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
246 views

If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

Provide details for the following alternative proof that if $u \in H^1(U)$, then $$Du = 0 \text{ a.e. } \, \text{ on the set } \{u=0\}. $$ Let $\phi$ be a smooth, bounded and nondecreasing ...
1
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1answer
100 views

If $u \in W^{1,p}(U)$, prove that $Du=0$ a.e. on the set $\{u=0\}$.

Assume $1 \le p \le \infty$ and $U$ is bounded. (a) Prove that if $u \in W^{1,p}(U)$, then $|u| \in W^{1,p}(U)$. (b) Prove $u \in W^{1,p}(U)$ implies $u^+,u^- \in W^{1,p}(U)$, and ...
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2answers
68 views

About an uncommon theorem of Morrey

I am searching for a reference with the following result: Let $\Omega \subset R^n$ an open bounded domain with smooth boundary . Let $2\leq p < n$ and $u \in W^{1,p}(\Omega)\cap ...
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1answer
46 views

help to prove $||u||_{W^{2,2}(\Omega) }\le C ||\Delta u ||_{L^2(\Omega )} $

Can some one give a reference or hint for proving $$||u||_{W^{2,2}(\Omega)} \le C ||\Delta u ||_{L^2(\Omega )} $$
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0answers
32 views

This Sobolev function is continuous?

Let $\Omega \subset R^n $ $(n \geq 2) $ an open bounded domain with smooth boundary $u \in W^{ 1,p}(\Omega)\cap L^{\infty}(\Omega)$ ($p \geq 2$ fixed). Suppose that exist $M > 0$ such that $$ ...
3
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1answer
87 views

$-\Delta u - \alpha u^{1/3} = 0$ implies $u \equiv 0$ if $\alpha$ is small

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ with smooth boundary. Let $u(x)$ be a $H^{1}(\Omega)$ solution of the equation $-\Delta u - \alpha u^{1/3} = 0$, $u|_{\partial \Omega} = 0$. The problem I ...
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0answers
25 views

Compactness of Pseudo-differential Operators on $H^s(\mathbb R^n)$?

The Sobolev space of order $s\in\mathbb R$ in $\mathbb R^n$, denoted by $H^s(\mathbb R^n)$, is defined as follows: $$H^{s}(\mathbb R^n):=\{u\in\mathscr{S}^{'}(\mathbb R^n): \exists f\in ...
1
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1answer
58 views

weak* convergence definition in Sobolev space

I have a question which might quite trivial but I would appreciate any assistance. Why does it follow that for Sobolev spaces, say $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$, it follows ...
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0answers
43 views

Verify that the unbounded function belongs to $W^{1,n}$ [duplicate]

Verify that if $n > 1$, the unbounded function $u = \log \log \left(1+\frac 1{|x|}\right)$ belongs to $W^{1,n}(U)$, for $U=B^0(0,1)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise 14. ...
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1answer
46 views

Bounded Right Inverse

If a linear operator between two Banach spaces is surjective and bounded, can we get any information about a right inverse? For example, is it bounded? Thanks, trying to understand trace operator ...
2
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1answer
68 views

Delta distribution as a bounded linear functional

Let $\Omega \subset \mathbb{R}^n$ be a domain with $0 \in \Omega$. For which $n \in \mathbb{N}$ is the Dirac-delta distribution a bounded linear functional on $H_{0}^{1}(\Omega)$, that is to say, an ...
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0answers
62 views

Proof of weak derivatives in Evans PDE?

In the textbook of Partial differential equation of Evans. Why from $\int_U(v-\overline v)\phi dx=0$ for all $\phi \in C_c^\infty (U)$, we can get $v-\overline v=0$ a.e.? How to prove it? ...
2
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1answer
47 views

Why does this completion of a Sobolev space contain constant functions? Please explain text.

