For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

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1answer
15 views

Cauchy sequence in $W^{1,2}$ is Cauchy in $L^2$?

I'm trying to show that the space $W^{1,2}$ is an Hilbert space, i found this answered question: Showing Sobolev space $W^{1,2}$ is a Hilbert space which offer a proof, but I'm having trouble with ...
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1answer
51 views

Show that this function is weakly differentiable

I need to show that the function \begin{equation} u(x_1,x_2) = 1-x_1^2 \quad x_1>0 \\ u(x_1,x_2)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear ...
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1answer
22 views

Weak derivative of maximum

At some point in a lesson in Calculus of Variations, my teacher had a function defined as: $$v(x)= \begin{cases} u(x) & x\notin B(x_0,3\delta) \\ \max\{R(2\eta-1),u\} & x\in B(x_0,3\delta) \...
3
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2answers
100 views

Why if $u_h \rightharpoonup u$ in $*w-W^{1,\infty}$ then exists a subsequence s.t. $u_{h_k}\rightarrow u$ in $L^{\infty}$?

I can't understand why the following fact holds: I consider a sequence $(u_h)\subset W^{1, \infty}(U, \mathbb{R}^N)$, with $U$ open bounded set in $\mathbb{R}^n$, such that $$u_h \rightharpoonup u$$ $...
1
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1answer
394 views

Simple heat equation, solution regularity

I have a small problem with a regularity result for a simple parabolic heat equation: Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
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0answers
24 views

I don't see why $W^{1, 2}(\partial D)$ being compactly embedded in $L^2(\partial D)$ lets us show an operator is Fredholm of index zero.

Let $D$ be a bounded Lipschitz domain. Let $A$ be the single layer potential which maps $L^2(\partial D)$ into $W^{1, 2}(\partial D)$ boundedly. $A$ is given by: $$ A_D[\phi] = \int_{\partial D}G(x-y)...
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2answers
34 views

$W^{1,1}\subseteq AC$ and a certain property implies BV: why?

Brézis states that the functions in $W^{1,1}(I)$, with $I$ a bounded interval, are absolutely continuous, and that, for $u\in L^1(I)$, if the following holds for some constant $C$: $$\left|\int_Iu\...
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0answers
74 views

Name of a Sobolev/Poincare type inequality

It is a basic analysis exercise to verify that for $u \colon [0,1] \times [0,\infty) \to \mathbb R$, if $u(0,t) = 0$ for all $t$, then $$ ||u(\cdot,t)||_\infty \leq ||u_x(\cdot,t)||_2,$$ where the ...
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1answer
26 views

Why is $W^{k,p}(\Omega)$ the completion of $(\widetilde{C}^k(\Omega)$ instead of $W^{k,p}(\Omega) = (\widetilde{C}^k(\Omega)$?

In the first answer to this question the users states that if we define a norm $$\|f\|_{k,p} = \left( \sum_{|\alpha| \leq k} \|D^{\alpha} f\|_p^p \right)^{1/p},$$ and write $$\widetilde{C}^k(\Omega) = ...
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1answer
57 views

Has the distributional Laplacian $\Delta f:C_c^\infty(\Omega)'\to C_c^\infty(\Omega)'$ a unique extension in $H_0^1(\Omega)'$?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega)$ and $$H=\overline{\mathcal D}^{\langle\;\cdot\;,\;\cdot\;\rangle_H}\tag 1$$ with $$\langle\phi,\psi\rangle_H:...
1
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1answer
14 views

Does the validity of Sobolev inequality on a domain imply it's a Sobolev extension domain?

Let $\Omega$ be an open subset of $\mathbb{R}^{n}$ and let $1\leq p<n$. It is well--known that if $\Omega$ is an extension Domain for $W^{1,p}(\Omega)$, then the Sobolev inequality holds in that ...
2
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0answers
37 views

The eigenfunction of modified 1-laplace equation

Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0. $$ ...
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0answers
21 views

The eigenvalue of Laplace problem on a domain with a line removed

Let $\Omega\subset \mathbb R^2$ be given such that $\Omega$ is open bounded with nice boundary. Let $\Gamma$ be a finite segment in $\Omega$. We also define $\Omega_1:=\Omega\setminus \Gamma$. ...
0
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1answer
32 views

Does $f \in W^{2,p}(a,b)$ imply $f \in C^1([a,b])$?

Let $f:[a,b]\rightarrow \mathbb{R}$, and suppose $f \in W^{2,p}$ for some real $p \geq 1$. Does that imply $f \in C^1([a,b])$? My attempt: since $f$ is twice (weakly) derivable, and continuity is a ...
1
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1answer
32 views

If $\mathcal H$ is the closure of the set $D$ of divergence-free smooth functions in $L^2$, then $H_0^1∩\mathcal H$ is the closure of $D$ in $H_0^1$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$\mathfrak D:=\left\{u\in\mathcal D:\nabla\cdot u=0\right\}$$ $H:=H_0^1(\Omega,\mathbb R^d)$...
1
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1answer
23 views

How to think of Sobolev spaces $W^{k, p}$ for a function that is no longer an element of $W^{k, p}$ for $p$ greater than some number?

