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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2
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1answer
96 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 ...
1
vote
1answer
51 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 ...
1
vote
0answers
23 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
votes
1answer
58 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 ...
1
vote
0answers
43 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
vote
1answer
53 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
votes
1answer
50 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
votes
1answer
47 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
vote
1answer
64 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. ...
0
votes
0answers
17 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
votes
1answer
92 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, ...
1
vote
0answers
24 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 ...
0
votes
1answer
23 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 ...
0
votes
1answer
39 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 ...
1
vote
0answers
31 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 ...
0
votes
0answers
77 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 ...
0
votes
1answer
45 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
votes
2answers
95 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
votes
1answer
56 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 ...
1
vote
0answers
52 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 ...
-1
votes
1answer
43 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 ...
1
vote
0answers
44 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
votes
0answers
85 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 ...
0
votes
2answers
76 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
71 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
60 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
votes
1answer
33 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
57 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
62 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
48 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 ...
2
votes
2answers
80 views

Dirichlet Principle in Sobolev Spaces

According to Zeidler, 1995, in his book "Applied Functional Analysis: Application to Mathematical Physics". Dirichlet problem is a problem to minimize $$F(u)=\frac{1}{2}\int_G(u')^2\ dx-\int_G fu\ ...
2
votes
0answers
88 views

The Co-area formula for $BV$ function V.S. the co-area formula for $C^\infty$ functions

I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial ...
1
vote
1answer
31 views

Estimation the $L^p$ norm of $u$ by using trace and gradient.

Given $\Omega\subset \mathbb R^N$ open bounded with nice boundary. Then for $u\in W^{1,p}(\Omega)$, $1\leq p\leq \infty$, we have $$\|u\|_{L^p(\Omega)}\leq C(\|T[u]\|_{L^p(\partial\Omega)}+\|\nabla ...
1
vote
1answer
47 views

Does convolution of two functions in $H^s(\mathbb{R})$ belong to $H^{2s}(\mathbb{R})$?

Let $f$, $g$ be two density functions and assume that $f,g\in H^s(\mathbb{R})$, $s>\frac{1}{2}$, where $${H^s}(\mathbb{R}) = \left\{ {u:\int_{ - \infty }^\infty {{{\left| {\hat u\left( t \right)} ...
1
vote
1answer
43 views

The essential sup over the boundary VS the trace of a Sobolev function

Given $\Omega\subset \mathbb R^N$ is open bounded, nice boundary, and $u\in H^1(\Omega)$. We say that $u\leq \alpha$ on $\partial \Omega$ for a constant $\alpha$ if $(u-\alpha)^+\in H_0^1(\Omega)$, ...
0
votes
1answer
103 views

Application of Fubini's theorem (in a proof of energy minimizing harmonic maps)

Let $u\in H^1(B_1,S^k)$, where $B_1$ is the open unit ball in $\mathbb{R}^n$ and $S^k$ is the unit sphere in $\mathbb{R}^{k+1}$. Suppose that $u$ is a minimizer for the Dirichlet energy functional $$ ...
2
votes
1answer
69 views

Is there an unbounded integrable function with integrable derivative in $(0,1)$?

I wonder if there is a differentiable unbounded function $f\in L^1(0,1)$ with $f'\in L^1(0,1)$. The elementary examples $x^\alpha$ or $\log x$ suggest that my question should be answered negatively. ...
1
vote
1answer
59 views

Intuitive question about the trace operator (Sobolev spaces)

Let $\Omega$ an open and bounded domain in $R^n$ . Let $u \in W^{1,p} (\Omega) \cap L^{\infty}(\Omega)$ $(2 \leq p < \infty)$ . Let $B \subset \subset \Omega$ a open ball and consider $u_1$ the ...
0
votes
2answers
54 views

A modified version of Poincare inequality

We know the general version of Poincare inequality: $$ \int_\Omega |u-u_\Omega|^2dx\leq C\int_\Omega|\nabla u|^2dx,\quad \forall u\in W^{1,2}(\Omega), $$ where $u_\Omega$ is the average of $u$ over ...
2
votes
1answer
55 views

maximum principle for $p$-Laplace equation

Consider $\Omega \subset R^n$ a bounded domain. Let $\varphi \in W^{1,p}(\Omega) \cap L^{\infty}(\Omega)$. Let $u \in W^{1,p}(\Omega)$ with $\Delta_p u = 0$ in $\Omega$ with $u - \varphi\in ...
6
votes
0answers
108 views

Sobolev space on $M \times [0,\infty)$, $M$ compact closed manifold

Consider a manifold of the form $M \times [0,\infty)$, where $M$ is a closed compact manifold without boundary. So $M \times [0,\infty)$ is a semi-infinite cylinder. I want to know information about ...
1
vote
1answer
49 views

Showing that $a(\cdot,\cdot)$ is coercive

I am working on a problem and I have the weak formulation of Poisson's problem in $2$ spatial dimensions i.e. $u = u(x,y)$: $$a(u,v) = \ell(v) $$ where $$a(u,v)=\int_{\Omega}\nabla u\nabla ...
1
vote
0answers
42 views

Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$ I've been ...
2
votes
1answer
68 views

Is $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$ dense in $H^1(\Omega)$?

Can it be true that the space $$\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$$ is dense in $H^1(\Omega)$? If so, please give me a reference to this. Every $u \in H^1$ has $\Delta u \in ...
0
votes
0answers
36 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
1
vote
0answers
74 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
0
votes
1answer
50 views

Trace map from $H^1$ into $H^{\frac 12}$, does this statement imply another?

Consider trace map $T:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ on a sufficiently smooth domain $\Omega$. It has a partial inverse $E$. If we have the statement $$F(u,Eu) = 0\quad\text{for all ...
3
votes
1answer
49 views

Help with proof that $H^{1}(\mathbb{R})$ is closed under multiplication.

Edit: Prove that if $u,v \in H^{1}(\mathbb{R})$ then $uv \in H^{1}(\mathbb{R})$. My idea is to approximate with functions in $C^{\infty}(\mathbb{R})$ with compact support. Let $u,v \in ...
0
votes
1answer
30 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
2
votes
1answer
79 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...