Sobolev spaces are function spaces generalising the Lebesgue spaces. Whereas elements of Lebesgue spaces have certain integrability condition imposed on them, functions in a Sobolev space have also differentiability conditions: that is, we require all partial derivatives of the function up to a ...
1
vote
2answers
86 views
sobolev space-equivalence of scalar product
Let $f \in L^2(\mathbb{R}^n).$ Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique solution $u \in H^1(\mathbb{R}^n)$?
2- Prove that there exist a constant $C \geq 0$ ...
0
votes
1answer
74 views
Fourier transform in $L^2$
I have a function $\phi\in L^2(\mathbb{R}^3)$ and I know that its Fourier transform satisfies the following equation:
$$(p^2-A)\hat{\phi}(p)=Q\frac{A+\lambda}{p^2+\lambda}$$
where $Q$ is a constant, ...
2
votes
1answer
89 views
question 9 - chap 5 evans PDE
The question is :
Integrate by parts to prove :
$$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$
for $ 2 \leq p < \infty$ ...
0
votes
2answers
52 views
Laplacian inequality in Sobolev space
Is the following assertion true?
For all $\alpha>0$ there exists some $\theta \in H^2(\Omega)\cap H_0^1(\Omega)$ such that $\|\frac{\Delta \theta}{\theta}\|_\infty \le \alpha.$
Thanks!
1
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1answer
84 views
Examples of truly abstract evolution PDEs?
Let $V \subset H \subset V^*$. Consider the parabolic PDE
$$y' = A(t)y + f$$
which is found in many books. Usually under some assumptions on $A(t)$ and $f$, there is a solution $y \in L^2(0,T;V)$ with ...
1
vote
1answer
22 views
sobolev spaces on cartesian products
We have a canonical isomorphism
$$ C^0(X,C^0(X,Y)) \simeq C^0(X \times Y, Z)$$
given by $f \mapsto \hat{f}$, where $\hat{f}(x,y) = (f(x))(y)$.
Is there a similar statement for Sovolev space?
For ...
0
votes
1answer
36 views
Elimination of a singularity
Let $z=a+ib$ with $b>0$; the function
$$f(x)=\frac{e^{iz|x|}}{|x|}$$ is in $L^2(\mathbb{R}^3)$; in fact
...
1
vote
0answers
54 views
Sobolev trace theorem for manifolds with boundary
Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality
$$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$
will hold.
...
0
votes
1answer
47 views
Sum of Banach spaces
Let $H^2(\mathbb{R}^3)$ the usual Sobolev space and consider the following set
$$X=\bigg\{u\bigg|u=\phi+\frac{Q}{|x|},\phi\in H^2,\,\, Q\in\mathbb{C}\bigg\}$$
I observe that the decomposition is ...
0
votes
0answers
48 views
Supremum of norms of line integrals
I have the following problem:
Let's say $\Omega\subset\mathbb{R}^2$ is a bounded open connected set with Lipschitz boundary and $f\colon\Omega\rightarrow\mathbb{R}$ is a function such that $f\in ...
2
votes
2answers
62 views
Boundedness of functions in $W_0^{1,p}(\Omega)$
Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and $p\in (1,\infty)$. Suppose that $u\in W_0^{1,p}(\Omega)$ and $u$ is locally essentially bounded. Does this implies that $u$ is globally ...
2
votes
1answer
52 views
Laplacian inequality in $L^\infty$
Let $\Omega$ be a bounded domain of $R^n$ and let $y\in
H^2(\Omega)\cap H_0^1(\Omega)$ such that the set $[x\in \Omega/
y(x)\ne 0]$ has non nul measure and $ \; \frac{\Delta y}{y}
1_{\{x\in \Omega/ ...
3
votes
0answers
71 views
A finely open set, not open up to polar set?
Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, which is not open up to a polar set (i.e. zero capacity), i.e., there does ...
1
vote
1answer
33 views
Does this embedding holds?
Could you please tell me whether the following continuous embedding holds?
$L^1(R^d)\hookrightarrow H^{-s}(R^d)$ $(s>\frac{d}{2})$
1
vote
1answer
23 views
How to show this equation holds for all $H^1_0$ by density? Don't understand the proof.
I read this:
Suppose $a$ is a bounded (in $H^1_0$) coercive bilinear form and it holds that
$$\langle u_t, w \rangle + a(u, w) = 0$$
for all $w \in E_M$, where $\cup_{M \in \mathbb{N}} E_M$ ...
1
vote
1answer
72 views
Trace inequality
Could you please give me a hint on how to prove the following inequality
$\|u\|_{L^2(\Gamma)}\le C\|u\|^{\frac12}_{L^2(\Omega)}\|u\|^{\frac12}_{H^1_0}$ , $\quad \forall u\in H^1_0$, $\quad ...
1
vote
0answers
17 views
about Sobolev imbedding theorem and dual sapce question
Let $D$ be an open and bounded subset of domain $\Omega$, let $f$ be a distribution on $\Omega$. Show that there is an integer $k$ such that the restriction of $f$ to $D$ is in $H^{-k}(D)$.
