The sobolev-spaces tag has no wiki summary.

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### Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...

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265 views

### Is there a generalization of Sobolev spaces for certain locally compact groups?

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square ...

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29 views

### Compact imbedding - reference request

I am looking for reference to the following imbedding theorem
Theorem
For any $s>1/2$ fractional Sobolev space $W^{s}_2(0,1)$ imbeds compactly into $C([0,1])$.
I know how to prove it but I need ...

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**1**answer

85 views

### Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary

I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form
$$|u(t)|_{L^p} \leq ...

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347 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 $K$.
...

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110 views

### Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...

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99 views

### A question on density of Lipschitz functions in weighted Sobolev spaces

Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$.
Let now ...

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137 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: ...

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145 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 ...

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30 views

### Approximating a superharmonic function, by smooth superharmonic functions

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$, $u\ge 0$ and $-\Delta u\ge 0$ in the sense of distributions ($u$ is superharmonic).
The standard ...

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76 views

### Existence and smoothness of convolutions of distributions in Sobolev spaces

Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative.
It is easy to show that $f *g$ is defined pointwise when ...

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229 views

### Which functions can be approximated by piecewise constant functions?

Let $\Omega \subset \mathbb{R}^d$ be a connected and bounded domain. We call a function $f\in L^\infty(\Omega)$ nice if for each $\epsilon>0$ there exist $n\in \mathbb{N}, a_1,\dots,a_n \in ...

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130 views

### Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required.
Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy
$$0 < a \leq g(x,t) \leq b < ...

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189 views

### Alternative representations of Sobolev space

Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...

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182 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 ...

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181 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 ...

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193 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 ...

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452 views

### Open sets and Poincaré's inequality

In many references, Poincaré inequality is presented in the following way :
Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...

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205 views

### Where can I find interpolation inequalities for derivatives of the following form?

Here they are:
$$||f||_{\infty} \leq C ||f||_q^{\frac {qk} {n+kq}} \left( \sum_{|\mu|=k} ||D^\mu f||_{BMO} \right)^{ \frac n {n+kq}}$$
and
$$||f||_{Lip_\alpha} \leq C ||f||_q^{\frac {qk} {n+kq} \frac ...

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### $L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?

Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact ...

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### Showing a normal-derivative operator is a (sort of) contraction (related to Crandall-Liggett and PDEs)

Denote by $\mathbb{E}(g)$ the solution of the PDE
$$\Delta v(x,y) = 0 \quad\text{in $\Omega$}$$
$$v(x,0) = g(x) \quad\text{on $\partial\Omega$}.$$
Let $X=L^1(\partial\Omega)$. Let $h(t)$ be a ...

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55 views

### Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...

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106 views

### Trace Theorem for $p=\infty$

I am considering the Sobolev space $W^{1,\infty}(\Omega)$ on a bounded Lipschitz domain $\Omega \subseteq \mathbb{R}^2$. I am wondering whether the trace theorem holds in this case with constant one ...

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### Almost a Green formula

Let $\Omega$ be the half-space $\mathbb{R}^{n-1}\times \{ x_n>0 \}$, let $v \in L^2(\Omega)$ and $\phi\in \mathcal{C}^{\infty}(\overline{\Omega})$ with compact support in $\overline{\Omega}$. What ...

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### Weak periodic solution of parabolic PDE

Take
$$
u_t(t) + A(t)u(t) = f(t),
$$
$$
u(0) = u(T),
$$
where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...

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110 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 ...

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113 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 ...

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119 views

### Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$

We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$
I want to show for $r \geq 2$ that
...

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122 views

### $b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$

This question stems from the proof of Theorem A.1 on page 425 of this paper.
Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in ...

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138 views

### Is this function space a “classical” Sobolev space?

I apologise if this is indeed classical but my functional analysis is quite rusty...
My work recently led me to the norm: $(\|u\|_p)^p=\int_D (|u|^p+|\Delta u|^p)d\lambda$ where $D$ is the unit disk ...

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### Is $\partial^\alpha$ a map $H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)$?

More precisely, the question is formulated as follows. Let $F$ be an arbitrary Banach space, especially not having the UMD−property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . ...

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### properties of frequency-uniform decomposition operator $\square_{k}^{\sigma}$

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...

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### Closure of partial differential operators on $L^2(\Omega)$

Let $\Omega\subset\mathbb{R}^2$ open set. Consider an $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$
$$
H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha
$$
...

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105 views

### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...

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38 views

### Well-posedness of a Stefan problem with Faedo-Galerkin approach

Given a domain $\Omega$ which is divided by $\Omega_1(t)$ and $\Omega_2(t)$ and the interface $\Gamma(t)$, does anyone have a reference to where a Stefan problem of the type
$$\frac{d}{dt}H(u) - ...

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### 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 ...

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219 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 ...

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### 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 ...

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### 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 ...

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253 views

### Relation between interpolation spaces and besov spaces

Consider the following two norms:
The interpolation norm:
1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x ...

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384 views

### What is the Schauder estimate on usual Hölder space for parabolic type equations

What is the Schauder estimate on usual Holder space for parabolic type equations ?

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### Density of 0-homogeneous functions in $H^1(\partial \Omega)$

Recall: A function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is called $0$-homogeneous if
$f(\lambda x)= f(x)$ for every $\lambda>0$ and every $x\in \mathbb{R}^n$.
Question: Let $B$ a convex balanced ...

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292 views

### Normal form transformation

Hi, my question is related to normal form transformations...One of the papers I would like to understand is Wu, S., "Well-posedness in Sobolev spaces of the full water wave problem in 2-D"
where the ...