# 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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### Multivariate polynomial approximation

Let $f$ be a function on $[-1,1]^d$ with some smoothness property, for example, it is in the Sobolev space $W^{k,p}$. Let $P_n$ is a space of polynomials with degree $n$. My question is what is the ...
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### Regularity and decay of Fourier-like series on a manifold

Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...
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### Trace theorem for $L^2([0,1]; H^k(S^2))$

Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer. Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
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### Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?

The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
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### Image of trace operator on $W^{2,1}(\mathbb{R}^2)$

It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$. For ...
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### Finding an element of Gelfand triple with a designated time derivative

Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that $$V \subset H \subset V'$$ where $V'$ is the dual of $V$ and the inclusions are ...
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### Prove J.L. Lions’s Lemma without using Fourier transform

When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states Let $\Omega \subset \mathbb R^n$ be a ...
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### $T$ trace, then $Tg(u)=g(T(u))$ for all $u$ on $W^{1,p}$

The trace operator $T$ is defined for bounded domain $U$ with $C^1$ boundaries as the linear, continuous operator $T: W^{1,p}(U) \rightarrow L^p(\partial U)$ such that $$Tu=u\;\text{ on }\partial U$$...
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### Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function

Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
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### A function whose derivatives belong to $BMO(\mathbb{R}^n)$

I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp. 539-558. In this paper he defines ...
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### How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$

Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...
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### Fractional Bochner spaces $H^s((0,\infty); V)$

I'm looking for "intrinsic" definitions of fractional Bochner spaces such as $H^s((0,\infty); V)$ for a Banach space $V$ and exponent $s \in (0,1)$. I have seen these spaces being defined as ...
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### Derivative of a functional involving integral and level set

Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional $$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$ where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
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### $H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)

This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao. There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...
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### Weighted version of the Gagliardo Nirenberg inequality

I'm searching for a weighted Gagliardo-Nirenberg inequality similar as in https://arxiv.org/pdf/1307.1363.pdf where the weight is a power of the last component. Is there an inequality of the form  \...
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