
- 1
1
vote
1
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145
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Convolution with the Jacobi Theta-function on "both the space and time variables" - still jointly smooth?
Let $\Theta(x,t)$ be the Jacobi-Theta function:
\begin{equation}
\Theta(x,t):=1+\sum_{n=1}^\infty e^{-\pi n^2 t} \cos(2\pi n x)
\end{equation}
Usually, the heat equation with the periodic boundary ...
- 2,858
0
votes
0
answers
102
views
Asking a reference about the $p$-Laplacian of $|\nabla u|^p$
It is well-known that for a harmonic function $u$, i.e.
$$ \Delta u=0, $$
the quantity $|\nabla u|^2$ is subharmonic, i.e.
$$\Delta (|\nabla u|^2) \geq 0. $$
Reason:
$$\Delta (|\nabla u|^2)= 2 \nabla (...
- 6,367
1
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1
answer
153
views
How to show that $ u $ is vanishing in $ \mathbb{R}^3\setminus B_1 $?
I come across an interesting question.
Let $ B_r=\{x\in\mathbb{R}^3:|x|\leq r\} $ be the ball in $ \mathbb{R}^3 $ with radius $ r $. Assume that $ u \in C(\mathbb{R}^3\setminus B_1) $ satisfies
$$
\...
- 6,035
1
vote
1
answer
60
views
Are there $f,h$ such that $h$ is Lipschitz, $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(h(t), x)$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Then
$$
\partial_t g(t, x)...
- 128k
1
vote
1
answer
93
views
Is there $f$ such that $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(Ct, x)$ for all $t>0$ and $x \in \mathbb R^d$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Then
$$
\partial_t g(t, x)...
- 128k
3
votes
1
answer
96
views
Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}{t_1} \le C_1 \frac{g(C_2 t_2, \cdot)}{\sqrt{t_2}}$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Let $0 < t_1 < t_2 &...
- 128k
8
votes
3
answers
701
views
Regularity of Newtonian potential along smooth boundary
Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define
$$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$
Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?
...
- 4,899
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118
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If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?
The question is as in the title.
Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
1
vote
1
answer
160
views
On an integral equation
Let $B: C^{\infty}([0,1]^3)$ satisfy
$$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$
Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation:
$$ \int_0^1 f(t,x)\,dx + \int_0^t\...
- 6,696
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0
answers
75
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Morse functions on subset $\bar \Omega$ of $\mathbb {R}^d$ and its level sets
Let $f$ be a $C^{2}(\bar{\Omega})$ Morse function, where $\Omega$ is a bounded open set of $\mathbb{R}^d$: this means that
$$
\begin{cases}
f(x) = 0 \\
\nabla f (x) \neq 0
\end{cases}\text{ on }\...
- 6,367
0
votes
1
answer
154
views
Finite dimensionality of a subspace
Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...
- 128k
4
votes
1
answer
172
views
Existence of a domain with simple Dirichlet eigenvalues
Let $g$ be a smooth Riemannian metric on $\mathbb R^3$ that coincides with the Euclidean metric outside a compact set $K$. Does there exist some domain $\Omega$ with smooth boundary such that $K \...
- 829
2
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0
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75
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On Dirichlet eigenfunctions of a domain
Given any bounded domain $\Omega\subset \mathbb R^n$, $n\geq 2$, with a Lipschitz boundary, let $\{(\lambda_k,\phi_k)\}_{k\in \mathbb N}$ be the Dirichlet eigenvalues and eigenfunctions of $-\Delta$ ...
- 4,143
3
votes
2
answers
382
views
Singular support: equivalent definition
Let $U\subset\mathbb{R}^{d}$ be an open set. The singular support of a distribution $u\in\mathcal{D}^{\prime}(U)$ is defined to be the compliment of the set of points, which have a neighbourhood in ...
- 7,817
0
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1
answer
72
views
Orthogonality to a one parameter family of eigenfunctions
Let $\rho>0$ be a smooth realvalued function such that $\rho=1$ outside the unit interval $(-1,1)$. For each $t>0$, let us denote by $\{\lambda_n(t)\}_{n=1}^{\infty}$ and $\{\phi_n(t;x)\}_{n=1}^{...
- 188k
6
votes
2
answers
326
views
Looking for references to study $U^p$ and $V^p$ spaces
I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?
Edited
The ...
- 1,075
5
votes
0
answers
417
views
All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
- 1,075
5
votes
1
answer
216
views
Bounds on dimension of a subspace
Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...
- 61.9k
3
votes
1
answer
275
views
Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?
$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
CommunityBot
- 1
2
votes
2
answers
281
views
Most general reverse Hölder inequality for polynomials
Theorem. Let $m$ be an integer and $P_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P_m$,
$$\|p\|_{L^\infty(a,b)} \...
- 981
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votes
1
answer
125
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Why is this integrability condition needed for uniqueness in the continuity equation?
I am reading about the uniqueness problem for the continuity equation $\partial_t \mu_t + div_x (b \mu_t)=0$ in the lecture notes by Ambrosio (here: https://warwick.ac.uk/fac/sci/maths/research/events/...
- 39k
0
votes
0
answers
142
views
Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1