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18
votes
4answers
789 views

When to use more exciting function spaces than ordinary Sobolev spaces?

In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions. For example, Sobolev spaces ...
2
votes
1answer
138 views

Compact embedding of weighted sobolev spaces in continous functions spaces

Let the weighted sobolev space $M^p_{s,\delta}$ be the completion of $C_0^\infty(\mathbb{R}^n)$ in the norm \begin{equation} \sum_{\left\vert\alpha\right\vert\leq s}\left\Vert(1+\left\vert ...
3
votes
0answers
186 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 ...
4
votes
1answer
143 views

Dependence of the constant in Korn's inequality on the domain

Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and $$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} ...
2
votes
1answer
163 views

Does a particular iteration produce a weak solution to a non linear pde?

Consider the following non linear pde in the unknown $v(x,y)$: $$ \frac{\partial v(x,y)}{\partial x} + \Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$ where $t$ is some fixed small ...
5
votes
0answers
151 views

Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function $\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted $L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions $f: ...
3
votes
0answers
213 views

Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is \begin{equation} \omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})} \end{equation} where ...
11
votes
0answers
368 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
4
votes
1answer
218 views

Caccioppoli-Leray Inequality for De Giorgi's theorem proof

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
4
votes
1answer
235 views

about smoothing pseudodifferential operators

Hi, I have a question which involves pdo. Let us consider a pseudodifferential operator $A:S(\mathbb{R}^d)\rightarrow S(\mathbb{R}^d) $ whose symbol $a(x,\xi)$ lives in the $S_{0,0}^0$ class : $$ ...
2
votes
0answers
125 views

compact embedding in duals of weighted Sobolev spaces

On the whole space $\mathbb{R}^d$ consider the weight $\omega(x)=\sqrt{1+|x|}$. Under which conditions on $k,q$ is the embedding $$ L^p(\mathbb{R}^d,\omega(x)dx)\subset\subset ...
0
votes
1answer
162 views

Is BV2 space closed in L2 space?

We define the BV2 space by $S = \lbrace f\in L^2:\textrm{TV}(f)<\infty\rbrace$, where $TV(f)=\sup_{g\in C_c^1,\|g\|_\infty\leq 1}\int f\cdot \textrm{div}g$. My question is: is $S$ closed in $L^2$? ...
0
votes
0answers
123 views

Adjoint operator in sobolev space

Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with norm $|u|=(\|u\|^2 +\|\Delta ...
5
votes
0answers
153 views

A finely open set, not open up to polar set?

I already asked this on M.SE, but get no answers. Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...
-1
votes
1answer
97 views

A suitable Sobolev-type space

Let $\Omega$ be a domain in $\mathbb{R}^3$, does there exist a vector-valued Sobolev-type space, or maybe space in other sense, $V$, satisfying the following: 1) $S:=\lbrace v\in ...
5
votes
1answer
351 views

Compactness in Sobolev spaces

I was wondering whether the set $\lbrace f\in H_0^1(\Omega)|\|f\|_{L^\infty(\Omega)}\leq 1\rbrace$ is compact in $H_0^1(\Omega)$ or not. Here $\Omega$ is a convex domain in $\mathbb{R}^3$ with ...
4
votes
1answer
273 views

Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions

This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f ...
1
vote
1answer
98 views

Nonintegrable inverse powers as distributions

I am working through Lieb/Loss's "Analysis", and have been stuck on one of the problems for a while; Suppose we are on $\mathbb{R}^n$ and define $f(x) = |x|^{-n}$. This is not a locally integrable ...
3
votes
1answer
219 views

Sharpness of the Sobolev embedding theorem

We know that $W^{k,p}\hookrightarrow C^{k-\lfloor\frac{n}{p}\rfloor-1,\gamma}(\bar{\Omega})$ with $kp>n,\gamma=\lfloor\frac{n}{p}\rfloor+1-\frac{n}{p}$, where $n$ is the dimension of $\Omega$, ...
1
vote
0answers
121 views

Trace Inequality question

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 ...
1
vote
2answers
260 views

A question on optimal Sobolev inequality.

Let us consider the Sobolev inequality $||u||_{L^p} \le C||u||_{H^1}$ for $2 <p< 2^*$, where the constant $C$ depends on $p$ and the domain. My question is, how can one see that the optimal ...
1
vote
0answers
259 views

A question on a variant of Hardy's inequality.

