All Questions
Tagged with fa.functional-analysis sobolev-spaces
652 questions
3
votes
0
answers
112
views
Is a relatively weakly compact subset of $W^{1,1}(\Omega)$ metrizable?
Let $\Omega$ be a domain with smooth boundary. Let $S\subset W^{1,1}(\Omega)$ be a relatively weakly compact set.
Is it true that $(S,w)$ is metrizable?
Since $S$ is relatively weakly compact, it ...
- 1,245
3
votes
0
answers
85
views
Existence for $-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$
Let $\Omega$ be a smooth bounded domain. Consider the equation
$$-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$$
$$u|_{\partial\Omega} = 0$$
where $f,g$ are smooth functions on $\Omega$ and $\varphi$...
- 251
3
votes
0
answers
465
views
Fractional sobolev spaces
On the whole space $\mathbb R^d$, the fractional Sobolev space
$H_s(\mathbb R^d)$ of order $s\in \mathbb R$ can be defined as the subspace of tempered distributions $T$ such that $\mathcal F T \in L^...
- 630
3
votes
0
answers
210
views
Meromorphic continuation of resolvent of free Laplacian on homogeneous Sobolev space
Let $n \ge 2$. Set $\dot{H}^1(\mathbb{R}^n)$ to be the homogeneous Sobolev space, defined as the Hilbert completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|^2_{\dot{H}^1} \...
- 481
3
votes
0
answers
789
views
What is the optimal constant for the injection of $H^1$ into $L^\infty$ on an interval?
Let $I\subset \mathbb R$ be an interval, $1\leq p\leq \infty$, and $W^{1,p}(I)$ the usual Sobolev space. It is known that the injection $W^{1,p}(I)\hookrightarrow L^\infty(I)$ holds, i.e. there exists ...
- 31
3
votes
0
answers
285
views
Error estimate on convolution of mollifiers
Given $u\in W^{1}_{p}(\omega)$ with $1\leq p\leq \infty$, and the mollifier $\rho\in C_0^{\infty}(R^d)$ with support $B_1$ is a unit ball centered at the origin, $\rho\geq 0$ and $\int_{B_1} \rho = 1$....
- 133
3
votes
0
answers
74
views
Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$
Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
- 251
3
votes
0
answers
392
views
Compact embedding of ${\rm L}^1_{loc}$ space
I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely:
Let $\Omega\subset{\bf R}^2$ be open and bounded and let $p\in\langle
1,2\rangle$. ...
- 165
3
votes
0
answers
72
views
Pointwise (a.e) evaluation of $\sum_{n \geq 0}(u,w_n)_{L^2}w_n$ and equalities in $L^2$
Let $w_n$ be a orthonormal basis of $L^2(\Omega)$. Given $u \in L^2$ we can write $$u=\sum_{n \geq 0}(u,w_n)_{L^2}w_n.$$
Suppose $w_n$ are the eigenfunctions of the Neumann Laplacian. We can write
$$\...
- 31
3
votes
0
answers
582
views
Lipschitz continuity of a composition operator
Let $M$ be a compact Riemannian submanifold of $\mathbb{R}^K$, $U\subset \mathbb{R}^K$ an open neighboorhood of $M$ such that the shortest point Projection $P_M\colon U\rightarrow M$ is well-defined ...
- 2,286
3
votes
0
answers
411
views
Bounded functions dense in Sobolev Spaces
Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...
- 9,853
3
votes
0
answers
185
views
Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)
Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...
- 93
3
votes
0
answers
224
views
Strong solution to $u_t - \Delta_p u = f$
For $p > 1$, consider the equation
$$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$
$$u(0) = u_0$$
$$u|_{\partial\Omega} =0$$
for all $v \in W^{1,p}(...
- 65
3
votes
0
answers
140
views
convergence of $e^{it\Delta}f$
I heard of a conjecture that
$e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$
but couldn't find a proper reference.
