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
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
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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 ...
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0answers
30 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
69 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
44 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
90 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
55 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
47 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
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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 ...
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0answers
40 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
84 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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2answers
66 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
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1answer
67 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
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1answer
59 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
32 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
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1answer
55 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
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0answers
61 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 $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
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1answer
47 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
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2answers
79 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
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0answers
79 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
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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
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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
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1answer
41 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
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1answer
102 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
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1answer
66 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
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1answer
57 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 ...
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2answers
52 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
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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
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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
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1answer
48 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 ...
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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
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1answer
67 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 ...
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0answers
35 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? ...
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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
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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
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1answer
48 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
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1answer
28 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
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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), ...
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0answers
31 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
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0answers
27 views

Second order equation and regularity

Let $U$ be an bounded open set in $\mathbb R^3$ and ${V}\subset \subset{W} \subset \subset {U}$. Let $u \in {H^1}(U)$, and $f\in L^{2}(U)$ satisfies $$\int_{\mathbb R^3} \nabla u(x). \nabla \varphi ...
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0answers
43 views

Integration by parts in Sobolev space

I'm looking for a reference of the following fact (if it is true...): if $u\in W^{1,1}(\Omega)$ and $v \in W^{1,\infty}(\Omega)$ ($\Omega$ a open subset of $\mathbb{R}^n$ ($n \ge 1$) with a regular ...
0
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1answer
46 views

calculate weak derivate of $|x-2|^2$

Let $u$ be a function with $u(x):=|x-2|^2$ on $I:=(-1,1)$. I want to test whether $u \in H^2(I) \backslash H^3(I)$. Let $\phi$ be in $C_0^\infty(I)$. Then: $T_u(\phi '') = \int_{-1}^1 |x-2|^2 ...
2
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0answers
41 views

Spaces of the derivative in a direction

I have two question regarding the spaces where the first, and second, directional derivatives of a functional are. Let $\Omega \in \mathbb{R}^2$ an open subset. Let the functional: $$\phi =L^p ...
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0answers
30 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
1
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1answer
30 views

Basic question about weak solution of p-Laplace equation

Let $p \geq 2$ and $B(x_0, d)$ an open ball in $R^n$ ($n \geq 2$). Let $u \in W^{1,p}(B(x_0,d))$ a weak solution of $\Delta_p u = f$ , $u = 0 , \ on \ \partial B(x_0,d)$, that is, $\int_{B(x_0,d)} ...
0
votes
1answer
37 views

Bump function on set

Given an arbitrary compact set $K \subset \mathbb{R}^n$ and an open set $V \subset \mathbb{R}^n$, I want to construct a bump function $\zeta$ that is $1$ on $K$ and has its support in $V$, where I ...
0
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0answers
35 views

Weak time derivative for functions $u \in L^2(0,T;L^2(\Omega))$

The weak time derivative of a function $u \in L^2(0,T;H^1)$ is defined to be $u' \in L^2(0,T;H^{-1})$ satisfying $$\int_0^T \int_\Omega u(t) \varphi'(t) = -\int_0^T \langle u'(t), \varphi(t) \rangle$$ ...
3
votes
1answer
52 views

Is fractional order Sobolev spaces reflexive?

Let $0<s<1$, we define $$ W^{s,p}(\Omega):=\left\{u\in L^p(\Omega),\,\frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\Omega\times\Omega)\right\} $$ with norm $$ \|u\|:=\left(\int_{\Omega} ...
0
votes
1answer
35 views

The Coercivity of uniformly positive definite Matrix of Sobolev function

For $u=(u^1,\ldots, u^N)\in W^{1,2}(\Omega,R^N)$ where $\Omega$ is bounded. We define $$ E[u]=\int_\Omega g_{ij}(u)\nabla u^i\nabla u^jdx$$ where $G=(g_{ij})_{1\leq i,j\leq N}$ is an given uniformly ...
0
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1answer
28 views

About a Morrey's type inequality

Let $\Omega \subset R^n$ an open bounded domain and consider $B_r(x_0) \subset \Omega$ an open ball. Let $u \in W^{1,p}(\Omega)$ ($p \geq 2$). Let $s > n$ and suppose that $\int_{B_r(x_0)} |\nabla ...