Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

3,523 questions
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Sobolev spaces on domains and manifolds, what is the difference?

What are the (technical) differences between Sobolev spaces on domains $\Omega \subset \mathbb R^n$ or (compact) manifolds such as two-dimensional spheres?
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Are 2 Hilbertspaces with different inner products identical if the associated norms are equivalent?

suppose we have the Sobolev space $H^1_0(\Omega)$ over a bounded domain $\Omega \subset \mathbb{R}^2$. With the standard inner product it sure is a Hilbert space. BUT: What if we equip $H^1_0$ with ...
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Do invariant functions form a Banach (sub)manifold in function spaces?

Let $G$ be a topological group, and $X$ some function space; preferably a Sobolev space $X=W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is some invariant subset ($g\Omega \subset \Omega$) or ...
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$u \in W_0^{1,2}(\Omega) \Rightarrow |u| \in W_0^{1,2}(\Omega)$

For $\epsilon>0$ define $g_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$. One finds $\nabla g_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u$ and $g_\epsilon(u)\in W_0^{1,2}(\Omega)$ . Then ...
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Deriving Variational Formulation

Determine the variational formulation of \begin{cases} -\Delta u+u=xy \quad& \text{in } \Omega\\ \nabla u\cdot \vec{n}+2u=3 \quad& \text{in } \partial\Omega \end{cases} What I have tried: \...
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Is the embedding of $W^{2,p}$ onto $C^1(\overline{I})$ compact?

We know that when $I$ is a bounded interval and $1<p\leq \infty$ that the injection $W^{1,p}\subset C(\overline{I})$ is compact. The proof of this fact uses the Arzela-Ascoli theorem on the unit ...
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Is every linear finite element space over a bounded domain a subspace of the sobolev space H^1?

Since my knowledge of functional analysis, $L^p$-, Sobolev- and Hilbert spaces is not very good, I thought I could ask... Suppose we have a domain $\Omega \subset \mathbb{R}^2$ which is continuously ...
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function is convex in Sobolev space

Let $u ∈ W^{1,1} (]0, 1[)$ and $$J_\varepsilon(u) =\frac{1}{2}\int_0^1(\varepsilon+x^\alpha)u'^2dx+\frac{1}{4}\int_0^1u^4 dx-\int_0^¹uf dx.$$ Prove $J(u)$ is convex. My argument is sum of 3 ...
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Question about the Sobolev space $W^{1,p}(I)$.

Let $\{u_n\}$ be a bounded sequence in $W^{1,p}(I)$ where $1<p\leq\infty$ and $I$ is bounded. Then there is a subsequence $\{u_{n_k}\}$ and $u\in W^{1,p}$ such that $\|u_{n_k}-u\|_{L^\infty}\to 0$ ...
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Interpolation of weak derivatives

Is it true that if a function $f \in L^p$ has the property that for all $\vert\alpha\vert=k$ we have $D^{\alpha} f \in L^p$ then $D^{\alpha} f \in L^p$ for all $\vert \alpha \vert \le k$? This is ...
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$u \in H_0^1(\Omega) \implies u^+ \in H_0^1(\Omega)$ (boundary not $C^1$)

I know how to prove that $u \in H^1(\Omega) \implies u^+ \in H^1(\Omega)$ since this is Exercise 5.18 in Evan's PDE book. However, I'm not sure how to extend this to $H_0^1(\Omega)$. My first ...
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Are the derivatives of the orthogonal polar factor locally integrable?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a real-analytic map, satisfying $\det df>0$ everywhere except on a set of Hausdorff dimension not greater ...
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Is it true that $\int_{\Omega} u^2\leq \alpha \int_{\Omega} |Du|^2$?

Let $u\in H^1_0(\Omega)$, where $\Omega$ is a bounded open set in $\Bbb{R}^2$. Is the following true: For some $\alpha>0$, $\int_{\Omega} u^2\leq \alpha \int_{\Omega} |Du|^2$ I remember reading ...
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Is the normalized derivative of a holomorphic function Sobolev?

Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$ be the closed unit disk, and let $f:B \to \mathbb{C}$ be holomorphic. More precisely, I assume that $f$ is holomorphic on the interior $\text{int}(B)$, and ...
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Am I allowed to integrate by parts?

I have the following situation: Consider a function $f\in H^1(\mathbb{R})$, and a uniformly bounded function $S\in\mathcal{C}^\infty(\mathbb{R})$ with well-defined limits at $\pm\infty$ and such that ...
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Sobolev-Gagliardo-Nirenberg: Why is $|f|^q$ continously differentiable?

I wanna understand a proof of the Sobolev-Gagliardo-Nirenberg inequality. Therefore, I need to know why $|f|^q \in C_c^1(\mathbb{R}^n)$ for $f \in C_c^1(\mathbb{R}^n)$ and $q>1$. Can eventually ...
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Technique to prove existence?

I would like to prove the existence of a T-periodic function $v$ in $H^1(R^N)$ s.t. $v=|v|e^{i\theta}$ for some T-periodic $\theta \in H^1(R^N)$ and s.t. one certain functional $I_c$ turns negative at ...
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If $v\in H^1(]0,1[)$, then $|v(\lambda)| \leq ||v||_{L^1(]0,1[)} + ||v'||_{L^2(]0,1[)}$

Let $v\in H^1(]0,1[)$. I want to prove for all $\lambda \in [0,1]$ that, $$|v(\lambda)| \leq ||v||_{L^1(]0,1[)} + ||v'||_{L^2(]0,1[)}$$ My idea : I defined $u(\lambda)= \int_0^{\lambda}v'(t)dt$ ...
Let $f(x,y)=\frac{x}{\sqrt{x^2+y^2}}$. Does $f \in W^{1,p}(B)$ for some $p \ge 1$, where $B$ is the open unit disk in $\mathbb{R}^2$? (I guess we can replace $B$ with a disk with arbitrarily small ...