- 31
3
votes
0
answers
201
views
Degree of a function in $H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$
We can define the degree of a function $f \in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$ as
$$\mathrm{deg} \hspace{1mm} f = \frac{1}{2\pi i} \int_{\mathbb{S}^1} f^{-1} \frac{\partial f}{\partial \...
- 31
2
votes
2
answers
235
views
$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
- 128k
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
2
votes
1
answer
94
views
Fredholmness of formal selfadjoint operator $AA^*$ and Fredholmenss of $A$
Let $X$ and $Y$ be Hilbert spaces with respective inner products $\langle , \rangle_{X,Y}$. Let $A:X \rightarrow Y$ be a bounded linear operator. Assume there is a non-degenerate sesquilinear product $...
- 123
2
votes
1
answer
387
views
Can we interpret fractional Sobolev spaces in terms of fractional derivatives?
Let $1 \leq p < \infty$, $0<s<1$, and $\Omega \subseteq R^n$ be a domain. The Banach space $W^{s,p}(\Omega)$ is defined as
$$W^{s,p}(\Omega) := \left\{ f \in L^p(\Omega) \colon \int_{\Omega \...
- 5,327
2
votes
2
answers
450
views
The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
2
votes
2
answers
650
views
Question on Sobolev spaces in domains with boundary
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$
$$||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.$$
...
- 21.8k
2
votes
1
answer
431
views
Sobolev imbedding
It is known that, for $n \ge 3, 2 < p< 2^*$, the imbedding $H^1(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is not compact. Let $G=O(n_1) \times O(n_2)\times\cdots\times O(n_k)$, with
$n_1+...
- 397
2
votes
1
answer
576
views
Riesz potential and homogeneous Sobolev spaces
Consider the Bessel potential and Riesz potential spaces $$L^{s,p}=\{f:f=(I-\Delta)^{-s/2}g,\, g\in L^p\},$$ $$\dot{L}^{\alpha,p}=\{f:f=(-\Delta)^{-s/2}g,\, g\in L^p\},$$
where $(I-\Delta)^{s/2}$ and $...
- 101
2
votes
1
answer
1k
views
Embedding of weighted Sobolev spaces
I define the following wheighted Sobolev spaces
$$L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$$
and
$$H^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\...
- 23
2
votes
1
answer
117
views
Special density on $L^2$
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
- 1,759
2
votes
1
answer
160
views
Minimum solution over closed ball of $H_0^1(\Omega)$
Since more than 4 months ago, I have posted a question on Mathstack and I haven't recieved any concrete answers. The link to the original post with the problem and my attempts are here.
To summarize, ...
- 123
2
votes
1
answer
91
views
Joining Hölder continuous functions on Whitney covering
Let $u$ be a bounded function and let a closed set $E$ be given. The compliment of $E$ can be covered with a Whitney type covering $B_i$ such that the following are satisfied:
1) $E^c \subset \...
- 455
2
votes
1
answer
232
views
Generalization of maximum principle to other norms
Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...
- 2,286
2
votes
1
answer
2k
views
Sobolev Space, "characteristic function" for the weak derivative
Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, working in the space $H_0^1(\Omega)$ with the inner product
$$(u,v)_{H_0^1} = \int_\Omega \nabla u \cdot \nabla v$$
for $u\in H_0^1$ and $\mu(...
- 485
2
votes
1
answer
128
views
Density of smooth functions in weighted Sobolev space
Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
- 23
2
votes
1
answer
103
views
Strong convergence of a sequence in $L^2((0,T); H^{s,2}(\Omega)) \cap C([0,T];H^{-s,2}(\Omega))$, $0<s<1$
Let $u_n$ be a sequence with $u_n \in L^2((0,T);H^{1,2}(\Omega))$ and $\frac{\partial u_n}{\partial t} \in L^2((0,T);H^{1,2}(\Omega)^*)$. Then, how could one get a subsequence of $u_n$ that strongly ...