Consider the function $u(x) = x^{\frac{1}{2}}$ on the domain $[0, 1]$. This function is an element of $W^{1, 1}$ and $W^{1, \infty}$ but not $W^{1, 2}$ as for $W^{1, 1}$, we have $\Vert\frac{\...
2
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0answers
13 views

Different definitions of Besov norm/space

I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent. Victor I. Burenkov - Sobolev Spaces On Domains. Given $f:\mathbb{R}^n \...
2
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1answer
33 views

Closure of an operator

I am wondering what is the closure of the domain of the operator $A_0:D(A_0)(\subset H)\to H$in $H=L^2(0,1)$ $$A_0= f^{(4)}-f^{(6)}$$ $$D(A_0)=\big\{ f\in H^6(0,1)\cap H_0^3(0,1) |f^{(3)}(1)=f^{(4)}(...
2
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1answer
22 views

Using scaling arguments to determine relationships between Sobolev spaces?

I was looking up how to find relationships between Sobolev spaces and I came across this post on MO in which the first comment talks about a scaling procedure for understanding the relationships: ...
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0answers
13 views

Curl pde with Dirichlet boundary condition in a simply connected domain

Let $\Omega$ be an open, bounded, connected, simply connected domain in $\mathbb{R}^3$, with boundary $\partial\Omega=\partial\Omega_1\cup \partial\Omega_2$. Suppose $\mathbf{H}\in H(\mbox{curl};\...
1
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1answer
23 views

Understanding the definition of $W_0^{k,p}$

Let $\Omega \subset \mathbb{R}^n$. The space $W_0^{k,p}(\Omega)$ is defined as the closure of $C_0^{\infty}(\Omega)$ in $W^{k,p}(\Omega)$. I can't say I fully understand the rationale of this ...
2
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2answers
38 views

Weak problem formulation for PDE and boundary conditions

Consider the following example: $$ - \Delta u = f \mbox{ in } \Omega, $$ $$ u = 0 \mbox{ on } \Gamma, $$ Here $\Gamma$ is boundary of $\Omega$. To produce weak formulation we multiply by arbitrary $v$ ...
0
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1answer
22 views

Conditions for membership of $H^2$

I'm looking for conditions for a function $u$ defined on a bounded domain $\Omega\subset\mathbb{R}^n$ to be an element of the Sobolev space $H^2(\Omega)$. I heard the other day that if $u$ is harmonic ...
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0answers
38 views

Relationship between the distributional Laplacian and the weak Laplacian

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the $L^2(\Omega)$- or $L^2(\Omega,\mathbb R^d)$-inner product (depending on the context) $\mathcal ...
1
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1answer
31 views

Proof verification for the Friedrich's inequality in $\mathbb{R}$

I'm trying to prove the Friedrich's inequality in $\mathbb{R}^n$, but I first want to prove it in $\mathbb{R}$ because using iterated integrals I can then use the same idea in $\mathbb{R}^n$. So let $\...
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0answers
17 views

Regularity of elliptic operators- no flux boundary conditions

It is known that if $U$ is a bounded smooth domain in $\mathbb{R}^n$ than for every $f\in C^{\infty}(\bar{U})$ the solutions to the problem $$ Lu=f $$ where $L$ is uniformly elliptic operator with ...
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0answers
19 views

integral constraint induce a manifold on Sobolev space

given the set $$ M:=\{u\in H^2(\Omega):\int_{\Omega}u=m\,\} $$ $m\in \mathbb{R}$, $\Omega $ is a bounded piecewise smooth domain in $\mathbb{R}^n$. also denote by $u(t)$ a map: $u(t):(0,T)\to M$ ...
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0answers
15 views

Yet another question about "characterization of $H^{-1}$ in Evans

I'm referring to Evans' "Characterization of $H^{-1}$ in his book Partial Differential Equations. It seems like all of this applies just as well if we take the dual of $H^{1}(U)$ instead of $H_{0}^{...
1
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1answer
24 views

Convergence of composition of functions in $L^p$

I am dealing with the proof of proposition $9.5$ given in Haim Brezis' Functional analysis, Sobolev Spaces and Partial differential equations. I quote it here: How does one conclude $G \circ ...
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0answers
28 views

The $L^2$ convergence of semi-$p$-lapace equation

This question is similar to the one I post early here. But this one might be more reasonable I think... Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with ...
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0answers
28 views

The convergence of $p$-laplace equation

Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with smooth boundary. Define, for $1<p\leq 2$, $$ u_p:=\operatorname{argmin}\left\{\int_\Omega|u-g|^pdx+\...
1
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1answer
49 views

Definition of the Laplacian as an operator from $H_0^1(\Omega)$ to $H_0^1(\Omega)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $f\in L^2(\Omega)$ and $$\langle f\rangle:=\left.\langle\;\cdot\;,f\rangle_{L^2(\Omega)}\right|_{\...
1
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0answers
13 views