Below, $\mathcal{C} = \Omega \times (0,\infty)$, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$, and $\Omega$ is a bounded smooth domain. $tr_\Omega:H^1(C) \to ...
3
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1answer
47 views

Using Mac Shane's Lemma

Let $I \subset \mathbb{R}^{N}$ be a convex, bounded open set with Lipschitz boundary $\partial I$. Let $\lbrace u_{n} \rbrace_{n}$ and $u$ be such that $$ u_{n} \rightharpoonup^{*} u~~ \text{ in }~ ...
1
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1answer
141 views

Estimate $L^{2p}$ norm of the gradient by the supremum of the function and $L^p$ norm of the Hessian

Prove $$\|Du\|_{L^{2p}(U)} \le C\|u\|_{{L^\infty}(U)}^{1/2} \|D^2 u\|_{L^p(U)}^{1/2}$$ for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise ...
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1answer
56 views

$\int_\Omega |\nabla u^+|^2 \, dx$ is not differentiable with respect to $u$ in $W_0^{1,2}(\Omega)$

Let $u \in W_0^{1,2}(\Omega)$, where $\Omega$ is some domain in $\mathbb{R}^N$, $N \geq 1$. Denote $u^+ := \max\{u, 0\}$. (It is know that $u^+$ also belongs to $W_0^{1,2}(\Omega)$ (see, e.g., ...
2
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0answers
37 views

Inequality involving $H^s$ and $L^2$.

I have this inequality which I don't see how to prove it. We have $F \in C^s$, and $u\in H^s$. I want to show that: $$\| F\circ u \|_{H^s} \leq C(\| F \circ u \|_{L^2}+\sum_{r=1}^s \sum_{j=1}^r ...
1
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1answer
70 views

Deriving $|u(x)-u(y)|\le|x-y|^{1-\frac 1p}\left(\int_0^1 |u'|^p \, dt \right)^{1/p}$

Assume $n=1$ and $u \in W^{1,p}(0,1)$ for some $1 \le p < \infty$. (a) Show that $u$ is equal a.e. to an absolutely continuous function and $u'$ (which exists a.e.) belongs to $L^p(0,1)$. ...
0
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1answer
75 views

The Trace Theorem for $W^{1,p}$ functions

I'm trying to understand the proof of the trace theorem. This is from a course I am taking, so I will write out what we have done explicitly. $\textbf{Trace Theorem}$ Suppose $\Omega ...
3
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1answer
95 views

Convergence in dual of Sobolev space

Hi please view the following question: Consider Sobolev space $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^{n}$ is bounded. We also have a mapping $a: \Omega \times \mathbb{R} \times ...
2
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1answer
97 views

The averages, near the boundary, of a function $u\in W_0^{1,p}(\Omega)$, converge to zero.

Let $E_\delta =[0,1]^{N-1}\times [0,\delta]$, $p\in [1,\infty)$ and $1/p+1/p'=1$. Let $\varphi\in C^1(E_\delta)$ such that $$\varphi(x)=0,\ \forall \ x\in [0,1]^{N-1}\times \{0\}.$$ By the ...
2
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1answer
64 views

A generalization of the problem: "$\|\Delta u\|_{L^2}$ is an equivalent norm for standard $H^2$ norm in space $H_0^1\cap H^2$

We know the norm $\|\Delta u\|_{L^2(\Omega)}$ is an equivalent norm of $H^2$ norm in space $H_0^1(\Omega)\cap H^2(\Omega)$ where $\Omega$ open bounded with smooth boundary. Now let's generalize this ...
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0answers
27 views

Proof of existence of trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ not using the Fourier transform

I'm looking for a proof of the existence of the trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ which does not use the Fourier transform. In particular, I want to prove the ...
0
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1answer
66 views

The Sobolev Space $H^{1/2}$

this is a very stupid question. In my course of linear PDEs, the professor used $H^{1/2}$ without defining, and I have looking on google to find a definition, but the only related thing I found was ...
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0answers
46 views

Is trace operator is invertible?

This question come to me when I try to find the weak solution of following problem \begin{cases} -\Delta u =f&x\in\Omega \\\ u=g&x\in\partial\Omega \end{cases} where $\Omega$ is open bounded ...
1
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1answer
55 views

A problem for laplace operator in Sobolev space

Suppose $u\in L^2(\Omega)$, then for any $\phi\in C_c^\infty(\Omega)$ we have $$ \int_\Omega v\,\phi\,dx=\int_\Omega u\Delta \phi\,dx $$ Then can I conclude that $u\in H_0^1\cap H^2(\Omega)$ and ...
0
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1answer
52 views

Poincare inequality on $H^1_0(\mathbb{R}^2_+)$?

