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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38 views

Relationship between norm of a function and its supremum

Let $u \in H^{1}(\Omega)$ , $0<\tau <1$ and $B_{\tau R}$ the open ball of radius $\tau R$ under what conditions (I) $||u||_{L^{p}(B_{\tau R})} = \sup\limits_{B_{R}} u$ and (II) $||u||_{L^{p}(...
2
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0answers
41 views

Hahn Banach Theorem extending distribution

For any given distribution $T\in D'(\Omega)$, could $T$ has a coutinuous extension $$\widetilde{T}:C_0(\Omega)\rightarrow R,\ \ \ \widetilde{T}\in(C_0(\Omega))'\ \ ?$$ Could you state a general ...
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0answers
23 views

Why the time derivative lives in ${H^1}^*$?

Assume we have a parabolic PDE. For sake of simplicity consider the heat equation defined on a "nice" domain: $$\partial_t u(x,t)-\Delta u(x,t)=0$$ We know that the weak form of this PDE is: $$\left&...
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1answer
21 views

Could we choose $\phi_m\in D(I-\{0\})$ to approximate $\phi\in D(I)$ ($\phi(0)=0$).

Suppose $\phi\in D(-1,1)$ ($C_c^\infty$ space with standard topology) with $\phi(0)=0$. Does there exist $\phi_m\in D(-1,1)$, such that the support of $\phi_m$ is away from $0$, and $\phi_m\rightarrow\...
1
vote
1answer
54 views

Can weak convergence in $V$ imply strong convergence in $H$?

In the proof of the existence of strong solutions of the stationary NSE in the setting of Hilbert spaces, the following argument is made in Constantin and Foias's Navier-Stokes Equations (p60): ...
2
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1answer
66 views

Minimizing a funtional in the Sobolev space $H_0^1$

I am trying to show that, given $f \in H^{-1}(U)$, there exists a unique $u \in H_0^1(U)$ such that: $$\int_U \nabla u\cdot\nabla v \, \mathrm{d}x= \langle f,v \rangle_{H^{-1}} \, , \quad \forall \, v ...
2
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1answer
85 views

What makes a norm “appropriate”? Why can't testfunctions be normed appropriately

I often hear the term "using an appropriate norm". Then I once read that the $C^\infty_0(\Omega)$ cannot be appropriately normed. Why is that? Furthermore, when doing some numerical analysis you often ...
3
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2answers
78 views

Closure of $C^\infty_0(\mathbb{R}^3\!\setminus\!\{0\})$ in the $H^2$-norm?

It is a standard fact (e.g., Lieb-Loss, Analysis, Theorem 7.6), that the closure of $C^\infty_0(\mathbb{R}^3)$, namely the space of (complex-valued) compactly supported smooth functions on $\mathbb{R}^...
2
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1answer
74 views

Example of an operator with continuous spectrum

Given $f \in L^2(\mathbb{R})$, let $u \in H^1(\mathbb{R})$ be the unique (weak) solution of the problem$$-u'' + u = f \text{ on }\mathbb{R},$$in the sense that$$\int_{\mathbb{R}} u'v' + \int_\mathbb{R}...
2
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1answer
73 views

What is the intersection of all Sobolev spaces of square integrable functions?

Let $U \subset \mathbb{R}^n$. If $$H^k=\{f: U \rightarrow \mathbb{R}: D^\alpha f \in L^2(U)\ \forall \alpha \in \mathbb{N}^n \ \text{with} \ \vert \alpha \vert \leq k \}$$ then how do I show that $$...
7
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4answers
126 views

Does there exist unique $u \in V$ satisfying integral equation?

Set$$V = \{v \in H^1(0, 1) : v(0) = 0\}.$$Given $f \in L^2(0, 1)$ such that ${1\over x}f(x) \in L^2(0, 1)$, does there exist a unique $u \in V$ satisfying$$\int_0^1 u'(x)v'(x)\,dx + \int_0^1 {{u(x)v(x)...
1
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2answers
77 views

Is a point evaluation a continuous linear functional on $H^1(0, 1)$?

