Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm that controls both a function and its weak derivatives in some Lebesgue space.

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11 views

How to prove that Sobolev space is a Banach Space?

Suppose fixed $V \hookrightarrow H$ separable Hilbert space, $I \subset \mathbb R$ open interval and $p \in [1,\infty)$. We denote $p'$ conjugate exponent of $p$. We define $W(I,P)=\{ {u \in L^p(I;V): ...
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13 views

Regularity of distributions with $L^2$ curl or divergence

Let $\Omega$ be a domain in $\mathbb R^n$. A classical result tells us that if a distribution $T \in \mathcal{D}'(\Omega)$ has partial derivatives (say gradient) in $L^2(\Omega)$ then $T \in L^2_{loc}(...
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14 views

A Sobolev embedding question

Is there a condition on $s$ for which it is true that the Sobolev space $H^s(\mathbb{R})\subset L^1(\mathbb{R})$ continuously? Note that $s\in\mathbb{R}$ here, possibly negative.
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Index of an elliptic operator acting on smooth sections vs acting on “Sobolev” sections

According to https://math.mit.edu/~rbm/18-155-F17/Chapter6.pdf, theorem 3.1, an elliptic operator is Fredholm both when considerd as action on smooth sections and when considered as acting on Sobolev ...
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1answer
49 views

Characterization of $H^{1/2}(\partial \Omega)$ norm?

I can't understand why $\|g\|_{H_{\partial}^{1/2}} = \|u\|_{H^1} $ where $u$ solves the problem \begin{equation*} - \Delta u + u = 0, \quad \Omega\\ u|_{\partial \Omega} = g \end{equation*} The weak ...
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1answer
18 views

partition of unity and $W^{2,2}_{\mathrm{loc}}$

Let $K$ be a compact subset of $\mathbb{R}^n$. Choose a finite open cover $\{B_i\}_{i=1,\dots,k}$ so that $K\subset \bigcup_{i=1}^k B_i$. Take a partition of unity $\{\zeta_1,\dots,\zeta_k\}$ so that $...
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1answer
33 views

Weak convergence in Sobolev space $W_0^{1, 2}(\Omega):$ how it works?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^N$ and let $J:W_0^{1, 2}(\Omega)\setminus\{0\}\to\mathbb{R}$ be a $\mathcal{C}^1$ functional. Let $(u_n)_n$ be a sequence such that $$ u_n\to u \...
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25 views

Sobolev space with negative index

The Dirac delta function is in the Sobolev space $H^{-1/2-\epsilon}(\mathbb{R})=W^{-1/2-\epsilon,2}(\mathbb{R})$ for $\epsilon>0$, but it is a distribution as opposed to a function in the ...
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1answer
26 views

Is a bounded function in $H^1(\Omega)$ also continuous for a Lipschitz domain $\Omega \subset \mathbb R^2$?

Let $\Omega \subset \mathbb R^2$ be a bounded Lipschitz domain (lets say $\Omega = (0,1)^2$). Let $u \colon \Omega \longrightarrow \mathbb R$ be a function in the Sobolev space $H^1(\Omega)$, that is $...
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46 views

Laplacian in a perforated domain with non-homogenous boundary conditions. Link with the existence of a suited extension function.

Let $A,Q$ be two regular bounded open subset of $\mathbb{R}^d$ such as $\overline{A}\subset Q$. Let $\varphi \in C^\infty(\overline{A})$ be fixed data. We want to give a meaning to the following ...
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1answer
16 views

A evolutionary triple relating to Sobolev space

In Brezis's functional analysis(p.291), it says the following $$W_0^{1,p}(\Omega) \subset L^2(\Omega) \subset W^{-1, p'}(\Omega) \quad \mbox{if} \quad 2N/(N+2) \le p \le 2$$ whether $\Omega$ is ...
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13 views

Interpolation between homogeneous sobolev spaces

Is it possible to interpolate between homogeneous Sobolev spaces $\dot{H}^{s,p}$ and $\dot{H}^{s,r}$ for a fixed $s\in\mathbb{R}$?
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2answers
35 views

Product of test function and function in first order Sobolev space is also in first order Sobolev space

Suppose $1 \leq p \leq \infty$ , $\phi \in C^{\infty}_c (\Omega) $ and $u \in W^{1,p}(\Omega)$ ,where $Ω$ is an open subset of $\mathbb{R}^n$. Prove that $\phi u \in W^{1,p} (Ω)$ and $D_{x_i}(\phi ...
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26 views

