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
57 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
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
32 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
89 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
32 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
28 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
52 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 ...
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1answer
51 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
43 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
52 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
39 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
48 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 ...
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0answers
52 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
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1answer
70 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
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1answer
43 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
35 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 ...
4
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1answer
53 views

A property of sobolev spaces

Let $W^{k,p}(\Omega):=\{y\in L^p(\Omega) : D^{\alpha}y\in L^p(\Omega)$ for all $|\alpha|\leq k\}$ I want to prove now that: (1) $u \in W^{1,2}(\mathbb R)$ is equivalent to (2) $u \in L^2(\mathbb ...
3
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0answers
51 views

Sobolev estimation of second derivative against Laplacian and higher terms

Given $u \in H^2(\Omega)$ (and $\Omega \subseteq \mathbb{R}^n$ with appropriate properties) is there a way to estimate the norm of the second derivative $\Vert D^2 u\Vert_{L^2(\Omega)}^2$ against the ...
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0answers
37 views

Besov norm in $W^{1,2}(\mathbb{R}^n)$

A well known result on Besov spaces is that $\Lambda_1^{2,2}(\mathbb{R}^n)=W^{1,2}(\mathbb{R}^n)$. One way to define this Besov space (without Fourier transform) is to consider $$ ...
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0answers
26 views

Norm of linear functional in $W^{1,2}$

I want to find the norm of the following linear functional: $\phi:W^{1,2}_{[0,1]} \rightarrow\mathbb{R}$ $\phi(f)=\int^1_0t\cdot f(t)+ f^{'}(t) dt $ The norm is: ...
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0answers
53 views

counterexample to strauss inequality

I am looking for a counterexample to Strauss inequality in dimension 1, where it supposedly fails. How can one construct an $H^1(\mathbb{R})$ function which does not decay at infinity, for which for ...
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0answers
17 views

If $\nabla \cdot (|\nabla u|^{p-2}\nabla u) \in L^2$ what space is $u$ in?

Define $\Delta_p u = \nabla \cdot (|\nabla u|^{p-2}\nabla u)$. I want to know, if $\Delta_p u \in L^2(\Omega)$, then what space is $u$ in? I am having trouble figuring it out. Take $p=2$. Then ...
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1answer
59 views

Existence of weak derivative

Can a uniformly continuous function have a weak derivative?. In other words can $C_{unif.~cont.}$ be continuously be embedded in $W^{1,2}(\Omega)$.?
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1answer
28 views

Is Hlawkas Inequality holds for sobolev space

im wondring is that inequality holds for any functionnal space such as sobolev space and if it's true how we can write it in that space /HlawkasInequality any help would be apperciated
3
votes
1answer
219 views

The Courant Min-Max theorem of elliptic pdes.

This is an exercise function Evans PDE book, Chapter 6. The theorem states that for $Lu:=-\text{div}(A\cdot\nabla u)+cu$ where $c\geq 0$, we have the eigenvalue of $L$ can be written in the following ...
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0answers
37 views

Extend concept of weak derivatives?

Let $f \in H^m(\mathbb{R}^n)$, then we have for $|\alpha| \le m$ that $$\langle \partial^{\alpha}f, \phi \rangle = (-1)^{|\alpha|} \langle f , \partial^{\alpha} \phi \rangle$$ for all test ...
3
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0answers
40 views

Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow ...
1
vote
1answer
56 views

Is this derivative somehow bounded?

I have a function $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is a test function and $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by $f(x) = \phi(\frac{\|x\|}{n})$. Now if I take any ...
2
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1answer
62 views

Approximate $C^{\infty}$ functions by test functions in the Sobolev space norm

I am looking for a way to approximate a function $f \in \mathbb{C}^{\infty} \cap H^m(\mathbb{R}^n)$ by test functions such that I approximate $f$ and all of $f's$ $m-$ derivatives in the canonical ...
1
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1answer
51 views

$\|u\| = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ where $\Omega$ is bounded

Let $\Omega$ be a bounded open set of $\mathbb{R}^N$, define $\|u\|_1 = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ for $2\leq p<\infty$, where $$\int |\nabla u|^p ...
1
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1answer
42 views

Lebesgue's points Sobolev functions

Given $u\in W^{1,p}_{loc}(U)$, define $$u_{x,r}:=\frac{1}{|B(x,r)|}\int_{B(x,r)}u(y)dy. $$ I proved that $$ \frac{d}{dr}u_{x_0,r}\le Cr^{\frac{\varepsilon}{p}-1} $$ for $r\in ...
4
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0answers
57 views

I want to proof that the lebesgue measure of the set below is positiva. Help me!

Let $\Omega$ be a domain limited with smooth boundary in $\mathbb{R}^{n}$ and consider the Sobolev space $H^{1}_{0}(\Omega)$ equiped with the norm $||u||=\int_{\Omega}|\nabla u|^{2}dx$. Let ...
1
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1answer
49 views

$H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$

When $\Omega$ is a bounded open set of $\mathbb{R}^N$ with the help of Poincare inequality, we know that $H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$ is a Hilbert space. ...
3
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1answer
180 views

Density of smooth compactly supported functions in Sobolev space over unbounded domain.