Like the question title suggests, is the mapping $u \mapsto u(0)$ from $H^1(0, 1)$ into $\mathbb{R}$ necessarily a continuous linear functional on $H^1(0, 1)$?
3
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0answers
33 views

Continuous inclusions Sobolev theorem, inequality

How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...
9
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0answers
92 views

Are Sobolev spaces $W^{k,1}(\mathbb R^d)$ and $H^{k,1}(\mathbb R^d)$ the same?

We consider the following spaces $H^{k,p}(\mathbb R^d)$, $k \geq 1$ is integer, $p \geq 1$ (Bessel potential spaces): $$ H^{k,p}(\mathbb R^d) = \bigl\{ f \in L^p(\mathbb R^d) \colon \mathcal F^{-1}[...
1
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2answers
67 views

Is the function $|x|$ in $W^{1,p}$?

I have the following question: We consider in the segment $I=]-1,1[$, the function $f(x)=|x|.$ The question is: For each value $p \in [1,+\infty[$ do we have $f \in W^{1,p}(I)$? My purpose is: We ...
0
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0answers
50 views

Proof of Morrey's Inequality in Evan's PDE

I see that there are a couple of topics on this question, but neither of them has the answer I am looking for. In general the goal is to prove the Morrey's inequality $\|u\|_{C^{0,\gamma}(\mathbb{R}^...
9
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1answer
86 views

Proof Nehari manifold of semilineal subcritical $-\Delta u = f(u)$ in $\Omega$ is not empty.

Given the problem $$ \left\{ \begin{array}{rll} -\Delta u& = f(u) & \text{in }\Omega \\ u & = 0 & \text{in } \partial\Omega \end{array} \right. $$ In a bounded domain $\Omega\subset ...
4
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2answers
57 views

Is $D(A)$ necessarily dense in $E$? Is $G(A)$ necessarily closed in $E \times E$?

Let $E = L^p(0, 1)$ with $1 \le p < \infty$. Consider the unbounded operator $A: D(A) \subset E \to E$ defined by$$D(A) = \{u \in W^{1, p}(0, 1),\text{ }u(0) = 0\} \text{ and }Au = u'.$$I have two ...
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1answer
49 views

Strangely defined ball compact in $L^p(I)$ or not?

Let $I = (0, 1)$ and $1 \le p \le \infty$. Set$$B_p = \{u \in W^{1, p}(I) : \|u\|_{L^p(I)} + \|u'\|_{L^p(I)} \le 1\}.$$When $1 < p \le \infty$, does it necessarily follow that $B_p$ is compact in $...
0
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1answer
31 views

If $A$ is the Laplacian on $H^2(0,1)∩H_0^1(D)$, then the fractional power space $\mathfrak D(A^{r/2})=H_0^r(D)$ for all $r\in\mathbb R$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
0
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0answers
28 views

If $G$ is the Green's function of the Laplacian $A$ and $L$ is the integral operator with kernel $G$, then $L$ is the inverse of $A$

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
1
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0answers
25 views

How do we compute the Green's function of the Laplacian?

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
3
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2answers
49 views

Why $u_n \to u$ in $H^1$ implies that $u_n \to u$ in $L^2$?

Why does a sequence $u_n \to u$ in $H^1$ imply that $u_n \to u$ in $L^2$? Is it because $H^1$ is continuously embedded inside $L^2$?
2
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0answers
20 views

Poisson equation, $L^2$ bounds

Consider a bounded domain $\Omega\subset\mathbb{R}^d$ and $u\in H^1_0(\Omega)$. I know that $$ \|u\|_{H^m}\leq C\|\Delta u\|_{H^{m-2}} $$ for $m\geq 1$. Is the same true for $m=0$, i.e. for the $L^2$ ...
2
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0answers
56 views

Does continuity imply weak differentiability?

