# Tagged Questions

**0**

votes

**0**answers

30 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 ...

**3**

votes

**2**answers

199 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

83 views

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

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), ...

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vote

**0**answers

72 views

+100

### 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 ...

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votes

**0**answers

101 views

### Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak formulation
$$a(u,v)+\int\limits_{\Omega}{f(u)vdx}=0,\,\forall v\in H_0^1(\Omega)\cap ...

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vote

**0**answers

34 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

49 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

260 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 ...

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vote

**0**answers

63 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 ...

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vote

**0**answers

64 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

90 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

15 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 ...

**2**

votes

**0**answers

65 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 ...

**3**

votes

**1**answer

89 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 ...

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vote

**1**answer

99 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 ...

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votes

**0**answers

57 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 ...

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vote

**2**answers

44 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 ...

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**0**answers

60 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.
...

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**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 ...

**3**

votes

**1**answer

79 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**

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**0**answers

48 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**

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**0**answers

57 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

42 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

188 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**

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**0**answers

50 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 ...

**5**

votes

**1**answer

89 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 ...

**7**

votes

**0**answers

46 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

71 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

140 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 ...

**0**

votes

**0**answers

64 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 + ...

**2**

votes

**0**answers

150 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 ...

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**0**answers

48 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
$$
...

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**0**answers

56 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 ...

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48 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

97 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 ...

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**0**answers

120 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

193 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 $p < \infty$)?

**7**

votes

**1**answer

496 views

### $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here:
Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...

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**0**answers

50 views

### Doubts regarding pre-compactness of bounded sequence of measure valued functions

Given:
$\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ .
...

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**0**answers

105 views

**1**

vote

**2**answers

129 views

### Convergence of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$.
Question 1: Can we ...

**2**

votes

**1**answer

69 views

### Convergence of energy of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W_0^{1,2}(B_0(1))$. For $r\in (0,1)$, define a function $f_r(x):[0,1]\rightarrow \mathbb R$ by
\begin{equation}
f_r(x):=
\begin{cases}
...

**1**

vote

**0**answers

55 views

### What is $(L^2(M), H^1_0(M))_{\frac 12}$ on a smooth manifold with boundary?

Let $M$ be a smooth compact manifold. If $M$ is closed, we have that the interpolation space
$$(L^2(M), H^1(M))_{\frac 12}=H^{\frac 12}(M)$$
(see Taylor's book on PDE for example). Suppose $M$ has a ...

**0**

votes

**0**answers

40 views

### Interior of positive cone in Sobolev space

In Sobolev spaces $W^{1,p}(\Omega),\Omega\subseteq \mathbb{R}^n,p<n$, positive cone has empty interior. How does one prove that? Is it a consequence of some more general result like the embedding ...

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votes

**1**answer

115 views

### Rellich-Kondrachov compacteness Theorem for the Euclidean space with Gaussian measure

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by
$$
\gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}.
$$
Let $d\gamma_n$ denote the following measure ...

**4**

votes

**1**answer

380 views

### A continuous path between two Sobolev functions

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in ...

**0**

votes

**0**answers

66 views

### Derivatives of Mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...

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votes

**0**answers

50 views

### Traces of fractional Sobolev spaces $W^{s,p}$ with $0<s<1/p$

I've stumbled upon a problem involving the trace of a function in a fractional Sobolev space of the form $W^{s,2}(H)$, where $H$ is a half-plane in $\mathbb{R}^2$. Would it be possible to define a ...

**1**

vote

**1**answer

100 views

### How to prove the Hölder continuity of a function $u$ by evaluating $\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx$?

I'm looking at a video on thin obstacle problem given by Arshak Petrosyan.
In his lecture, he uses the following results:
Let $0<\alpha<1$, and $B_1$ be the unit ball centered at origin in ...