Bound of mollified in $H^{-2}$

Let $f\in L^2((0,T); H^2) $ with $ \partial_t f \in L^{2}((0,T);H^{-2}) $ and let $ \eta_{\varepsilon} $ a standard mollifier sequence in $ (t,x) $, then there exists a constant $ C $ independent of $ ...
12
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1answer
317 views

Intuition behind losing half a derivative via the trace operator

This is an informal question, but here goes: For a function $f \in H^s(\Omega)$ ($s > 1/2$), there is a well-defined operator (the trace) $T$ such that $Tf = f\vert_{\partial \Omega}$ if $f \in C^\...
0
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1answer
17 views

Show: If $u \in W^{1,p}(B_4(0))$ then $\int_{B_4(0)} \frac{|\nabla u|^p}{|x-a|^{n-1}} dx < \infty$ for a.e. $a \in B_1(0)$.

I'm currently working through the article "Topology and Sobolev Spaces" by Brezis and Li and as basis for the proof of an important result the following fact is used: If $u \in W^{1,p}(B_4(0))$ ...
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1answer
74 views

Is a weakly differentiable function differentiable almost everywhere?

I am working with Sobolev spaces. Let's suppose $\Omega \subset \mathbb{R}^n$ is an open set. A function $u: \mathbb{R}^n \to \mathbb{R}$ in $L^1(\Omega)$ is said to be weakly differentiable if there ...
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1answer
11 views

Dimension of boundary of a bounded domain; what to use for Sobolev inequalities

Let $\Omega$ be a bounded (at least) Lipschitz domain in $\mathbb{R}^{N}$. Its boundary $\partial\Omega$, if I'm right, is $(N-1)$-dimensional object embedded in $\mathbb{R}^N$. We can define Sobolev ...
1
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1answer
75 views

On convergence of particular functions in Sobolev space

I'm having trouble showing the following fact: Suppose $1\le p<\infty$, $\ \varphi\in C^\infty(0,\infty)$ such that $$\varphi(s)= \begin{cases} 0, & s\leq 1/2\\ y\in[0,1],& s\in(1/2,...
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1answer
25 views

I'm confused of the test problem of Weak Derivatives

But why the problem says that "prove it" with such special $\varphi$?
0
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1answer
28 views

Can we talk about the adjoint of a linear operator defined on a distribution space?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ and $$...
1
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1answer
34 views

Trace Theorem and Neumann boundary.

I've been studying Trace Theorem. From PDE Evans, we have THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator $$T : W^{1,p}(U) \...
3
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1answer
910 views

Poincaré inequality in unbounded domain

Help me please, how can I show that Poincaré inequality doesn't hold in an unbounded domain? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
2
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1answer
86 views

Weak solution for equation $-\Delta u = f$.

Suppose $f \in L^{2}$. I know that $$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0, \partial\Omega \end{array}\right.,$$ where $\Omega \subset \mathbb{R}^{N}$ is open ...
2
votes
1answer
30 views

Approximation estimates in Sobolev spaces

Let's consider a bounded domain $\Omega \subset \mathbb{R}^d$, $d =2,3$, and let $\varphi$ be in $H^1(\Omega) \cap W^{1,\infty}(\Omega)$. Is it there a smooth (at least $W^{2,4}(\Omega)\cap W^{1,\...
2
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1answer
35 views

Is Laplacian of product of functions (one is smooth) square integrable?

Let $\Omega\subset \mathbb{R}^d$: open, connected. Suppose $u\in L^2(\Omega)$ and $\Delta^n u\in L^2(\Omega)$ for $n=1,\dotsc,N$, where $\Delta$ is in the weak sense. Let $\zeta\in C_c^{\infty}(\...
2
votes
1answer
61 views

Definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis

The following is the definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis: Here a regular distribution is a tempered distribution $T_f$ such that it is given by $$ T_f(\varphi)=\...
0
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1answer
24 views

Fourier transform and the Sobolev space $H_k$ in Folland's Real Analysis

The following is an excerpt from Folland's Real Analysis: Would anybody elaborate why the estimate in the first box implies the second and third boxes?
2
votes
1answer
48 views

How to show that a Schwartz distribution is in a Lebesgue or Sobolev space?

It is known that all $L^p$ spaces (and, consequently, all $W^{s,p}$ spaces) can be embedded in the space of Schwartz distributions $\mathcal D '$. There is a problem, though: how do I check whether ...
1
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0answers
30 views

Prove that $\left.F\right|_{\left\{ϕ∈C_c^∞(Ω,ℝ^d):∇⋅ϕ=0\right\}}=0⇔∃p∈C_c^∞(Ω)$ with $F=∇p$, for all $F∈H_0^1(Ω,ℝ^d)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$, $$H:=\...
0
votes
0answers
16 views

Testing heat equation with $\log(u)$?

On a bounded domain, let $u \in L^2\left(0,T;H^1\right) \cap H^1\left(0,T;H^{-1}\right)$ be weak solution to the heat equation $$u_t - \Delta u = 0$$ with the BC $\partial_\nu u = 0$ and some initial ...