I have recently been reading about weak derivatives. I have found few examples of only weakly differentiable functions and they were all continuous. Is there an example of a continuous function which ...
0
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1answer
45 views

Proof that bilinear form in $H_0^1$ is coercive

Let $$B(u,v)=\int_I uv + \int_I u'v'$$ where $u,v\in H_0^1(I)$ for a given interval $I=[a,b]\subset\mathbb{R}$. How can I prove that the bilinear form $B$ is coercive, i.e., that $$B(u,u)\ge C\Vert u\...
7
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1answer
96 views

Is it true that $\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over 2}$?

Is it true that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over 2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$?
3
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2answers
60 views

Exists $C$ where $\int_{\mathbb{R}^n} {{u^2}\over{|x|^2}}\,dx \le C \int_{\mathbb{R}^n} |Du|^2\,dx$, $u \in H^1(\mathbb{R}^n)$?

For each $n \ge 3$, does necessarily exist a constant $C$ so that$$\int_{\mathbb{R}^n} {{u^2}\over{|x|^2}}\,dx \le C \int_{\mathbb{R}^n} |Du|^2\,dx$$for all $u \in H^1(\mathbb{R}^n)$? Ideas. I ...
5
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2answers
183 views

If $u \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $u \in L^\infty(\mathbb{R}^n)$?

How do I use the Fourier transform to see that if $u \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $u \in L^\infty(\mathbb{R}^n)$, with the bound$$\|u\|_{L^\infty(\mathbb{R}^n)} \le C\|u\|_{H^s(\...
3
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0answers
25 views

Does it necessarily follow that $C^{n, \gamma}(\overline{X})$ is a Banach space? [duplicate]

Let $X$ denote an open subset of $\mathbb{R}^n$. Suppose $n \in \{0, 1, \dots\}$, $0 < \gamma \le 1$. Does it necessarily follow that $C^{n, \gamma}(\overline{X})$ is a Banach space?
11
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2answers
145 views

Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$.

Consider the function$$f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R},$$with $0 < \alpha < 1$. How do I see that $f \in W^{1, p}(\mathbb{R})$ for all $p \in [1/\alpha, \...
8
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1answer
113 views

Two questions about a function in $W^{1, p}(0, 1)$.

Let $f \in W^{1, p}(0, 1)$ with $1 < p < \infty$. If $f(0) = 0$, then does it necessarily follow that$${{f(x)}\over x} \in L^p(0, 1)$$and$$\left\|{{f(x)}\over x}\right\|_{L^p(0, 1)} \le {p\over{...
0
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0answers
18 views

Are spherical harmonics a basis for $H^1$?

We know that spherical harmonics are a complete orthonormal system for $L^2(\mathbb{S}^2)$. Is it true that they are also a complete orthonormal system for $H^1(\mathbb{S}^2)$? Furthermore, is it ...
3
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1answer
27 views

Do we have that $f \in W^{1, 1}(0, 1)$?

This is a follow up to my previous question here. How do I see that the function $$f(x) = \begin{cases} x \sin(1/x) & 0 < x \le 1,\\ 0 & x = 0 \end{cases} $$ is continuous ...
8
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1answer
123 views

Inequalities involving Sobolev spaces.

Let $I = (0, 1)$. I have two questions. Let $p > 1$. For all $\epsilon > 0$, does there necessarily exist $C = C(\epsilon, m, p)$ such that$$\sum_{j = 0}^{m-1} \|D^j u\|_{L^\infty(I)} \le \...
2
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1answer
55 views

Do we necessarily have that $W^{2, p}(I) \subset C^1(\overline{I})$ with compact injection?

Let $I = (0, 1)$ and $p > 1$. Do we necessarily have that$$W^{2, p}(I) \subset C^1(\overline{I})$$with compact injection?
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0answers
21 views

Behaviour of functions in weighted sobolev spaces

If $f$ and $Df$ are in $L^2(\mathbb{R}, e^{u^2} dx)$, can we say $f(u)e^{\frac{u^2}{2}}$ is bounded. Here $Df$ distributional derivatie of $f$. That is, If $\int_{\mathbb{R}} \lvert f(u) \rvert^2 e^{...
2
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1answer
50 views

Proof of equivalence of $\lambda$ norms in Sobolev space $H_0^1(\Omega)$

Consider the following metrics in $H_0^1(\Omega)$ with $\Omega$ a bounded domain: $$\| u\|_\lambda=\left( \int_\Omega |\nabla u|^2+\lambda\int_\Omega u^2\right)^{\frac{1}{2}}$$ and $$\| u\|_0=\left( \...
4
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2answers
50 views

Followup question, does $\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$ still hold when $p = 1$?