Let $$\mathbb{R}^2_+ = \{(x,y) \in \mathbb{R}^2 : y > 0\}$$ which has a boundary $B = \{(x,0) : x \in \mathbb{R}\}$. Does the Poincare inequality hold on $H^1_0(\mathbb{R}^2_+)$: $$\lVert u ...
2
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1answer
51 views

Obtaining this estimate

How do I obtain this following estimate: $$\max_{0\le t \le T} \| \mathbf{u}(t) \|_{L^2(U)} \le C(\|\mathbf{u}\|_{L^2(0,T;H_0^1(U))}+\|\mathbf{u'}\|_{L^2(0,T;H^{-1}(U))}), \tag{10}$$ the constant ...
1
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1answer
83 views

Trouble understanding proof of Sobolev trace theorem

I am trying to understand the proof of the Sobolev trace theorem. I am stuck at the bit where the boundary is flattend out using partitions of unity. See the following text (from the book of James C. ...
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0answers
18 views

A function that is in $H^s$, what can be said on its boundedness?

assume we have a function $u\in H^s$, what can I say on its $\| u \|_{L^{\infty}}$? Where a function is in $H^s$ iff $ \|(1+|y|^s)\hat{u} \|_{L^2} < C < \infty$, where $\hat{u}$ is its Fourier ...
3
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1answer
102 views

The regularity of elliptic equation

Define $$ Lu:=-\partial_j(a_{ij}\partial_iu)+b_i\partial_iu+cu $$ where $a_{ij}$, $b_i$, and $c$ are all of $C^\infty(\bar \Omega)$ and we assume $\Omega$ is open bounded in $R^N$, smooth boundary, ...
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0answers
29 views

$W^{1,p}(\Omega)$ V.S. $W^{1,p}_0(\Omega)$

Generally we know that $W^{1,p}(\Omega)$ is bigger then $W^{1,p}_0(\Omega)$ for arbitrary $\Omega\subset \mathbb R^N$ and also we have $W_0^{1,p}(\mathbb R^N)=W^{1,p}(\mathbb R^N)$. Today I found on ...
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1answer
24 views

If $u\in W^{1,p}(\Omega)$ has support compactly inside $\Omega$, then $u\in W^{1,p}_0(\Omega)$

I am trying to prove if $u\in W^{1,p}(\Omega)$ has support compactly inside $\Omega$, then $u\in W^{1,p}_0(\Omega)$, where $\Omega\subset \mathbb R^N$ is open. Intuitively this is true. Assume ...
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1answer
42 views

Consequence from Banach Theorem on $H^1(\Omega)$

I need to prove the following proposition: Any continuous linear functional on $H^1(\Omega)$ is of the form $v\mapsto\displaystyle\int_\Omega\left\{\sum_{i=1}^nq_i\,\dfrac{\partial v}{\partial ...
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0answers
37 views

Gigliardo-Nirenberg-Sobolev inequality for functions in $W^{k,p}$, without zero trace.

The G-N-S inequality can be stated as follows: Let $U\subset\mathbb R^d$, open bounded, with $C^1$ boundary, then for any $w\in W^{k,p}_0(U)$, $p<d$ $$\|w\|_{L^{p^*}(U)}\le C(d)\|\nabla ...
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0answers
107 views

chain rule for weak derivatives

First I apologise for not writing properly, but I'm using a cell phone. We know that having a Sobolev function $u$, if we have a good enough function $f$, then $f\circ u$ is Sobolev and the chain ...
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1answer
61 views

Verifying that the Sobolev space is a Banach Space

In PDE Evans 2nd edition (pages 262-263), I am trying to understand a proof of a theorem, which states: THEOREM 2 (Sobolev spaces as function spaces). For each $k=1,\ldots$ and $1\le p\le\infty$, ...
3
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2answers
106 views

Weak formulation for nonhomogeneous problem $-\Delta u = 0$

I am wondering about the definition of weak solution to the nonhomogeneous problem $$-\Delta u = 0 \text{ in }\Omega$$ $$u = g \text{ in }\partial\Omega$$ given $g \in H^{\frac 12}(\partial\Omega)$. ...
4
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1answer
60 views

The constraint subset of $H_0^1(\Omega)$ is a $C^1$-submanifold.

