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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.

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How to show this Sobolev inequality?

I am currently studying numerical analysis and stumbled upon the following task: Prove that for any $q \in [2, +\infty)$, there exists $C \gt 0$ such that $\Vert f\Vert_{L^{q}(\mathbb{R}^2)} \le C\...
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Weak Convergence of sequence in a Sobolov Space.

Consider the question asked in here. I understood most of the answer in the question but the part about the weak convergence I did not get. To show that $u'_{n_k}$ converges weakly to $u$ in $L^p$, ...
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Norm Equivalence in Sobolev Space

I am attempting to demonstrate that for all $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$ \begin{align*} \int_U |u|^p \ dx \leq C\int_U |Du|^p \ dx \end{align*} where $U$ is some open subset in $\mathbb{R}^n$....
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How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
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Equivalence of Gradient and Hessian Norms in Sobolev Space

I am attempting to demonstrate that for $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$ \begin{align*} \int_U |Du|^p \ dx \leq C\int_U |D^2u|^p \ dx \end{align*} where $U$ is some open subset of $\mathbb{R}^n$. ...
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understanding Hilbert Sobolev spaces

We denote the space $\dot{H}^k$ for the homogeneous Sobolev space and $H^k$ for the inhomogeneous Sobolev space where $H^k=W^{k,2}$. It is true that de dual of $\dot{H}^k$ can be identified with $H^k$...
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Sobolev imbedding theorem $H^{1,p}(\mathbb{R}^n)$ contained in $L^{{np}/(n-p)}(\mathbb{R}^n)$ (Taylor Michael)

i. I can not make sense of the following: For (2.4) I imagine that it is the fundamental theorem of the calculation but I can not prove it formally. Neither will it be understood how to arrive at ...
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Using Hille-Yosida to show existence of solution to differential equation.

Suppose $I=(0,l)$ is an interval and $v\in C(\overline{I})$. Consider the differential equation $$\begin{cases}u_t=u_{xx}+v(x)u_x,\\ u(0,t)=u(l,t)=0,\;\;\\u(x,0)=u_0.\end{cases}$$ where $u=u(x,t):I\...
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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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Schauder regularity

My question is: If $\Omega$ is bounded with smooth boundary... Let $f \in C^{0,\alpha}(\overline{\Omega})$, and let $u$ be a weak solution of the Poisson equation $-\Delta u = f$. Then $u \in C^{2,\...
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$f(x)=\sin(x)\in H_{0}^{1}(\Omega)$ with $\Omega=(0,\pi)$?

We know that $f\in C^{\infty}(0,\pi)$ and $f(0)=f(\pi)=0$. But $supp\ f=[0,\pi]$. From the definition, $H_{0}^{1}(0,\pi)$ is the clusear of $C_{0}^{\infty}(0,\pi)$ in $H^{1}(0,\pi)$. Since, $f$ is not ...
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We could identity $H^{-1}$ with $H_0^1$ but we don't. Why?

Today in our lecture on partial differential equations while discussing dual spaces of Sobolev spaces: We could identify $H^{-1}$ with $H_0^{-1}$ by the $H_0^1$ inner product (Riesz) But won't and ...
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Density of smooth functions in the weighted Sobolev space $W^{1,p}((0,1),w)$, where $w(x)=x^{\lambda}$, $\lambda>0$

Let $1<p<\infty$ and $w(x):=x^{\lambda}$, where $\lambda>0$. One way to define the weighted Sobolev space $W^{1,p}((0,1),w)$ is the following (see [1]): we say that $u\in W^{1,p}((0,1),w)$ if ...
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Why is $f_\epsilon(u) \in W_0^{1,2}(\Omega)$?

For $\epsilon>0$ let $f_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$ One calculates that $\nabla f_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u $ , for $\epsilon$ to 0 this term goes to $\...
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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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$||.||_{\infty}$ in sobolev space

Let $u_n ∈ W^{1,1} (I), I=(0,1)$ defined by: $u'_n (x) = n$ if $x < 1/n$ $u'_n (x) = 0$ if $x > 1/n$ $u_n (0) = 0$. Find $||u_n − 1||_∞$. My attempt: We have $u_n(x)=u_n(0)+\int_0^{1/n}ndt=...
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Estimates for Poisson's equation

I am studying Calderón-Zygmund estimates ($L^p$ estimates) for Poisson equation. What I already know is: Let $\Omega$ be a bounded domain, $f \in L^p(\Omega)$, $1<p<\infty$, and let $w$ be the ...
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1answer
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The dual space of the Sobolev space $W^{k,p}(\Omega)$.