The hint ...
1
vote
0answers
46 views
How to eliminate the regularizing function item?
Let $\mathbf{f} \in W^{1,q}_{loc}(\mathbb{R}^n)$,
$J \in L^p (\mathbb{R}^n)$, with $1 \leq p,q \leq \infty$,
and $\frac{1}{p}+\frac{1}{q} = 1$, $\rho_{\epsilon}$ be a regularizing kernel for ...
2
votes
1answer
28 views
Inequality in Sobolev Space
Given $\Omega \subset \mathbb{R}^3$, prove $\forall u, v, w \in H^{1,2} (\Omega)$ it holds that
$
| \int_{\Omega} u \frac{\partial v}{ \partial x} w dx | \leq \| u \|_{1,2,\Omega}\|v \|_{1,2,\Omega}\| ...
2
votes
1answer
37 views
Estimating Poincare constant for unit interval
I want to show that the Poincare constant for the $W^{1,2}_0(0,1)$ is smaller than $1$. More specifically, I want to show that there is a constant $C<1$ such that for any $f\in C^\infty_c(0,1)$ ...
2
votes
1answer
38 views
Doubt about Sobolev space norm
I consider the space $H^2(\mathbb{R}^3)$. I have a function and I have to verify that it belongs to this space. In the text I'm reading the author verifies that the function and its Laplacian are in ...
3
votes
1answer
45 views
Is $H^1_0(\Omega)$ dense in $L^2(\Omega)$?
Is $H^1_0(\Omega)$ dense in $L^2(\Omega)$ for bounded domains? It is true for $H^1$ functions of course but what about this subset?
Sorry for the elementary question but I never see this so I think ...
1
vote
1answer
26 views
Why the properties of $W^{m,p}(\Omega)$ cannot be extended to $W^{m,p}(\mathbb{R}^d)$
The text books on the topic of Sobolev Spaces and PDE etc., they treate the case $W^{m,p}(\Omega)$ with $\Omega\subset \mathbb{R}^d$ and $\Omega = \mathbb{R}^d \text{ or }\mathbb{R}^d_+$ separately.
...
0
votes
1answer
40 views
Does elliptic regularity result depend on boundary conditions?
Let $\Omega$ be a domain with boundary $\partial\Omega$. Suppose I am given a weak formulation:
$$b(u,v) = (f,v) \quad\forall v \in H^1(\Omega)$$
Assuming $b$ is nice enough, does the elliptic ...
2
votes
1answer
73 views
Using Galerkin method for PDE with Neumann boundary condition?
I am wanting to show existence of solutions to
$$u_t +L(u) = f \;\;\text{on}\;\; \Omega$$
with initial condition $u|_{t=0} = u_0$ and Neumann boundary condition $\nabla u\cdot \nu = 0$ on ...
2
votes
1answer
90 views
Is this set dense in $H^1(\Omega)?$
Is $$V_1 = \{v \in H^1(\Omega) \;:\;f(v) = 0 \text{ on } \partial \Omega\}$$
dense in $H^1(\Omega)$ with the same norm as $H^1(\Omega)?$
Here $f$ is some linear functional so that $V_1$ is also ...
0
votes
1answer
35 views
Poincare inequality on $H^1_0(M)$
Is it possible to deduce the Poincare inequality for functions in $H^1_0(M)$ from the Poincare inequality for functions in $H^1(M)$ with mean value 0?
$M$ is a hypersurface with non-empty boundary.
1
vote
1answer
35 views
Is the gradient of a function in $H^2_0$ in $H^1_0$?
Suppose we have $f\in H^2_0(U)$, so $f$ is the limit of some sequence $(g_n)$ of smooth compactly supported functions on $U\in\mathbb{R}^n$ (assume bounded & smooth boundary) and $f$ is in the ...
4
votes
1answer
67 views
Poincaré inequality and Rellich Theorem in one dimensional weighted Sobolev space
Consider the weighted Sobolev space $W^{1,2}\big((0,R),r^{N-1}\big)$, $N=2,3,\ldots$ and its subspace $W_0^{1,2}\big((0,R),r^{N-1}\big)$. Anyone knows if the Poincaré inequality is true in this case?
...
2
votes
1answer
55 views
Proving that a weak solution of a BVP satisfies the boundary condition
I am given the smooth function $u$ which satisfies $\int_U
(\nabla u \cdot \nabla v +uv)\,dx = \int_U
fv\,dx$ for all functions $v$ in the Sobolev space $H^1(U)$, where $f\in ...
0
votes
0answers
51 views
Almost every restriction is absolutely continuous
I'd like to prove the following: let $B_r(x)$ be the open disk of centre $x$ and radius $r$ contained in $\mathbb{R}^2$, and let $f \in H^1(B_1(0))$ ($H^1 = W^{1,2}$). Fix $\rho < 1$.
Then, for ...
