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A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.

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inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
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22 views

Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
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59 views

Does this chain rule in Sobolev spaces hold?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded set. Let $S \subseteq \mathbb{R}^k$ be an open dense smooth submanifold of $\mathbb{R}^k$. Let $u \in W^{1,p}(\Omega,\mathbb{R}^k) \cap C(\...
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17 views

Approximation in parabolic Sobolev spaces

I am given the following function: fix any $p$ and $\beta \in (0,1)$ $$ u \in L^{p-\beta}(0,T;W^{1,p-\beta}(\Omega)) \quad \text{and} \quad \frac{du}{dt} \in L^{\frac{p-\beta}{p-1}}(0,T; W^{-1,\frac{...
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2answers
112 views

Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$. Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
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50 views

Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$ The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
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54 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 ...
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1answer
61 views

Steklov averages and negative parabolic sobolev spaces

Suppose one is given a function $$ w \in L^p(0,T;W^{1,p}(\Omega)) \qquad \text{and} \qquad \frac{dw}{dt} \in L^{p'}(0,T; W^{-1,p'}(\Omega)) $$ I am interested if the following holds: Denote the ...
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0answers
46 views

Sobolev embeddings for vector-valued functions

I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space. In particular, let $\Omega \subset \mathbb{R}^n$ be a ...
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41 views

PDE satisfied by projection of a function onto a subspace

Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE $$ \begin{cases} -\Delta_p u=f\;\text{in $D$}...
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44 views

Approximating norms using numerical integration? [closed]

I have a sequence of functions $u_m$ in $H^1(\Omega)$, where $\Omega$ is Lipschitz such that $u_m(x)=\int_{|y|\le \epsilon} \, f_m(x,y) \, dy$, but the integral cannot be expressed in terms of ...
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0answers
51 views

Fast growing unbounded functions in the Sobolev space $H^1(\Omega)$

I am looking for unbounded functions that grow rapidly fast near the origin, but are in the Sobolev space $H^1{(\Omega)}$, where $\Omega$ is a unit square centered at the origin. I already know about ...
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1answer
156 views

Estimates for the Sobolev inequality

How to prove the Sobolev estimate: If $\Omega$ is a bounded open subset of $\mathbb R^N$, then for any $q>1$ $$ \|u\|_{L^{q}(\Omega)} \leq C|\Omega|^{1/q} q^{1- 1/N}\| \nabla u \|_{L^{N} (\Omega)...
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1answer
99 views

Weak closedness of the extremal set to a linear inequality

Let $\mathcal{H}$ be a Hilbert space, let $p\ge 2$, and consider a bounded linear operator $ T\colon \mathcal{H}\to L^p(\mathbb R^d). $ Is the set $M=\{f\in \mathcal H\ :\ \|Tf\|_{L^p}= \|T\|\|f\|...
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0answers
112 views

Intersection of Sobolev space with the space of continuous functions

While doing some problems, I came across the space $H=H^1(\Omega) \cap C(\Omega)$, where $\Omega$ is subset of $\mathbb{R^n}$. So far, by definition of these subspaces, We know that none of these are ...
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3answers
211 views

Smallness of cut-off functions at critical Sobolev regularity

Consider the class of functions $$X:=\{f\in \mathcal{C}_0^{\infty}(\mathbb{R})\;s.t.\;f\equiv 1 \mbox{ in a neighbourhood of}\;\;x=0\}$$ Is it true that, for every $\varepsilon > 0$, I can find $...
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1answer
168 views

Regularity of the Jacobian of a $W^{2,n}$ Sobolev mapping

Given a mapping in the Sobolev space $f\in W^{2,n}_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the Sobolev regularity of the Jacobian $J_f=\operatorname{det} Df$. It is well ...
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1answer
314 views

A Poincaré-type inequality: proof or counterexample

The following is a simplified version of a Poincaré-type inequality that I'm studying; I'd like to prove it (the inequality) or find a counter example. Consider a function $f:[0,1]^2\rightarrow\mathbb{...
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0answers
64 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 $\varphi$...
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0answers
37 views

Is a Sobolev map with smooth minors smooth on the whole domain?