Let $\Omega$ be a nice domain in $\Bbb R^n$. It is known that any element $T\in\left( W^{k,p}(\Omega)\right)^*$ admits a (possibly non-unique) representation of the form $$ Tu = \sum_{|a|\le k} \int_\...
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Orlicz space is complete under the Luxemburg norm

I want to prove that Orlicz space is complete. Here is my attempt Let $(X,\mu)$ be a measurable space and suppose $\phi(x)$ is a young function. Define $$L^{\phi}=\{f\hspace{0.2cm} \text{measurable:}\...
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Question on proper choice of test function

Let $\Omega$ be a $\mathcal{C}^2-$compact manifold in $\mathbb R^2$ and consider the non-homogeneous heat equation: $$ \partial_t v - \Delta v = f \quad \text{in } \ \Omega \times (0,T), $$ where $f \...
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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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Sobolev embedding when $p=n$: $W^{1,p}(\mathbb{R}^{n}) \hookrightarrow L^{q}(\mathbb{R}^{n})$ for $q: p \leqslant q < \infty$

In class, aside from the standard Gagliardo-Nirenberg-Sobolev and Morrey inequalities, my professor also covered the case when $p=n$. In particular, if $p=n$, then $W^{1,p}(\mathbb{R}^{n}) \...
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Compact embedding of fractional space

Is the space $H^\lambda((0,T); H^1(K))$ for $0 <\lambda <1$ where $K$ is compact subset of $\mathbb{R}^n$ compactly embedded in $L^2( (0,T) \times K)$? $H^\lambda((0,T); H^1(K))\hookrightarrow \...
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1answer
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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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1answer
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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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Lipschitz constant in a bounded open set and infinity sobolev derivative

Let $V$ be a bounded open set. Let $f:V\rightarrow\mathbb{R}$ such that $f\in C(\overline{V})$ and $f$ is Lipschitz function on V ($Lip(f,V)<+\infty$). I would like to show that $Lip(f,\overline{V})...
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Is it true that: $u \in C^0(\bar U) \cap C^{\infty} (U) \Rightarrow u \in W^{k-1/p,p}(\partial U)$?

Let $U$ be an open, bounded subset of $\mathbb R^3$ with a $C^2-$boundary. For $k\ge 2$ and $1\le p<\infty$ is it true that: If $u \in C^0(\bar U) \cap C^{\infty} (U)$ then $u \in W^{k-1/p,...
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1answer
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Sobolev space identification

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$. It is well known that $L^2(\Omega\times (0,T))$ can be identified with $L^2(0,T;L^2(\Omega))$. Now, let us consider $$H^{1}(0,T;H_{0}^{1}(\Omega))=\{...
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Precise Definition of Embedding Theorem in Sobolev Spaces

Rellich Embedding Theorem Let $\Omega$ be a bounded domain, then $H_{0}^{1}(\Omega)$ is compactly embedded in $L^{p}(\Omega)$ (denoted by $H_{0}^{1}(\Omega)\subset\subset L^{p}(\Omega)$) for $1\leq p &...
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1answer
36 views

Density in $H^1[0,1]$.

Why is $\{u \in C^2[0,1] | u'(0)=u'(1)=0 \}$ dense in $H^1[0,1]$?
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A Poincare-like inequality

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set with continuous boundary. Prove that for each $\epsilon>0$ there is a constant $C(\epsilon)>0$ s.t. $$ \int_{\Omega}|f(x)|^pdx\leq C(\...
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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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About an equality of fractional Laplacian on a bounded domain

Let $0<s<1$. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. We know that $$\|(-\Delta)^{s/2}u\|_{L^2(\mathbb{R}^n)}^2=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+...
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1answer
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A convenient redefiniton of the Sobolev norm

I am dealing with the Sobolev space $W^{m,2}[0,1]$, i.e. the space of functions on $[0,1]$ with absolutely continuous $m-1$st derivative and square integrable $m$th derivative. I am using the norm $$|...
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1answer
14 views

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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26 views

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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2answers
29 views

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 $u(\lambda)= \int_0^{\lambda}v'(t)dt$ ...
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1answer
52 views

Is the projection onto the unit circle Sobolev?

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 ...
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1answer
42 views

How is this operator well defined? $\frac{D}{(1+D^2)^{1/2} }$.

Let $D_+ = \partial_x +x, D_-=-\partial_x+x$. $$D= \begin{pmatrix} 0 & D_- \\ D_+ & 0 \end{pmatrix} $$ which acts on a dense subspace $C_c(\Bbb R) \oplus C_c(\Bbb R)$ of $L^2(\Bbb R) \...
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9 views

Discrete equivalent of Sobolev norms and numerical experiment

I am solving a boundary value problem (BVP) that involves a system of equations (similar to the Euler or Navier-Stokes equations) for which, at this moment, there exists no sufficient theory to define ...
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1answer
32 views

Convergence of Sequence of Solutions to Elliptic Equation

Consider the standard uniformly elliptic equation on a domain $\Omega \subset \mathbb{R}^d$: $$ \mathrm{div}(A(x)\nabla u) = f $$ for $u \in H^{1}(\Omega)$ , $f\in H^{1} (\Omega)$, $a_{ij}(x)$ ...