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0
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0answers
59 views

Continuous inclusions in Hilbert-Sobolev space $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ [on hold]

I have to prove that $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ with $s \in \mathbb{R}$, $k \in \mathbb{N}$ and $s-k > n/2$, where $\mathcal{E}^k(\mathbb{R}^n):=\lbrace u: ...
8
votes
2answers
512 views

Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
2
votes
0answers
136 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 $f$ ...
0
votes
1answer
313 views

Condition to obtain a not compact embedding

I have the two spaces $W_0^{1,p}$ with the norme $$||u||^p=||u||^p_{L^p}+||\nabla u||^p_{L^p}$$ and $$L^{p^*}_{\alpha}=\{ u~\text{measurable}, \int_{\Omega} (|x|^{\alpha} u(x)|)^{p^*} dx<\infty\}$$ ...
2
votes
0answers
66 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 ...
0
votes
0answers
37 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 $$ ...
5
votes
1answer
202 views

Chain rule for weakly differentiable functions

Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ...
1
vote
0answers
40 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
0answers
47 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 ...
7
votes
1answer
445 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 ...
5
votes
1answer
76 views

Trace spaces on convex polyhedra: compatibility conditions at edges

Let $\Omega$ be a convex polyhedron in $\mathbf{R}^3$ with boundary $\partial\Omega$ consisting of $N$ polygons $\{\Gamma_j\}_{j=1}^N$. It's well known that in general $$ H^s(\partial \Omega) \neq ...
3
votes
0answers
92 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
0answers
106 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 ...
3
votes
0answers
87 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 $p < \infty$)?
2
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0answers
46 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 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ . ...
7
votes
1answer
815 views

Isoperimetry and Poincaré Inequality

What are the known relations between isoperimetric and Poincaré inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
2
votes
1answer
68 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
2answers
124 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 ...
1
vote
1answer
70 views

Compact embedding and fractional Sobolev spaces in unbounded domain

It is known that there exists a compactness results involving fractional sobolev spaces in bounded domain. What about unbounded domain? More precisely, Under which conditions, we can extend the ...
1
vote
0answers
52 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 ...
4
votes
1answer
374 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
0answers
31 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 ...
3
votes
1answer
100 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
2answers
223 views

Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem. Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...
5
votes
1answer
378 views

traces of sobolev spaces under additional assumptions

Let $p\in [1,\infty]$, $\Omega$ an open bounded domain with (smooth, if necessary) boundary $\partial \Omega$. Is there a subspace $X\subset L^p(\Omega)$ - a simply describable space, ideally a ...
0
votes
0answers
54 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 $b(t,x)$ ...
3
votes
0answers
46 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
1answer
94 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 ...
2
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1answer
61 views

Definitions of negative order Sobolev spaces

I am having a problem with the definition of the space $W^{-k,p}$. I use Adams's definition $$ W^{-k,p} = \left\{T \in D'(\Omega) \ \middle| \sum \limits_{0 \leq |i| \leq k} (-1)^{|i|} \int_{\Omega} ...
12
votes
2answers
343 views

When is the closed unit ball in a smaller Banach space closed in a larger Banach space?

Recently I saw an interesting lemma: For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u_j\in H^s$ and $\|u_j\|_{H^s}\le 1$. Suppose $u_j\to u$ in ...
2
votes
1answer
81 views

Getting an estimate of the form $\lVert u(t+h)-u(t) \rVert_{L^1(\Omega)} \leq \frac{Ch}{t}$ on solution of PDE

Let $u$ be a weak solution (i.e. $u \in C([0,T];L^1(\Omega))$ of some degenerate or nondegenerate parabolic equation $u' - Au = f$ on a bounded domain. (For my purpose it is enough to have this for ...
0
votes
0answers
18 views

Difference between $W^{k,p}([0,1]^d)$ and $W^{k,p}(\mathbb{T}^d)$

Let $\mathbb{T}^d \sim \mathbb{R}^d/\mathbb{Z}^d$. I know that $$W^{k,2}(\mathbb{T})\equiv\{f\in W^{k,2}([0,1]); f^{(i)}(0) = f^{(i)}(1), i = 0, \ldots, k-1\}.$$ Are there similar characterizations ...
2
votes
0answers
34 views

Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$

Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
0
votes
0answers
44 views

The properties of the solution pf minimizing problem with different parameters

I asked an similar problem before but received no respond. Here I modified the problem, add in more informations and assumptions, and with an extra question... Let $\Omega\subset \mathbb R^2$ be open ...
3
votes
0answers
106 views

Compact embedding of ${\rm L}^1_{loc}$ space

I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely: Let $\Omega\subset{\bf R}^2$ be open and bounded and let $p\in\langle 1,2\rangle$. ...
1
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0answers
25 views

The jump set of $SBV$ function over a hyper surface

Assume $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. Also assume $S\subset \Omega$ is a smooth hyper surface such that $0<\mathcal H^{N-1}(S)<+\infty$. Now, given a positive ...
1
vote
0answers
66 views

Regularity of solutions to $u' + Au = f$ for nonlinear monotone operator $A$

Consider the equation $$u' + Au = f$$ $$u|_{\partial \Omega} = 0$$ $$u(0) = u_0$$ where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). ...
0
votes
0answers
51 views

Compact embedding for Bochner spaces

I was reading some topics about the Bochner integral, and this question arises: Assume we have a sequence $u_n \in H^s(0,2\pi,L^2(\mathbb{T}))$ converging weakly to $v$ for all $s \in (\frac12, ...
0
votes
0answers
42 views

Equivalence of Sobolev--Slobodeckii and interpolation space on boundaries

Let $s \in (0,1)$. Given a sufficiently smooth hypersurface $\Gamma$ in $\mathbb{R}^n$, one can define the Sobolev--Slobodeckii space with the norm $$|u|_{H^s(\Gamma)}^2 = \int_\Gamma |u|^2 + ...
6
votes
0answers
146 views

Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
2
votes
0answers
59 views

Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in ...
1
vote
0answers
92 views

A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
2
votes
0answers
181 views

One parameter family of elliptic equations

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = ...
2
votes
0answers
80 views

Rellich Embedding Theorem for the $2$-Sphere

I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the $2$-sphere. To be precise, for $S$ the spinor bundle of $S^2$; $L^2(S^2)$ the space of square integrable ...
5
votes
3answers
268 views

A space of distributions vanishing on the boundary

The revised question After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let $$\mathcal ...
2
votes
1answer
100 views

Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$. Here, $F$ denotes the ...
2
votes
0answers
71 views

elliptic regularity for Neumann BVP on square

I am interested in the regularity of ellitpic equations like $$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ ...
2
votes
1answer
68 views

interpret of Picone inequality for non-regular functions

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set. There is a well-known picone identity that says Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
1
vote
0answers
33 views

Sobolev spaces, Finite Element Error Analysis

Let $\Omega\subset \mathbb{R}^{2}$ be a bounded, convex, polygonal domain and $H=\{(u_{1},u_{2})\in H^{\epsilon-\frac{1}{2}}(\Omega)\times ...