This question is related (but not identical) to this question. Let $d>2$ be an integer and let $2 \le k \le d-1$ be a fixed integer. Suppose that at least one of $k,d$ is not even. Let $\Omega$ be ...
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0answers
32 views

Fourier Lapalacian over periodic end

This is a technical question on Taubes' paper: Gauge theory over periodic end. on Page 378. Recall that: Let $Y$ be a closed manifold, with $b_1=1$, and $\tilde Y$ be the $\mathbb Z$-covering of $...
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1answer
49 views

Relation between a norm and norm of Besov spaces

Let $(H, \|\cdot\|)$ be a Hilbert space, $A \colon D(A)\subset H \longrightarrow H$ generates an analytic semigroup $T(t)$ on $H$. We define the following Banach space with the respect norm $$F=\{x\in ...
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38 views

Shifting Sobolev norms in a hyperbolic estimate

Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate: $$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...
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0answers
197 views

Is a Sobolev map with smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $d>2$ be an integer and let $2 \le k \le d-1$ be a fixed integer. Suppose that $k,d$ are both even. Let $\Omega$ be an open subset of $\mathbb{R}^d$, and let $f \...
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1answer
158 views

Continuity of a multiplication operator in fractional Sobolev space

Let $\Gamma$ be a regular boundary of a $C^{k,1}$ domain $\Omega$ and $H^s(\Gamma)$, $s\in(0,1)$, denote the fractional Sobolev space on $\Gamma$. Suppose I define a multiplication operator $M_\phi:H^...
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0answers
64 views

Exponential decay of a convolution

Let $z=(x,y) \in \mathbb{R}^N \times (0,+\infty)$, and let $$ P_m(z)=y^{2s} |z|^{-\frac{N+2s}{2}} K_{\frac{N+2s}{2}}(m|z|), $$ where $N \geq 3$ is an integer, $0<s<1$ and $K_{\frac{N+2s}{2}}$ ...
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64 views

How does the constant in Sobolev trace theorem depend on the domain geometric property?

Consider a domain $\Omega$ with Lipschitz boundary, $\Omega \subset \Re^N$ we can define fraction Sobolev seminorm $|\cdot|_{1/2, \partial \Omega}$: $$ |g|_{1/2,\partial \Omega}^2=\int_{\partial \...
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1answer
73 views

Fractional Sobolev spaces of order 0

For $1\leq p <+\infty$, $0<s<1$ and $\Omega\subset R^n$ domain, the fractional Sobolev space $W^{s,p}$ is defined as $$W^{s,p}(\Omega):=\big\{f \in L^p(\Omega)\colon \int_{\Omega} \int_{\...
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1answer
187 views

Regularity of solutions to $-\Delta u = \operatorname{div} F$, $F\in L^1$

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary. What are the regularity results for solutions to $$ -\Delta u= \operatorname{div} F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? $$...
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52 views

Regularity of level sets of Sobolev derivatives

I am interested in the regularity of the sets $$U_{\lambda}:=\{x: |\nabla^k u(x)|> \lambda \}$$ for a function $u\in W^{k,p}(R^d)$ with compact support. We can choose a lower semicontinuous ...
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1answer
152 views

Can I approximate a function of bounded variation with orthogonal polynomial?

Let function $u\in BV(\Omega)$ be a function of bounded variation and $\Omega\subset \mathbb R^2$ be a smooth domain. I know it is possible to approximate function $u$ with polynomials, i.e., $$ u = \...
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1answer
103 views

Is the maximal function bounded on the Besov space?

The Hardy-Littlewood maximal function of a function $f$ is defined by $$ M f(x):=\sup_{0<r<\infty}\frac{1}{|B_r|}\int_{B_r}|f(x+y)|dy, $$ where $|B_r|$ denotes the Lebesgue measure of the ball $...
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61 views

Coercivity of $\int (\Delta u + u)^2$ on a subspace of $H^2$?