Identify $W^{k,p}(\mathbb{S}^{d-1})$ as $W^{k,p}(\partial\Omega)$ with $\Omega := B_1(0)$

I want to work with Sobolev spaces $W^{k,p}(\mathbb{S}^{d-1})$ on the unit sphere. There a general concepts for Sobolev spaces on Riemannian Manifolds (e.g. "Sobolev Spaces on Riemannian ...
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26 views

Generalization of Poincaré Inequality

Poincaré Inequality Let $\Omega \subset \mathbb{R^n}$ be a bounded set, then $\exists c=c(\Omega) \gt 0$ s.t $||u||_{L^2(\Omega)} \le ||\nabla u||_{L^2(\Omega)}, \forall u \in H^1_0(\Omega)$ We have ...
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21 views

Convergence of normal derivative implies that of Dirichlet trace for the solutions of Laplace equation?

Suppose $\{u^n\}_{n=1}^{\infty}$ is a sequence of functions which (i) are bounded in $H^1(\Omega)$ with the bound independent of $n$, (ii) converge weakly (as $n\to\infty$) to zero in $H^1(\Omega)$, (...
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66 views

Gagliardo-Nirenberg-Sobolev inequality on a half-space

I'm looking for a reference about the Gagliardo-Nirenberg-Sobolev interpolation inequality on a half space $\mathbb{R}^d_+:= \lbrace x \in \mathbb{R}^d \mid x_d > 0 \rbrace$ : Let $1 \leq p,q,r \...
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22 views

On order of distributions and operators between Sobolev spaces

Denote by $H_{s}(\mathbb{R}^{n})$ the $s$-th Sobolev space on $\mathbb{R}^{n}$. Fix a real $s$ and a non-negative integer $k$ and let $A: H_{s}(\mathbb{R}^{n}) \rightarrow H_{s-k}(\mathbb{R}^{n})$ be ...
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33 views

Special case of Rellich-Kondrachov Theorem

I am studying Sobolev Spaces, and I am following the classic book "Sobolev Spaces" by Robert A. Adams and John. J. F. Fournier, 2d edition. In the page 168, the hypotheses of theorem 6.3 (...
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1answer
26 views

Embeddings and Imbeddings

I am studying Sobolev Spaces and I am very confused about the following: What are the difference between the Embeddings and Imbeddings of spaces? If I have that: $W^{k,p}\to W^{m,q} $ is a compact ...
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1answer
105 views

How to prove this subspace of $W^{2,2}$ is Hilbert?

Let $I=(l_0,l_1)\subset \mathbb{R}$ $$H^2_{l_0}=\{u\in H^2(I): u(l_0)=u'(l_0)=0\},$$ endowed with the inner product $$\langle u,v\rangle_{H^2_{l_0}}:=\left\langle u'',v''\right\rangle_{L^2(I)}=\int_I ...
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48 views

“Moral” difference between Poincare and Sobolev inequalities

I wonder what are the "moral" differences between Poincare and Sobolev inequalities. Let me state them (hopefully without errors) using Wikipedia as source: Poincare inequalities Let $\Omega$...
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2answers
44 views

A general question about inequalities

I am studying Sobolev spaces, and always appear inequalities like the following $$ ||u||_{\mathcal{X}}\leq C ||u||_{\mathcal{Y}}, $$ where $\mathcal{X}=L^\infty $ and $\mathcal{Y}=W^{1,p}$ for ...
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1answer
33 views

$L^\infty$ and Sobolev spaces

I am studying Sobolev spaces, and my professors give us the following exercise, but I do not understand how is possible conclude it. Let $1\leq p< \infty$ and $v\in C_c^\infty(\mathbb{R})$. If $$ |...
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21 views

How to prove that if $f \in H^{1,2}(\mathbb{R}^n)$ and $\nabla f \in H^{1,2}(\mathbb{R}^n)$, then $f \in H^{2,2}(\mathbb{R}^n)$?

