**0**

votes

**0**answers

15 views

### Splitting the region and estimating fractional Sobolev norms

x-post from math.stackexchange (http://math.stackexchange.com/q/1836766/349671), since the question arose from reading through a scientific paper:
I've been reading the paper "On the Bourgain, Brezis,...

**1**

vote

**0**answers

70 views

### Hausdorff Dimension of Exceptional Set for Carleson's Theorem

In Mattila's book Fourier Analysis and Hausdorff Dimension, Mattila presents a result of Barcelo, Bennett, Carbery, and Rogers about convergence of solutions of the Schrodinger equation to the initial ...

**5**

votes

**1**answer

62 views

### Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that
$$
\lvert u\rvert_s = \...

**5**

votes

**3**answers

160 views

### Morrey's inequality for Sobolev spaces of fractional order

Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that
$$
\|u\|_{H^s}^2=\sum_{k\...

**3**

votes

**0**answers

53 views

### $W^{1, p}(M, N)$ path-connected if and only if $C^0(M, N)$ is path-connected

I'm asked to show that for compact, smooth Riemmanian manifolds $M$, $N$ we have that $W^{1,p}(M,N)$ is path-connected if and only if $C^0(M,N)$ is path-connected. The theorem (0.1) is taken from "...

**4**

votes

**1**answer

167 views

### Density argument with Schwartz functions?

I was wondering whether the Schwartz functions are also dense in
$$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$
...

**0**

votes

**0**answers

69 views

### compact injection

Put:
$D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$
Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...

**0**

votes

**0**answers

43 views

### stokes-equation estimate in $L^2(0,T,L^\frac{3}{2}(\Omega))$

I'm interested in the default Stokes-system, e.g.
$ \frac{\partial}{\partial t} u - \Delta u + \nabla p = f \; \text{in} \; \Omega$
$ \nabla \cdot u = 0 \; \text{in} \; \Omega$
$ u = 0 \; \text{on} ...

**3**

votes

**0**answers

50 views

### Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...

**3**

votes

**1**answer

185 views

### Critical case of Sobolev Embedding

I got stuck in the following lemma:
Lemma: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q>1$.
As we know this is ...

**1**

vote

**0**answers

47 views

### Sobolev norm of a composition with a singular homeo

Let $H_p^t(\mathbb{R})$ be a fractional Sobolev space with the standard norm. The with $p>1$, $0<t<1$. Take some smooth $\phi$ from this space. Let $T$ be an ivertible homeomorphism of $\...

**4**

votes

**1**answer

88 views

### Does the Nash inequality hold on manifolds with Lipschitz boundary?

Let $N$ be a smooth manifold without boundary of dimension $n$. $M$ is a manifold with Lipschitz boundary if $M \subset N$, $M$ and $N$ are of the same dimension, and in the charts of $N$, the ...

**0**

votes

**0**answers

51 views

### Is $\Delta u+f\in (H^1(\Omega))^*$ with $u\in H^1_0(\Omega)$ and $f\in L^2(\Omega)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...

**0**

votes

**0**answers

60 views

### Inverse trace theorem for partial trace

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that $u|_{\...

**3**

votes

**2**answers

236 views

### Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...

**2**

votes

**1**answer

93 views

### Are compactly supported continuous functions dense in the Continuous functions of Sobolev space? [closed]

I have a question about Sobolev space.
Let $\Omega$ be an open subset of $\mathbb{R}^{d}$,
we consider the Sobolev space
$H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), j=1,...

**1**

vote

**0**answers

107 views

### Regularity of a Dirichlet form

I have a question about Dirichlet form.
Let $\Omega$ be an Euclidean domain of $\mathbb{R}^{N}$ and
$X=\bar{\Omega}$. The measure $m$ on the Borel
$\sigma$ algebra $\mathcal{B}(X)$ is given by $m(A)...

