Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm that controls both a function and its weak derivatives in some Lebesgue space.

Sobolev spaces are function spaces generalizing the Lebesgue spaces. Whereas elements of Lebesgue spaces have certain integrability condition imposed on them, derivatives of functions in a Sobolev space are also required to be sufficiently integrable: that is, we require all (weak) partial derivatives of the function up to a certain order belong to a certain fixed Lebesgue space.

In more detail, let $U \subseteq \mathbb{R}^n$ be an open set. A weak $\alpha$th partial derivative $D^\alpha f$ of $f$ is a function $g\in L^1_{\mathrm{loc}}(U)$ such that $$\int_U f D^\alpha \phi \, dx = (-1)^{|\alpha|} \int g\phi \, dx$$ for each compactly supported smooth function $\phi \in C^\infty_c(U)$. the Sobolev space $W^{k, p}(U)$ consists of those functions $f\in L^p(U)$ such that for every multi-index $\alpha$ of length at most $k$, every weak partial derivative $D^{\alpha}f$ exists and is an element of $L^p$. The Sobolev spaces are equipped with norms defined by

$$\|u\|_{W^{k,p}(U)} = \begin{cases} \left( \sum_{|\alpha| \le k} \|D^{\alpha} u\|_{L^p(U)}^p \right)^{1/p} & p < \infty, \\ \max_{|\alpha| \le k} \|D^{\alpha}\|_{L^{\infty}(U)} &p=\infty .\end{cases}$$

$(W^{k,p}(U),\|\cdot\|_{W^{k,p}(U)})$ are Banach spaces for each $k\in\mathbb N$, and each $p\in[1,\infty]$. The norms measure both the size and the regularity of a function.

The basic fundamental result of Sobolev spaces is the Sobolev embedding theorem. In words, it says that (1) if $kp<n$, having $k$ weak derivatives in $L^p$ places your function in a better Lebesgue space $L^{p^*}$, where $\frac1{p^*} = \frac1p - \frac kn$, and (2) if $kp>n$, then your function is not only in $L^\infty$ but also has a continuous representative in some space of Hölder continuous functions. In particular, sufficiently many weak derivatives means that your function is in fact classically differentiable.

Reference: Sobolev space.

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Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $C^\infty(M)$ to be those functions which have all…
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Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have $\lVert uv\rVert_s \leq C \lVert u\rVert_s \lVert…
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Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient algebraic manipulations, but is there more behind it…
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Why are Sobolev spaces useful?

Why are Sobolev spaces useful, and what problems were they developed to overcome? I'm particularly interested in their relation to PDEs, as they are often described as the 'natural space in which to look for PDE solutions' - why is this? How do weak…
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The Sobolev Space $H^{1/2}$

In my course on linear PDEs, the professor used $H^{1/2}$ without defining it, and I have been looking on google trying to find a definition, but the only related thing I found was $H^{-1/2}$ as being the dual space to $H^{1/2}$ which does not…
Löwe Simon
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dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a Hilbert space is itself (in the sense of…
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What does "test function" mean?

I am trying to learn weak derivatives. In that, we call $\mathbb{C}^{\infty}_{c}$ functions as test functions and we use these functions in weak derivatives. I want to understand why these are called test functions and why the functions with these…
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Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form $$ \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} $$ It is well known that if $f$ is continuous and furthermore Lipschitz continuous in an…
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Why do we need semi-norms on Sobolev-spaces?

I have been studying Sobolev spaces and easy PDEs on those spaces for a while now and keep wondering about the norms on these spaces. We obviously have the usual norm $\|\cdot\|_{W^{k,p}}$, but some proofs also use the semi-norm $|\cdot|_{W^{k,p}}$…
dinosaur
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Suppose that there exist a set $\Gamma$ of positive measure such that $\nabla u=0, a.e.\ x\in\Gamma$.

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 $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in C_0^{\infty}(\Omega)$$ Suppose that there exist a set…
Tomás
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Are polynomials dense in Gaussian Sobolev space?

Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)$ under the inner product $$\langle f,g…
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How to prove the spectrum of the Laplace operator?

How can I prove that the spectrum of the Laplace operator $$\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$$ is $\sigma(\Delta)=[-\infty,0]$?
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Dual space of $H^1$

It holds that $W^{1,2}=H^1 \subset L^2 \subset H^{-1}$. This is clear since for every $v \in H^1(U)$, $u \mapsto (u,v)_{H^1}$ is an element of $H^{-1}$. Moreover for every $v \in L^2(U)$, $u \mapsto (u,v)_{L^2}$ is an element of $H^{-1}$. But I also…
tba
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Intuition behind losing half a derivative via the trace operator

This is an informal question, but here goes: For a function $f \in H^s(\Omega)$ ($s > 1/2$), there is a well-defined operator (the trace) $T$ such that $Tf = f\vert_{\partial \Omega}$ if $f \in C^\infty \cap H^s(\Omega)$ (so that it agrees with…
BaronVT
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Poincaré inequality for a subspace of $H^1(\Omega)$

The following is the well known Poincaré inequality for $H_0^1(\Omega)$: Suppose that $\Omega$ is an open set in $\mathbb{R}^n$ that is bounded. Then there is a constant $C$ such that $$ \int_\Omega u^2\ dx\leq C\int_\Omega|Du|^2\ dx\quad…
user9464
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