For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

learn more… | top users | synonyms

2
votes
1answer
61 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
votes
4answers
119 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 ...
1
vote
2answers
65 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
votes
0answers
30 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
votes
0answers
85 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 ...
1
vote
2answers
64 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
votes
0answers
44 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 ...
9
votes
1answer
79 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
votes
2answers
55 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 ...
1
vote
1answer
45 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
votes
1answer
23 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
votes
0answers
26 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
vote
0answers
24 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
votes
2answers
47 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
votes
0answers
19 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
votes
0answers
45 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
votes
1answer
36 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 ...
7
votes
1answer
92 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
votes
2answers
55 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
votes
2answers
167 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 ...
3
votes
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
votes
2answers
118 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
votes
1answer
110 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 ...
0
votes
0answers
16 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
votes
1answer
26 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 ...
5
votes
0answers
48 views

$u \in W^{1, 1}(0, 1)$, $u' \in L^1(0, 1)$, $u \in BV(0, 1)$ all equivalent? [closed]

Let $u \in C^1((0, 1))$. Are the following conditions are equivalent? (a) $u \in W^{1, 1}(0, 1)$ (b) $u' \in L^1(0, 1)$ (where $u'$ denotes the derivative of $u$ in the usual sense) (c) $u \in ...
8
votes
1answer
119 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
votes
1answer
53 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?
1
vote
0answers
20 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 ...
2
votes
1answer
46 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
votes
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)} + ...
5
votes
2answers
81 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
19 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
33 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
60 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 $u \in W^{1, p}(0, 1)$?
2
votes
1answer
19 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
25 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
vote
0answers
33 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
33 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
21 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 ...
1
vote
1answer
22 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
21 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( ...
2
votes
0answers
40 views

The convergence rate of the derivative of a sequence of function

Let $v_\delta$ be a sequence of continuously differentiable functions on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and ...
1
vote
2answers
37 views

Finite Element Theorem help

There is a general theorem (Ciarlet for example) that: For $v$ in a finite dimensional space $V_{h}$, and for an element $K$, $v|_{K} \in H^{1}(K)$ for all $K$ and $v \in C^{0}(\bar{\Omega})$ implies ...
0
votes
1answer
35 views

Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
1
vote
1answer
35 views

Find the strong form of a PDE from the weak form.

I'm having a little difficulty understanding how to find the strong form of a PDE given the weak form. For example, I have the weak form as: $\displaystyle\int_\Omega [a(x)\nabla u\cdot\nabla ...
0
votes
1answer
35 views

Is this element of $H^1(\Omega)^*$ actually in $L^2(\Omega)$?

Let $\Omega$ be a smooth bounded domain. Let $v \in H^2(\Omega)$ satisfy $-\Delta v = 0$ on $\Omega$ with $\partial_\nu v = g$ where $g \in H^{1/2}(\partial\Omega)$ is normal derivative data. ...
1
vote
0answers
32 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u ...
0
votes
0answers
12 views

Fractional Sobolev space on a compact 1-D segment

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L^p(\mathbb R)$. Here, $F$ denotes the ...
1
vote
1answer
39 views

An equivalent theorem for Sobolev spaces in infinite dimensions

There is a proposition which states: Let $f\in W^{1}(U)$ be real valued and $h\in C^{1}(\mathbb{R})$ with $h'\in C_{b}(\mathbb{R})$. We then have $h\circ f\in W^{1}(U)$ and $$\partial_{j}(h\circ ...