This is a followup to this question. Let $p > 1$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, p)$ such that$$\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(...
6
votes
2answers
89 views

Does it follow that $u_n \rightharpoonup 0$ weakly in $W^{1, p}(\mathbb{R})$ for all $p \in (1, \infty)$?

Fix a function $\varphi \in C_c^\infty$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Does it follow that $u_n \rightharpoonup 0$ weakly in $W^{1, p}(\mathbb{R})$ for all $p \in (1, \...
0
votes
1answer
20 views

neumann boundary conditions meaning

under context of PDE's Given a bounded set $\Omega$ and a smooth function $f:\Omega\to \mathbb{R}$ what does it mean to define $\frac{\partial f}{\partial n}=0$ on $\partial \Omega$? the function $f$ ...
0
votes
1answer
39 views

Euler-Lagrange equation and unknown coefficients

I want to show that the nonlinear functional $$ J(u) = \int_0^1 (u'(x))^2 + b(x)u^2(x) + f(x) u(x) \,\textrm{d}x $$ attains its minimum in exactly one point of the Sobolev space $W_0^{1,2}(0,1)$. ...
4
votes
1answer
61 views

Exists $C = C(\epsilon, p)$ where $\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $u \in W^{1, p}(0, 1)$?

Let $p > 1$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, p)$ such that$$\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$$for all $u \in W^{1, p}(0, 1)$?
3
votes
1answer
21 views

Question about sequence being bounded in $W^{1, p}$?

Fix a function $\varphi \in C_c^\infty(\mathbb{R})$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Let $1 \le p \le \infty$. Do we have that $(u_n)$ is bounded in $W^{1, p}$?
0
votes
1answer
27 views

dot comma notation in functions spaces

When I have a PDE $$ u_t+\Delta u=f, \ \ x\in\Omega $$ with $H:=H^2(\Omega) $ $f:[0,T]\to H $ what does it mean that $f\in L^2(0,T;H)$ for every $T>0$? (what is dot comma?) what does it mean ...
1
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0answers
35 views

Characterizing Bounded Symmetric Bilinear Functions on Hilbert Spaces

Context: I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem ...
3
votes
1answer
47 views

If $u \in H^1(\Omega) \cap L^\infty(\Omega)$, is $u|_{\partial\Omega} \in L^\infty(\partial\Omega)$?

Let $\Omega$ be a bounded Lipschitz domain. Let $u \in H^1(\Omega) \cap L^\infty(\Omega)$, and suppose that $\lVert u \rVert_{L^\infty(\Omega)} \leq A$. Let $T:H^1(\Omega) \to L^2(\partial\Omega)$ ...
1
vote
1answer
24 views

What is this Sobolev inequality called, or where can I find its proof?

Let $\Omega$ be a bounded Lipschitz domain. Can someone tell me what this inequality is called, or how to prove it: $$\lVert u \rVert_{L^{r_1}(0,T;L^{q_1}(\Omega))} + \lVert u \rVert_{L^{r_2}(0,T;L^{...
1
vote
1answer
27 views

Weak formulation with non homogeneous Dirichlet

I have to find the Weak formulation oh this problem: $$ \left\{ \begin{gathered} u'' = f{\text{ on }} \Omega =\left] {0,1} \right[ \hfill \\ u(0) = \alpha \hfill \\ u(1) = \beta \hfill \\ \...
0
votes
0answers
23 views

functions in sobolev spaces

Find all $k \in \mathbb N \cup \{0\}, p \in [1, + \infty) $, such that $f(x) = \frac{1}{1 + \sqrt{|x|}}$ belongs to the Sobolev space $W^{k, p}(\mathbb R)$. We can easly show that $f \in L^p( \mathbb{...