**1**

vote

**0**answers

36 views

### Trace of $u$ on bottom edge of a square if $u_x=0$ inside the square

I want to show that:
Let $\Omega =(0,1)\times (0,1)$. For $u \in H^1(\Omega)$, if $u_x=0$ a.e. in $\Omega$, then the trace of $u$ on bottom edge $y=0$, i.e., $u\left|_{y=0}\right.$, is a constant.
...

**1**

vote

**1**answer

64 views

### The dependence of constant in a trace theorem on the diameter of domain

The trace theorem says for a nice domain, say bounded Lipchitz domain, $\Omega$, we have
$$\left\Vert Tu \right\Vert_{H^{1/2}(\partial \Omega)} \leq C \left\Vert u \right\Vert_{H^1(\Omega)},$$
where $...

**4**

votes

**1**answer

263 views

### $\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction ...

**1**

vote

**0**answers

71 views

### How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?

Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$.
Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in H^{1/2}(\...

**1**

vote

**0**answers

68 views

### An H2 estimate for Helmholtz equation

How to show the following statement?
Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation,
$$
-\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u \...

**2**

votes

**1**answer

117 views

### condition for $f \in H^{1/2} \cap C^0$

I need to show that for $f \in H^{1/2}(S^1) \cap C^0(S^1)$ we have :
$$ \iint\limits_{S^1 \times S^1}{ \frac{|f(x)-f(y)|^2}{\sin^2(\pi(x-y))}dx dy}< + \infty $$
and that, conversely, if $f$ is a ...

**1**

vote

**0**answers

23 views

### Approximation error estimate

I would like to find a good reference for the following or a similar, probably well-known, approximation error result:
Let $\Omega\subset \mathbb{R}^d$ be bounded, $p\in [1,\infty]$, $l, m\in \mathbb{...

**2**

votes

**0**answers

75 views

### Getting an a priori energy estimate from PDE weak formulation

On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying
$$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int uv$...

**3**

votes

**1**answer

93 views

### $L^p$ norm of boundary values of holomorphic function

I am looking for an estimate of the following form:
Suppose that $D\subset \mathbb{C}$ is a simply connected domain. Suppose that $F$ is holomorphic and bounded on $D$ and can be holomorphically ...

**1**

vote

**1**answer

107 views

### Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan.
I have a problem understanding one step ...

**0**

votes

**0**answers

60 views

### Sobolev function and capacity

It seems that there is a theorem which claims that
If $u\in L_1^2(B_1\setminus T)$ and $T\subset B_1$ is of capacity zero (in some sense, since the author says $(1,2)$-capacity, of which I don't ...

**1**

vote

**2**answers

66 views

### List of tensor product spaces with uniform crossnorms

Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products
$H:=\bigotimes_{j=1}^n H^{(j)}$ and $G:=\...

**1**

vote

**0**answers

62 views

### Properties of a Sobolev bound

I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...

**0**

votes

**0**answers

40 views

### trace sobolecv inequality for $q=2(n-1)/(n-2)-\varepsilon$ in half space

Can I do the following inequality, for $ u\in D^{1,2}(R^n_+)$, we have
$(\int _{R^{n-1}} |u|^{2(n-1)/(n-2)-\varepsilon}dx')^{\frac{ 1}{2(n-1)/(n-2)-\varepsilon } }\leq C ( \int _{ R^n_+}| \nabla u|^...

**3**

votes

**1**answer

85 views

### Sobolev inequality involving summing from $j = 0$ to $m - 2$, exists constant

Let $I = (0, 1)$ and $1 \le q < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|D^{(m - 1)}u\|_{L^q(I)} + \sum_{j = 0}^{m - 2} \|D^ju\|_{L^\infty(I)} \le \...

**2**

votes

**0**answers

50 views

### Constant in a trace Sobolev theorem for concave domains

I wonder is the following inequality is true/known:
Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then
$$
\int_{\partial\Omega} |u|^2 ds
\...