Let $\Omega = [0,L] \times [0,2\pi]$ and split its boundary into $\Gamma_d = \{0,L\} \times [0,2\pi]$, $\Gamma^1_p = [0,L] \times \{0\}$, $\Gamma^2_p = [0,L] \times\{2\pi\}$. Consider the following ...
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1answer
67 views

Constructive definitions of the Sobolev trace

I have a function $u\in H^1_0(\Omega)$, $\partial\Omega$ being bounded and as nice as you need, and I need to prove that $u^k\in H^1_0(\Omega) $ for a fixed $k>1$ (integer if this can help). ...
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3answers
968 views

Research topics in distribution theory

The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of ...
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1answer
107 views

Solving classical parabolic equation by using Littlewood-Paley theory

Consider the following classical PDE in $R^n$: $$ \partial_tu(t,x)+\Delta u(t,x)+b(t,x)\cdot\nabla u(t,x)=f(t,x),\quad u(0,x)=0. $$ Is there any references on solving the above equation by using the ...
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1answer
110 views

Comparison of Bessel Capacities

The Bessel kernels $G_{\alpha},\, \alpha>0$ are defined by their Fourier transform $ \hat G_{\alpha}(\xi):= \frac 1 { (1+4\pi ^{2}\vert \xi \vert ^{2})^{\alpha/2}}. $ Bessel $(\alpha, p)$-capacity ...
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1answer
303 views

Proving the inequality $|\nabla |\nabla^r \psi|| \le |\nabla^{r+1} \psi|$

Following Aubin's book "Some nonlinear problems in Riemannian geometry", we use the notation $$ |\nabla^r \psi|^2 = \nabla_{\alpha_1}\cdots \nabla_{\alpha_r}\psi \nabla^{\alpha_1}\cdots \nabla^{\...
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0answers
72 views

Is the product of two Sobolev functions in L^p?

Assume that $f\in W^{\alpha-1,p}(R^n)$ with $0<\alpha<1$ and $p>2n/\alpha$. Given another function $ g\in W^{\beta,p}(R^n)$ with $\beta>0$. Under what conditions on $\beta$ can we get ...
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1answer
121 views

Morrey condition (integral condition) and (local) Holder condition

Let $x \in \mathbb{R}^n$ and $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$) $$\limsup_{r \to 0} r^{-\alpha \beta}\frac{...
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0answers
78 views

Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space

It is known that for $\alpha\in(0,1)$ and $p>1$, the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by $$ W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
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1answer
174 views

Is the Delta distribution a continuous functional on $H^1(\mathbb{R})$? [closed]

While it is easy to see that $H^1(\mathbb{R})$ are Hölder $1/2$-continuous, I started wondering whether this implies that $\delta_x(\varphi)=\varphi(x)$ is continuous as a functional $$\delta_x:H^1(\...
13
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1answer
274 views

Sobolev and Poincare inequalities on compact Riemannian manifolds

Let $M$ be an $n$-dimensional compact Riemannian manifold without boundary and $B(r)$ a geodesic ball of radius $r$. Then for $u\in W^{1,p}(B(r))$, the Poincare and Sobolev-Poincare inequalities are ...
3
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1answer
201 views

Gagliardo-Nirenberg inequality for bounded domain

For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that $$ \|u\|_4\le C \|u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}. $$ This is also true if $u\in W^{1,2}_0(\Omega)$ for a ...
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1answer
101 views

Sobolev Embedding on Finite Tube

Let $B=[0,T]\times Y$, here $Y$ denotes a closed manifold. Suppose we use the product metric on this finite tube. We can have the following inequality $$\|s\mid_{t\times Y}\|_{L^2_1(Y)}\leq Const \|...
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0answers
200 views

Traces of Sobolev spaces

Is there a simple proof of the following fact? Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset W^{1-\frac{1}{n},n}(\...
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0answers
28 views

Production of $H^s$ singularities in the strictly hyperbolic Cauchy problem

This question is a spin-off from Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$ as I am trying to find a solution without ...
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0answers
17 views

Hidden regularity for the coupled wave equation with dynamaic boundary condition

We have the equation \begin{equation} \left\{ \begin{array}{rrrr} u_{tt}-\Delta u=0,&\text{in} & \Omega \times ]0,T[ & \left( 1.1\right) \\ u=0, & \text{on } & \Gamma _{0}\...
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0answers
66 views

Trace embedding and unbounded domain

Let $D\subset\mathbb{R}^d$ be an open domain and let consider the open cylinder $D\times (0,T)\subset\mathbb{R}^{d+1}$ where $T\in (0,+\infty)$ arbitrary. Let $H^{1}(D\times (0,T))$ be the Sobolev ...
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0answers
28 views

Proving the existence of solutions of a coupled wave equation with dynamical boundary conditions

I want to prove the existence of the solution of this system by using the Faedo-Galerkin approximation method, I have to choose a basis for working on and I don't know how to do it in this case, I ...