- 101
2
votes
1
answer
159
views
A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...
- 4,135
2
votes
1
answer
253
views
A question about equivalence of weighted Sobolev space norm in S. Benzoni-Gavage and D. Serre's book
This question may not be at the research level, but it has really bothered me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic ...
- 187
2
votes
1
answer
2k
views
Norms in Sobolev space $W^{1,\infty}$
Let $n\in\mathbb{N}$ and consider the Sobolev space $W^{1,\infty}(\mathbb{R}^n)=\lbrace u\in L^{\infty}(\mathbb{R}^n):\partial_iu\in L^{\infty}(\mathbb{R}^n) \rbrace$. A function is in $W^{1,\infty}$ ...
- 237
2
votes
1
answer
1k
views
Weak derivatives and Sobolev spaces on Riemannian manifolds
I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.
Let $(M,g)$ be smooth ...
- 319
2
votes
1
answer
205
views
Estimates for an elliptic PDE
Let's say I have an equation of the form $\Delta A = J$ where $J=u\nabla u + A|u|^2$ (Clarification: We are on $\mathbb{R}^3$ and $u$ is assumed to be in $H^1(\mathbb{R}^3)$). Then I could simply ...
- 469
2
votes
1
answer
1k
views
Mixed (anisotropic) Sobolev spaces
Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...
- 23
2
votes
2
answers
402
views
Sobolev trace map: is the fractional seminorm bounded by just the gradient?
Let $M$ be a compact Riemann manifold. Consider the trace map $T:H^1(M) \to H^{\frac 12}(\partial M)$. Is it always the case that
$$|Tu|_{H^{\frac 12}(\partial M)} \leq C\lVert \nabla u \rVert_{L^2(M)...
- 29
2
votes
2
answers
753
views
About a completion of a Sobolev space
Let $\Omega$ be a bounded smooth domain and define $\mathcal{C} = \Omega \times (0,\infty)$. Below, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$. The map $\...
- 65
2
votes
1
answer
766
views
reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$
I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
- 5,380
2
votes
1
answer
330
views
functions of bounded variation and gradient vector measure
I want to prove a function of bounded variation on some domain $D\subset R^n$, $f\in BV(D)$, has the property that there is a constant $C$, such that
$$
\lim_{r\rightarrow 0}\frac{C}{r^{n+1}} \int_{...
- 21
2
votes
1
answer
412
views
Convergence in norm of Sobolev spaces
I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\lbrace{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\rbrace}$ and I have to show that the function $$f(x)=\...
- 21
2
votes
1
answer
188
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
- 75
2
votes
1
answer
620
views
On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$
Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$.
I have found in several research ...
- 101
2
votes
1
answer
176
views
Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?
Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, ...
- 820
2
votes
1
answer
351
views
References on duality of fractional order Sobolev spaces
I would like to ask you for any good references regarding fractional order Sobolev spaces. I know Hitchhiker's guide to the fractional Sobolev spaces is a very popular one, and I found it to be quite ...
- 101
2
votes
1
answer
912
views
Weak convergence in $H^1_{\mathrm{loc}}$
$\newcommand{\loc}{\mathrm{loc}}$Let $\Omega$ be a bounded open set (smooth as we wish if necessary) in $\mathbb{R}^n$, $(\omega_k)$ a sequence of open subsets whose closure is contained in $\Omega$ ...
- 364
2
votes
1
answer
404
views
Trace of a function
Let $T,L> 0$ two real numbers and we consider the Sobolev space $X := L^2(0,T; H^1(0,L))\cap H^{1}(0,T;H^{-1}(0,L))$. My question is:
Given $f \in X$, the trace $ t \mapsto f(t,L)$ belongs to what ...
- 29
2
votes
1
answer
168
views
A question about series involving a Sobolev functions
Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda_j$ and $e_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\...
- 339
2
votes
2
answers
251
views
inequality involving the fractional Sobolev space
Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality
$$|u(x)...
- 407