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4
votes
1answer
135 views

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 ...
2
votes
1answer
296 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 ...
1
vote
1answer
87 views

Strong maximum principle for weak solutions

Suppose I have a linear parabolic equation with solutions in the Bochner-Sobolev spaces (eg. $L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$). Is it possible to obtain a strong maximum principle with a proof that ...
0
votes
1answer
107 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 ...
11
votes
0answers
354 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 $K$. ...
6
votes
0answers
113 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) ...
5
votes
0answers
106 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 ...
5
votes
0answers
141 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: ...
5
votes
0answers
151 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 ...
3
votes
0answers
80 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 ...
3
votes
0answers
238 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 ...
3
votes
0answers
140 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 < ...
3
votes
0answers
193 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 ...
3
votes
0answers
183 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 ...
3
votes
0answers
192 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 ...
2
votes
0answers
50 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 ...
2
votes
0answers
62 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 ...
2
votes
0answers
112 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 ...
2
votes
0answers
93 views

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 ...
2
votes
0answers
115 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 ...
2
votes
0answers
194 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
0answers
455 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 ...
2
votes
0answers
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 ...
1
vote
0answers
105 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.
1
vote
0answers
95 views

Compact embedding

Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact. Is it true that the embedding $H^1_0(\Omega) \rightarrow ...
1
vote
0answers
67 views

$L^p$ 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 ...
1
vote
0answers
147 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 ...
1
vote
0answers
93 views

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 ...
1
vote
0answers
54 views

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 $1\le p\le +\infty$ . ...
1
vote
0answers
109 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) - ...
1
vote
0answers
98 views

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. ...
1
vote
0answers
115 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
0answers
221 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 ...
0
votes
0answers
125 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 ...
0
votes
0answers
127 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 ...
0
votes
0answers
35 views

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 ...
0
votes
0answers
61 views

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 $L^2(\Omega,\mathbb{C})$ $$ H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha $$ ...
0
votes
0answers
39 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) - ...
0
votes
0answers
114 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 ...
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 ...
0
votes
0answers
134 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
0answers
267 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 ...
0
votes
0answers
388 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 ?
0
votes
0answers
231 views

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 ...
0
votes
0answers
293 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 ...