I'm trying to learn about Sobolev spaces as completions and I was pondering the following question. Let $\Omega \subseteq \mathbb{R}^n$ be a domain. For $k \in \mathbb{N}$ and $1 \le p < \infty$, ...
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1answer
49 views

Injections of Sobolev Spaces

I am studying Sobolev Spaces and appear the following question, that look simple, by I do not know how answer it: If $T>0$ and $k\geq 2$, with $k\in \mathbb{N}$, then $ H^{k-1}(0,T) \subset W^{k,\...
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15 views

Characterization of the dual space of $H^1_0(\Omega)$.

I want to solve the following exercise: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. For a vector field $F \in L^2\left(\Omega \right)^n$ we define the functional $\mbox{div} F: H^1_0(\...
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1answer
27 views

Riesz' representative $v\in H^1$ of the functional $H^1\ni u\mapsto\langle u,f\rangle_{L^2}$ for $f\in L^1\setminus H^1$

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$. If $f\in L^2(\Omega)$, then $$\langle u,\varphi\rangle:=\langle u,f\rangle_{L^2(\Omega)}\;\;\;\text{for }u\in H^1(\Omega)$$ is clearly a bounded ...
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9 views

How to show localised Sobolev space norm is finite?

Assume that $m: (0, \infty)\to \mathbb R$ satisfies the following equation $$ |m^{(j)}(x)| \leq C x^{-j}\quad0 \leq j \leq k,\; k>\frac{d}{2}. $$ Let $0\neq \chi \in C_{0}^{\infty} (\mathbb R^{+})$ ...
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21 views

Equivalence of norms of Sobolev spaces

let $\Omega$ be a bounded domain with Lipschitz boundary $\Gamma =\Gamma_1 \cup \Gamma_2 \cup \Gamma_3$. $V=\{v\in H^1\left(\Omega\right)^d \mid v=0 \text{ sur } \Gamma_1\}.$ Are the norms $L^2\...
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21 views

Boundness in $W_0^{1, p}(\Omega)$ of the weak limit

Let $\Omega$ be an open bounded domain in $\mathbb{R}^N$ and $p\geq 2$. Let $(u_n)_n\subset W_0^{1, p}(\Omega)$ such that $u\in W_0^{1, p}(\Omega)$ exists so that $$ u_n\rightharpoonup u \quad\mbox{ ...
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26 views

Approximation theory of functions in Sobolev space

Consider f(x) is a function in $H^{1}(0,1)$,i.e. $W^{1,2}(0,1)$. Does there exist a sequence of function $f_{n}(x)$ in $C^{1}$such that $f_{n}(0)=f(0), f_{n}(1)=f(1)$ ,$|| f_{n}-f||_{H^{1}(0,1)}\to0$ ?...
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1answer
30 views

Question about Trudinger-Moser type inequality

Let $N\geq 2$ and $\Omega$ be an open bounded subset of $\mathbb{R}^N$ and consider the Trudinger-Moser inequality $$ \sup_{\substack{u\in W_0^{1, N}(\Omega)\\\|u\|_{W_0^{1, N}}\leq 1}}\int_{\Omega} e^...
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16 views

Weak convergence of derivatives of a rescaled function in $L^p$

Let $Q = (-\frac{1}{2}, \frac{1}{2})^n$ be the $n$-dimensional unit cube centered at $0$, let $u \in C_0^\infty(Q)$. For a given integer $j$ divide $Q$ into subcubes of sidelendgth $2^{-j}$, there are ...
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1answer
19 views

If grad f is in (H^(-1)(U))^3 is f in L^2(U)?

I know that if $f \in L^2(U)$ then $\nabla f \in H^{-1}(U)$. However, is the converse true? I was thinking that maybe I could use the characterization of $H^{-1}(U)$ that says that if $f \in H^{-1}(U)$...
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26 views

Confusing dot product and inner product in a weak formulation

I have been struggling with this for a while. Here, as you can see, they define the weak formulation of the Poisson equation as: $-\int_{\Omega}\nabla u\cdot\nabla v\,ds = \int_{\Omega}fv\,ds \equiv -\...
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1answer
42 views

Inequality involving magnetic derivative

I need to understand the following inequality: \begin{align*} \|\partial_j \psi\|_2 &\leq \|D_j\psi\|_2 + \|A_j\|_6\|\psi\|_3 \\ &\leq \|D_j\psi\|_2 + C \sum_{k=1}^3\|\partial_k A_j\|_2\sum_{k=...
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2answers
26 views