Prove that $C^{\infty}_ {c}(\mathbb{R}^n)$ is dense in $W^{k,p}(U)$ for any open $U\subset \mathbb{R}^n$ with $\partial U\in C^1.$ In which $p\in [1,\infty)$ Note: In Lawrence Evans's PDE text, the ...
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2answers
54 views

Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$?

For the weak formulation of the biharmonic equation on a smooth domain $\Omega$ $$ \Delta^2u=0\;\text{in}\;\Omega\\ u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega $$ why does one take ...
1
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1answer
27 views

Integral over the unit ball in $\mathbb{R}^n$

Let $f(x)=|x|^r$ on $B_1(0)$ real valued function.Where $B_1(0)$ is the unit ball in $\mathbb{R}^n$. I am trying to show that if $r>1-n$ f has a weak derivative. ATTEMPT: I know from the ...
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1answer
102 views

Semilinear equation (PDE)

I've found this hard exercise on chapter 6 of Evans' book. I have no idea on how to proceed. Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi ...
2
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1answer
63 views

Poincare type inequality on compact manifold

I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact manifold with or without boundary. The inequality I am looking for is the equivalent of $ ...
0
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1answer
53 views

for a compact manifold $M$, is the dual space of $H^1(M)$ equal to $H^{-1}(M)$?

Let $M$ be a compact Riemannian manifold. Is it true that $$(H^1(M))^* = H^{-1}(M)?$$ is there some intuitive explanation why? Or some reference? Thanks Here $H^1$ is the usual Sobolev space of $u ...
2
votes
3answers
51 views

Strongly convergent to zero in $L^2$ but $H^1$ norm not vanishing

Let $\Omega$ be some open, bounded, smooth subset of $\mathbb{R}^n$. I'm wondering whether it is possible for a sequence of functions $f_n:\Omega \rightarrow \mathbb{R} $ to be strongly convergent to ...
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1answer
40 views

Equivalence of dual spaces of Sobolev Spaces

I have a quick question: Is the following equivalence true for Sobolev Spaces $(W^{1,p}(\Omega))^{*} = W^{-1,p}(\Omega) = (W^{1,p}_{0}(\Omega))^{*}$ where $W^{1,p}_{0}(\Omega)$ is the closure of ...
3
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0answers
69 views

Function with divergence, curl and normal trace on boundary equals zero is zero

Let $u\in H^1(\Omega)$ with $\nabla\times u=0$ in $\Omega\subset\mathbb{R}^3$ (open bounded domain), $u\times n=0$ on $\partial\Omega$ (where $n$ is a a normal vector to $\partial\Omega$), ...
0
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1answer
41 views

Choosing a clever “test function” in Sobolev spaces.

Given $\mathbf{f}$ with $f_1,...,f_N\in L^2(\Omega)$ $$\int_\Omega \mathbf{f} \cdot \nabla v = 0 \quad\forall v \in H_0^1(\Omega)$$ we have $\mathbf{f} = \mathbf{0}$ a.e. since $\mathbf{f} \in ...
0
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0answers
56 views

Neumann eigenvalue problem for the Laplacian

Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$. Consider the Neumann eigenvalue problem $$ \Delta u =\lambda u \, \text{at} \, \Omega \\ \frac{\partial u}{\partial \vec{\nu}}=0 ...
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2answers
118 views

Interpolation inequality in sobolev space

Let $U$ be a bounded, connected open subset of $\mathbb R^n$ with $C^1$ boundary $\partial U$. Asume $|\beta| \leq k-1$ and $k$ is a integer. Show that for each $\epsilon >0$ there exists a ...
1
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1answer
37 views

Bounded variation of function $(f-M)^+$ and the measure of the set where it is concentrated

I read this statement in the book by Evans & Gariepy, page 215, last two lines. Here $f\in BV(R^n)$ and for fixed $\epsilon>0$ and $N>0$, we define $$A_\epsilon^N :=\{x\in \mathbb ...
0
votes
1answer
50 views

Convergence in $L^p$, Cauchy in $L^\infty$

If $u_n$ is a convergent sequence in $L^p$ with $u_n \to u$, and $u_n$ is convergent is $L^\infty$, is it true that the limit in $L^\infty$ must be $u$? Is it true if $u_n$ are all test functions, ...
0
votes
1answer
44 views

Question about difference quotient in Sobolev space

Let $u\in W^{1,p}(R)$ be given, $1\leq p<\infty$. We define $$ \tau_h(u)(x):=\frac{u(x+h)-u(x)}{h} $$ be the difference quotient. We all know that up to a subsequence $\tau_h(u)\to u'$ in the ...
2
votes
1answer
30 views

Trace on $H^1(\Omega_1\cup\Omega_2)$ (one little question in the conclusion of my proof)

Let $\Sigma$ a smooth surface that separates $\Omega_1$ and $\Omega_2$ (open and bounded sets) and let $(q^n_1)_{n\in\mathbb{N}}$ and $(q^n_2)_{n\in\mathbb{N}}$ sequences in $H^1(\Omega_1)$ and ...
0
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0answers
14 views

Prove that the subespace is closed [duplicate]

We consider $\Omega\subset\mathbb{R}^3$ (open bounded with smooth surface) and a surface $\Sigma\subset\Omega$. $\Sigma$ divides $\Omega$ in $2$ open bounded subsets: $\Omega_+$ and $\Omega_-$. ...