I would like to know a proof of a variant of Hardy's inequality below. Could anyone introduce me a reference or give me a proof? Thank you very much for your assistance. Set ...
4
votes
1answer
193 views

projection of sobolev spaces onto cones

Consider the Sobolev space $W^{k,p}(\Omega)$ for $k\in \mathbb N$, $p\in [1,\infty]$ and some open domain $\Omega\subset \mathbb R^n$ $^*$. Then it is known that $W^{k,p}(\Omega)$ is an ordered Banach ...
4
votes
2answers
217 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 - ...
0
votes
0answers
115 views

Is this function in the weighted Sobolev space $H^{2,-s}$?

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 ...
1
vote
1answer
358 views

Doubt on norm of the Sobolev space $H^2(\mathbb{R}^3)$

If I consider the Sobolev space $H^2(\mathbb{R}^3)$ I have the norm $$\Vert u\Vert_{H^2(\mathbb{R}^3)}=\bigg(\sum_{|\alpha|\leq 2}\Vert D^\alpha u\Vert^2_{L^2(\mathbb{R}^3)}\bigg)^\frac{1}{2}.$$ Is ...
0
votes
1answer
399 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 ...
-1
votes
1answer
137 views

Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense $$\lim_{|x-y|\to 0} f(x)$$ ? ($y$ is a fixed point) If i have $f$ in $H^2$ I can say that $$\lim_{|x-y|\to 0} ...
-1
votes
1answer
88 views

Every function in W^{1,1}(0,1) is continuous on (0,1)

I am trying to prove that if $u:(0,1)\to\mathbb{R}$ lies in $W^{1,1}(0,1)$, then $u\in C(0,1)$. Is there any help anybody can offer? Thanks.
20
votes
1answer
961 views

What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities: If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...
2
votes
1answer
199 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 ...
0
votes
0answers
137 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
0
votes
2answers
578 views

Interior regularity for elliptic equations

The monograph "Non-Homogeneous Boundary Value Problems and Applications" by Lions and Magenes is infamous for developing a truly extensive regularity theory for elliptic problems on domains, and for ...
8
votes
2answers
996 views

Characterizing the Dual of $W_0^{s,p}$

I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and ...
2
votes
3answers
651 views

How to define Laplacian on $L_2$

This might be a dumb question, but I thought the Laplacian (classical) is defined for $C^2$ functions. How do we extend that to be a self-adjoint operator on all of $L_2$? Is it the so called ...
3
votes
1answer
277 views

on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
2
votes
1answer
554 views

Norm of differential operator between Sobolev spaces

It is easy to check that the differential operator $\partial^a$ (where $\alpha\in \mathbb{N}_0^n$) is continuous between the Sobolev spaces (with usual norms) $W^{m,p}(U)\to W^{m-|\alpha|,p}(U)$, ...
9
votes
3answers
382 views

Boundedness of the derivative of the trace of an H^1 function

As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical ...
7
votes
1answer
202 views

Convergence of Schwartz kernels implies convergence of operators

Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
3
votes
1answer
1k views

Classical Derivative, Weak Derivative and Integration by Parts

Hello, While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated. QUESTION Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function ...
1
vote
1answer
62 views

Continuity of an extension map

Suppose $\delta\in (0,1)$ and $r<1+\delta.$ Suppose moreover we are given a sequence of functions $u_m\in H^{1/2,2}(\partial B_r(0))$, where $B_r(0)$ denotes the euclidean $n-$dimensional ball. ...
3
votes
2answers
151 views

Regularity properties of H(-1/2)

Arising as the traces of $H(div; \Omega)$, I am wondering if the space $H^{-1/2}(\partial \Omega)$ has any regularity properties? (Containment in BV would be wonderful, although I doubt it holds.) It ...
5
votes
1answer
383 views

Books about Capacity theory

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
3
votes
0answers
195 views

$\tilde{u}=0,\ a.e.\ x\in\Gamma$?

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in ...
2
votes
1answer
290 views

Weak divergence implies weak differentiability of components?

Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$. We say that $\sigma$ has weak divergence if there exists ...
1
vote
2answers
383 views

Sobolev-type inequality.

Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $ \left \| \int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} } ...
6
votes
4answers
596 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be smooth compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by \begin{equation} ...
2
votes
1answer
289 views

Nash inequality on a compact domain?

I have come across a few papers that make use of the Nash inequality for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the ...
6
votes
1answer
298 views

Compactness of Sobolev embedding for domains of finite measure

Let $\Omega \subset \mathbb{R}^d$ be a domain of finite Lebesgue measure, not assumed to be smooth or bounded. Is it true that the embedding of, say, $W^{1,p}_0(\Omega)$ (Sobolev functions with zero ...
2
votes
1answer
344 views

Definition of Sobolev spaces as a space of sections of certain type

I want to define Sobolev spaces for sections on a vector bundle, basically I want that a section will belong to the Sobolev space $W^{k,p}$ if its coordinates in any aceptable patch belong to the ...