This problem comes from the constraint problem in CoV. (the lagrange-multiplier case) Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. We define the sub-manifold $$ M:=\{u\in ...
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0answers
72 views

The best constant of Poincare inequality can be determined by eigenvalue of Laplace operator

Given $\Omega\subset \mathbb R^N$ be open bounded and smooth boundary. Then for $u\in H_0^1(\Omega)$ we have well-known Poincare inequality $$ \int_\Omega \lvert u\lvert^2dx\leq C\int_\Omega ...
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votes
1answer
45 views

Let $a, b \in \mathbb{R}, a < b$. Prove that if $\;v \in H^1_0(a,b),\,$ then $\,v(a)= v(b)= 0$ [closed]

Let $a, b \in \mathbb{R}$, such that $a < b$. How do we prove that if $\;v \in H^1_0(a,b),\,$ then $\,v(a)= v(b)= 0$? By definition, $H^1_0(a,b)$ is the completion of smooth compactly supported ...
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0answers
49 views

The $p$-Laplacian is strongly monotone

I am studying the solution of $p$-Laplacian by finding the minimizer of the following energy, among the space $W_0^{1,p}(\Omega)$, $p\geq 2$, $$ E[u]=\frac{1}{p}\int_\Omega \lvert\nabla ...
4
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0answers
95 views

The equivalent definition of $W_0^{1,\infty}(\Omega)$

Usually, for $1\leq p<\infty$, we define $W_0^{1,p}(\Omega)$, where $\Omega$ is open bounded smooth boundary, by taking the closure of $C_c^\infty(\Omega)$ under $W^{1,p}$ norm. However, we don't ...
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votes
2answers
91 views

Counter example for Poincare inequality does not hold on unbounded domain

The Poincare inequality states that if domain $\Omega$ is bounded in one direction by length $d>0$ then for any $u\in W_0^{1,p}(\Omega)$ we have $$ \int_\Omega|u|^p\,dx\leq ...
3
votes
1answer
79 views

What is the dense subset in $H_0^1(\Omega)\cap H^2(\Omega)$

I came across this problem when I try to prove that for space $H(\Omega):=H_0^1(\Omega)\cap H^2(\Omega)$, where $\Omega$ is open bounded with nice boundary, then the norm $\|u\|_1:=\|\Delta u\|_{L^2}$ ...
0
votes
1answer
62 views

An linear elliptic PDE, why it has these properties?

See this image (from this work) The existence is done through Lax-Milgram (at least for $\sigma = \frac{1}{2}$), I think. However, why the author only includes the gradient in defining $H^1$? Is it ...
0
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1answer
35 views

uniform constant and radius bound for poincare type inequalities On compact manifolds

I have a clarification question. If we have a Riemannian compact manifold $M$, then there exists constants c and $r_{0}$ such that for any radius r < $r_{0}$ we have $$ \bigg(\frac{1}{|B_{r}(x)|} ...
1
vote
1answer
71 views

Are all functions in the Sobolev space $W_0^{1,2}(\Omega)$ continuous and bounded?

Are all function in $W_0^{1,2}(\Omega)$, $\Omega$ being a bounded domain in $\mathbb{R}^n$, $n \geq 2$, continuous and bounded w.r.t. $|.|$?. In other words, given $u\in W_0^{1,2}$ can one say that ...
3
votes
0answers
70 views

What is a good way to prove the weak form of the Sobolev embedding theorem?

From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11. Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
5
votes
1answer
55 views

Geometric Interpretation of Weak Derivative

As we know, classic derivative $f'(x)$ of a function $f(x)$ can be interpreted as the rate of change of function $f$ in each point $x.$ How about weak derivative? Since it is defined through integral ...