# Questions tagged [sobolev-spaces]

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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### A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
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### Well-posedness in modified H2 space

Can I modify the $H^2$ space such that: $$\tilde{H}^2 := \left[ u(\Omega): ||u||^2_{L_2(\Omega)} + ||\nabla u||^2_{L_2(\Omega)} + ||\Delta u||_{L_2(\Omega)}^2 < \infty \right]$$ and then use the ...
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### Trace inequality normal derivative

For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality: $$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$ which can be found in many places in the ...
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### Inclusion $H^1_0(\Omega)\cap C^k(\Omega)\subset C^0(\overline\Omega)$?

Let $\Omega\subset {\bf R}^d$ be a bounded domain with a Lipschitz boundary. Assume that a function $u$ is $C^k$ inside $\Omega$ and that $u$ also belongs to $H^1_0(\Omega)$. Can one conclude that $u$ ...
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### Is there any nontrivial characterization of weakly differentiable functions?

When $f\in L_\text{loc}^1$, it's distributional derivative can be defined as $D_{f'}\in\mathfrak{D}'$, such that $D_{f'}(\varphi)=-\int f\varphi'$ for all $\varphi\in\mathfrak{D}$, where $\mathfrak{D}$...
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### Compact embedding of space of signed Radon measures into Sobolev space $W^{-1,q}$ from Evans paper; Does it work in one space dimension?

Background: I work on a PDE problem where I have some approximating sequence of measure-valued functions and I need to compactly embed it into some negative Sobolev space $W^{-m,q}$ on the bounded ...
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### Analogous $H^1$-space for pseudo inner products

Perhaps this is a naive question but I could not find anything related to this. Imagine we are on a bounded and regular open subset $\Omega$ of $\mathbb{R}^3_1$, i.e, $\mathbb{R}^4$ is considered ...
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### Covering number for the unit ball in a reproducing kernel Hilbert space

I am looking for a reference for an upper bound on the covering number for the unit ball $\{ f \in \mathcal{H}: ||f||_{\mathcal{H}} || \leq 1\}$, where $\mathcal{H}$ is a reproducing kernel Hilbert ...
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### Compatibility between the source and the boundary condition for an Helmholtz-type equation

Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
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### Limit case of Sobolev space in $1$-D

This might look too an elementary question, but I am confined and is not able to find a textbook which answers the following question. I have a function $f:{\mathbb R}\rightarrow{\mathbb R}$, such ...
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### Approximation of a Sobolev map with fixed singular values by smooth maps with the same singular values

Let $0<\sigma_1<\sigma_2$, and let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Let $f \in W^{1,\infty}(D,\mathbb{R}^2)$, and suppose that the singular values of $df$ are a.e. equal to ...
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### Second order differentiability of convex functions

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...
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### Stability of fractional Sobolev spaces under diffeomorphisms

Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the ...
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### Orlicz-Sobolev Spaces

let $A$ an N-function and suppose that $$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty$$ we denote by $\widehat{A}$ an N-function equal to A near infinity and $\widehat{A}$ ...
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### On a interpolation inequality for the Schrödinger unitary group (NLS)

I'm trying to understand scattering for the classical nonlinear Schrödinger equation and for that i'm studying a scattering criterion on Tao's paper. At Lema 3.1 he states that \left\|e^{it\Delta}f\...
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### Extension Operator for $W^{1,\infty}(U,X)$

I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
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### Non-convergence to a Gaussian

Let $f_n: \mathbb R^2 \rightarrow \mathbb R$ be a family of probability distributions with the property that they vanish on the diagonal $f_n(x,x)=0.$ I would like to know: Can we show that a ...
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### Approximate constant function

Let $f:[0,1]^2 \rightarrow \mathbb C$ be an $H^1$ function with the property that $f(x,x)=0$ and $\Vert f \Vert_{L^2[0,1]}=1.$ Does there exist a constant $c>0$ such that any such function ...
### Fourier transform for $H^2(\mathbb{R}^N)$, $N\geq 5$
How i can prove that if $u\in H^2(\mathbb{R}^N)$ then $u\in \mathcal{F}(L^{p^*}(\mathbb{R}^N))$, where $1/p+1/{p^*}=1,$ $2\leq p<2N/(N-4)$?
Let $f\colon\mathbb{R}^n\rightarrow\mathbb{C}$ be a bounded function with a bounded first derivative. Then the multiplication operator $H^1(\mathbb{R}^n)\ni x\mapsto A_f x:=fx\in H^1(\mathbb{R}^n)$ is ...