2
votes
0answers
46 views
Sobolev Spaces: The difference between $W^{k,p}$ and $W^{k,p}_0$
Let $U$ be an open set in $R^d$. I am confused about the differences between
$$W^{k,p}(U):=\{u\in L^p(U): D^{\alpha}u\in L^p(U) \text{ for all } |\alpha|\le k\}$$ and ...
2
votes
1answer
32 views
Regularity and the Varitational Inequality
Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
1
vote
1answer
51 views
Does $Du=0$ a.e. implie $u=c$ a.e.?
Let $W^{1,p}(U)$ be the Sobolev space. Suppose that $U$ is connected bounded domain in $\mathbb{R}^n$ and $u \in W^{1,p}(U)$ satisfies $Du=0$ a.e. in $U$. How can I prove that $u$ is constant a.e. in ...
2
votes
0answers
61 views
A question on weakly convergence and norm convergence.
Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. ...
2
votes
1answer
69 views
A question on a bounded sequence in $H^1(\mathbb{R}^n)$.
Let $r>0$, and $2 \le q \le 2^*$.
Suppose that $\{u_k\}_k$ is a bounded sequence in $H^1(\mathbb{R}^n)$ and
$\lim_{k\to \infty} \sup_{y \in \mathbb{R}^n} \int_{B_r(y)}|u_k|^q dx \rightarrow 0.$
...
6
votes
2answers
167 views
Trace regularity result $\lVert n \times u\rVert_{H^{-1/2}}$
There is a result in a paper I am reading :
Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that
$$\lVert n \times u\rVert_{H^{-1/2}(\partial ...
0
votes
1answer
39 views
theorem in capacity theory
I am trying to understand the proof of a theorem of capacity theory
the theorem can be found in the book "nonlinear potential theory of degenerate elliptic equations" the authors are Juha Heinonen, ...
2
votes
3answers
60 views
Sobolev inequality in $W_0^{1,p}$
If $\Omega \subseteq \mathbb{R}^N$ is an open bounded domain and $1<p<N$, then the classical Sobolev Inequality:
$$\| u\|_{p^*,\Omega} \leq C\ \| \nabla u\|_{p,\Omega}$$
holds with ...
0
votes
1answer
34 views
$L^2(U)$ compact embedded in $H^{-1}(U)$?
Let $U$ be an open subset of $R^d$. We already knew that $L^2(U)$ is a subset of $H^{-1}(U)$. Question: is this a compact embedding?
3
votes
2answers
55 views
Decomposition of functionals on sobolev spaces
It is a well known theorem, that every signed measure can be split into its positive and negative parts (Hahn-Jordan-Decomposition). My question is, if something similar is possible for functionals on ...
4
votes
2answers
107 views
about weak derivative ( Sobolev Spaces )
the following afirmation is true ?
Consider $\Omega $ a bounded and smooth domain . Let $u \in W^{1,p} ( \Omega)$ ( p>1). Supose that $u \geq 0$. let $\alpha >1$ . Then $\nabla u ^{\alpha} = ...
1
vote
0answers
36 views
Finding an optimal $p$ such that $u \in L^p$
We have an $L^2$ function $u$ defined on $\mathbb{R^2}$ with compact support such that $u \in H^{2/3}$ (H stands for Sobolev spaces, as always), $\partial_y u \in L^2$, and $(x\partial_y - ...
3
votes
1answer
67 views
showing $W^{s,p}(\mathbb{R}^n) \subseteq W^{r,p}(\mathbb{R}^n)$ for $r < s$
I'm aware of a way of doing this using pseudo-differential operator theory. One can easily reduce to showing that $W^{t,p}(\mathbb{R}^n) \subseteq L^p$ for $t > 0$. This in turn follows because ...
1
vote
0answers
62 views
Solving a Sturm-Liouville differential equation variationally
This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows:
Let $q ...
0
votes
1answer
41 views
Paradoxical argument when applying Lax-Milgram theorem to a piecewisely defined elliptic problem
$\newcommand{\v}{\boldsymbol}$
To make the problem easier, simply consider a smooth simply-connected domain $\Omega\subset \mathbb{R}^2$. $\overline{\Omega} = ...
1
vote
1answer
70 views
simple exercise - Sobolev Spaces
Let $\Omega \subset R^n$ a bounded and open set . Let $\psi : \Omega \rightarrow [ - \infty , + \infty]$ a function . Let $\eta \in H^{1,p} (\Omega)$ $( 1\leq p< \infty )$ . Let $u \in H^{1,p} ...
0
votes
0answers
31 views
Some ideas about $H=W$
Meyers and Serrins theorem says that $H=W$. ie $H^j_p(\Omega) = W_p^j(\Omega)$ .
Here the norm of $$\|u\|_{H^j_p(\Omega)} = (\int_\Omega\sum_{|\alpha|\le j}|D^\alpha u|^pdx )^{1/p}$$
where ...
3
votes
0answers
76 views
Is this function in the Sobolev space $H^{2,-s}(\mathbb{R}^3)$?
I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order ...
3
votes
1answer
77 views
$W^{1,p}$ compact in $L^\infty$?
Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