Derivative of a function and Sobolev space

Let function $f$ belongs to the Sobolev space or order $\beta$ defined by $$ \mathcal{S}^{\beta}(\mathbb{R}) = \left\{u \in L^2(\mathbb{R}): \int_{\mathbb{R}}(1+|\xi|^2)^{\beta}|\hat u(\xi)|^2d\xi <...
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26 views

Trace Theorem when $p= 1$

In Evans' PDE, there is Trace Theorem in Chapter 5. Everything is fine except for when $p=1$ since I doubt if I could use Gauss-Green Theorem and differentiation. More specifically, a part of the ...
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23 views

Real vs Complex Sobolev Spaces theoretical question

What are the differences betweens real and complex Sobolev spaces $W^{k,p}(\Omega)$ where $\Omega\subseteq\mathbb{R}^N$? Are there some properties that holds just for real or just for complex Sobolev ...
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21 views

Operator on Sobolev Space w.r.t. Gaussian measure

Let $X=H_r^p(\mathbb{R},\mu)$ be the Sobolev space of order $r$ with respect to the Gaussian measure $\mu$. Is the operator $d\colon H_1^p(\mathbb{R},\mu)\to L^p(\mathbb{R},\mu)$ given by $df(x)=-f'(x)...
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27 views

The regularity for elliptic problems

Assume $$\Omega=\{(r,\theta):0<r<1,0<\theta<\omega\},$$ here $\omega>\pi$. That means $\Omega$ is not a convex domain. We solve Laplace problem in $\Omega$. Solution $u$ will behave ...
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31 views

Second existence theorem for weak solutions (Evans chapter 6)

The author defines $L_{\gamma}^{-1}$ in the Theorem 4.(the first step in the first picture ) . Proof .1. Choose $\mu=\lambda$ as in theorem 3 and define the bilinear form $$B_{\gamma}[u,v]:=B[u,v]+\...
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1answer
40 views

How can I find the difference between weakly converge in Sobolev spaces and all orders weak derivative weakly converge in $L^p$

$U$ is a open subset in $\mathbb{R}^n$, $u_k$ is a sequence in $H^1(U)$ (Sobolev space). $u_k$ weakly converges to $u$ in $H^1(U)$. $u_k$ and $Du_k$ weakly converge to $v$ and $Dv$ in $L^2(U)$. How ...
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1answer
42 views

Div-Curl lemma and precompactness in $H^{-1}$

I am trying to understand $\operatorname{div-curl}$ lemma. An important requirement to apply $\operatorname{div-curl}$ lemma is the precomapctness of the sequences $\operatorname{div}(A_n)$ and $\...
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0answers
29 views

On a compact imbedding problem

Denote $$\mathcal{H}=\{u\in H^1(\mathbb{R}^3):\int_{\mathbb{R}^3} V(x)|u(x)|^2\,dx<\infty\},$$ where $V(x)\in L_{loc}^\infty (\mathbb{R}^3)$, $V(x)\geq0$ and $\lim_{|x|\rightarrow \infty} V(x)=\...
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1answer
49 views

Is $\log u\in W^{1,p}_{loc}(\Omega)$ if $u\in W^{1,p}_{loc}(\Omega)~~u>0$??

I trying to show that if $u\in W^{1,p}_{loc}(\Omega)~~u>0$ then $\log u\in W^{1,p}_{loc}(\Omega)$. My attempt Let $D\subset\subset\Omega$ and $\varphi\in C^{\infty}_0(D)$ $$\int_{D}\frac{\partial \...
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19 views

The definition of the Sobolev spaces on $n$-dimensional torus

In Evans' book, the definition of the Sobolev space is $W^{k,p}(U)$ consists of all locally summable functions $u : U \to \mathbb{R}$ such that for each multiindex $\vert \alpha\vert \le k$, $D^{\...
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0answers
35 views

References for magnetic Sobolev spaces

I am working on PDE. Recently I have been studying magnetic Sobolev spaces. While the theory is clear to me, having very little knowledge about physics, I have almost no idea how these spaces help ...
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10 views

Variational formulation of the transportation of fluid in a capillary tube.

Let the following boundary problem: \begin{eqnarray*}\label{eq1} -\frac{d}{dx}\left[c_1 (x)\frac{du}{dx}(x) \right] + c_2(x)\frac{du}{dx}(x)&=&f(x); \quad \forall x\in\Omega=(0,1),\\ u(x)&=...

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