**3**

votes

**0**answers

59 views

### Modify the jump set of $BV$ function

Let $u\in BV(\Omega)$ be a function of bounded variation where $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. We use $Du$ to denote the weak derivative of $u$. (So $Du$ is a Radon ...

**2**

votes

**1**answer

58 views

### What is the function space $H^1_{m, \sigma}$?

I am reading Hildebrandt's and Widman's 1975 paper on "Some regularity results of quasilinear elliptic systems of second order".
Theorem 3.1 is the first time in their paper that the function space $...

**6**

votes

**1**answer

196 views

### Simplicity of eigenvalues

Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where $p \in C([0, 1])$, $p \ge \alpha > 0$ on $I$, and $q \in C([0, 1])$. No further assumptions are made; in ...

**4**

votes

**0**answers

51 views

### Sobolev spaces defined on non-compact Lie groups

In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and ...

**7**

votes

**1**answer

100 views

### If $u \in H^1(U)$, then $Du = 0$ almost everywhere on the set $\{u = 0\}$, auxiliary result

Let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := \epsilon \phi(u/\epsilon).$$Do we necessarily have that$...

**7**

votes

**0**answers

48 views

### Continuous inclusions Sobolev theorem, question [closed]

How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...

**1**

vote

**1**answer

77 views

### $L^\infty(0,T;X) \cap C([0,T];Y) \subset C([0,T];X)$ for $X \subset Y$ dense?

is the Inclusion stated in the title true? In my case the spaces (essentially) are $X = H^1(\Omega)$ and $Y = L^2(\Omega)$ for $\Omega \subset \mathbb{R}$ bounded. My first try was to show
$\lim_{t_1 ...

**1**

vote

**3**answers

146 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,...

**0**

votes

**0**answers

66 views

### Boundedness of a function that satisfies a PDE-type inequality

Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$.
Suppose that
$$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + \int_{...

**2**

votes

**0**answers

153 views

### $L^\infty$ bound on solutions of linear parabolic equations

We work on a closed Riemannian manifold $M$. Let $u$ and $v$ be the non-negative weak solutions of
$$au_t - 2d\,\Delta au = cv - f$$
$$bv_t - d\,\Delta bv = f$$
$$u(0)=u_0, \quad v(0)=v_0$$
where $f$ ...

**2**

votes

**0**answers

74 views

### Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$,
$\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$
where the scalar complex function ...

**1**

vote

**0**answers

54 views

### Trace theorem for magnetic Sobolev spaces

I am considering a smooth vector field $A \colon \mathbb{R}^N \to \mathbb{R}^N$ and I define the magnetic Sobolev space $H_A^1$ as the closure of $C_0^\infty(\mathbb{R}^N)$ with respect to the norm
$$
...

**1**

vote

**0**answers

65 views

### Wavelet characterization of Sobolev spaces

We know that there exist wavelets generating orthonormal bases in Sobolev spaces $W^{p,s}(\mathbb R^n)$, where $p$ is the index of integral and $s$ is the index of smoothness. Consider the orthonormal ...

**0**

votes

**0**answers

49 views

### Approx the jump point of a $BV$ function from both hand side

Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as
$$
u(x)=
\begin{cases}
0,&\text{ if }x\in(-1,0)\\
1,&\text{ if }x\in(0,1)
\end{cases}
$$
Clearly, we have $u\in ...

**3**

votes

**0**answers

104 views

### Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...

**2**

votes

**0**answers

129 views

### $L^\infty-L^1$ smoothing effect for the heat equation

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.
Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation
$$u_t - \Delta u = 0$$
$$u(0) = u_0$$
where $u_0$ is bounded initial data. Here ...

**5**

votes

**1**answer

211 views

### Are Sobolev spaces on non-compact manifolds separable?

If $M$ is a Riemannian manifold that is not compact, is it true that the Sobolev spaces on $M$, $W^{k,p}(M)$, still be separable (for $p < \infty$)?