The sobolev-spaces tag has no usage guidance.

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### A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} ...

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### Rellich Embedding Theorem for the $2$-Sphere

I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the $2$-sphere. To be precise, for $S$ the spinor bundle of $S^2$; $L^2(S^2)$ the space of square integrable ...

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### Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$.
Here, $F$ denotes the ...

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### elliptic regularity for Neumann BVP on square

I am interested in the regularity of ellitpic equations like
$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ ...

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### Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in ...

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### A space of distributions vanishing on the boundary

The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...

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54 views

### interpret of Picone inequality for non-regular functions

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...

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### Sobolev spaces, Finite Element Error Analysis

Let $\Omega\subset \mathbb{R}^{2}$ be a bounded, convex, polygonal domain and $H=\{(u_{1},u_{2})\in H^{\epsilon-\frac{1}{2}}(\Omega)\times ...

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### How to define Biharmonic operator for second order sobolev spaces

I am studying an article Link of Article. There author assumes that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Some where in the paper we have
$$ \Delta^2 (\cdot) - \frac{\lambda}{|x|^4} (\cdot) : ...

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### Characterization of Sobolev spaces

Assume that for some smooth bounded open subset $\Omega$ in $\mathbb{R}^n$ and some $u\in H^{1,2}(\mathbb{R}^n)$ we know that $\chi_{\Omega} u\in H^{1,2}(\mathbb{R}^n)$. Is it then true that $u\in ...

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### When is the identity Hilbert-Schmidt between weighted Sobolev spaces?

Set $w(x) = (1 + |x|^2)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space
$$H_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert_{s} := ...

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### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

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### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & ...

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### Fractional Sobolev spaces and extension by zero

The all-wise Wikipedia suggest (https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero) that for $s > 1/2$ any function from $H^s_0(\Omega)$ can be extended by zero
to whole $\mathbb{R}^n$ ...

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### Weighted Sobolev spaces over open/closed intervals

I am escalating this question from SE, as I have been unable to obtain any guidence in relation to a possible solution.
Some context, I am working with weighted Sobolev spaces of the form ...

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### Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, ...

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215 views

### The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on ...

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### Extension by zero in Sobolev spaces

Let $\Omega$ be an open bounded set of $R^n$, and let $\omega$ be an open subset of $\Omega$ s.t $\overline{\omega} \subset \Omega.$
For $f\in H_0^1(\omega)$, it is known that the extension of $f$ to ...

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### The Best Korn's constant for bounded deformation

I am studying the following version of Korn's inequality. For $u\in BD(\Omega)$, $BD$ denotes the bounded deformation space, we have, there exists a $r(u)\in \operatorname{ker}\mathcal E$ such that
...

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### Does this time-dependent trace space have a name?

This question is a follow up to this question.
Let $\Omega \subset \mathbb{R}^d$ be an open connected set. For each $t\in \mathbb{R}^+$ let $u_d:\partial\Omega \to \mathbb{R}$ be in ...

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### Second order differentiability of subharmonic function almost everywhere?

The following general definition of subharmonic function comes from the classical text book [elliptic partial differential equations of second order] by Gilbarg and Trudinger.
We call a function $u$ ...

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### Is this function $u\in SBV(\Omega)$ also belongs to $L^\infty(\Omega)$?

Some early discussion can be found here.
My question:
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Also $\mathcal H^{N-1}(\partial\Omega))<\infty$.
Let $u\in ...

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### Is this set of function belongs to $L^\infty$?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal ...

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### Points are removable for weakly differentiable functions

If $\Omega \subseteq \mathbb{R}^N$ is an open set and $N \ge 2$, then any point $a \in \Omega$ is removable for weakly differentiable maps: for each function $u \in W^{1, 1} (\Omega \setminus \{a\})$, ...

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### One parameter family of elliptic equations

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = ...

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### Are functions whose partial derivatives are simple functions dense in $W^{1,\infty}$?

In a 2D domain, are the functions whose partial derivatives are simple functions dense in $W^{1,\infty}$ ?

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### Are piecewise linear functions dense in $W^{1,\infty}$?

Are piecewise linear functions dense in $W^{1,\infty}$ ?

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### biharmonic equation with L^1 data and Navier Condition

I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...

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### Reference request: The compactness and compact embedding in Besov Space?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...

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### The best constant in Poincare-liked inequality in $BV$ and $BD$ space

This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention.
Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...

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### Regularity of weak solution

I have also posted the question here. Let me explain what difficulties I have. In fact, one may write
\begin{equation}
\partial_1(f-\partial_1 u)=0
\end{equation}
in $\Omega$. Then one may have the ...

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### “Friedrichs extension Laplacian” vs “Weak Laplacian” and fractional powers

Take $\Omega$ to be a bounded domain and consider Neumann BCs.
In some works, I see that a Laplacian $(-\Delta_D)^{\frac 12}$ is defined as an operator with domain $H^1(\Omega)$, and in other works, ...

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### Equivalence of Sobolev--Slobodeckii and interpolation space on boundaries

Let $s \in (0,1)$. Given a sufficiently smooth hypersurface $\Gamma$ in $\mathbb{R}^n$, one can define the Sobolev--Slobodeckii space with the norm
$$|u|_{H^s(\Gamma)}^2 = \int_\Gamma |u|^2 + ...

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### Trace spaces on convex polyhedra: compatibility conditions at edges

Let $\Omega$ be a convex polyhedron in $\mathbf{R}^3$ with boundary $\partial\Omega$ consisting of $N$ polygons $\{\Gamma_j\}_{j=1}^N$. It's well known that in general
$$ H^s(\partial \Omega) \neq ...

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### About norm on $H^{\frac 12}(M \times \{0,1\})$

Let $X=M \times \{0,1\}$ with $M$ a smooth compact manifold without boundary.
Define the fractional Sobolev space $H^{\frac 12}(X) = (L^2(X), H^1(X))_{\frac 12}$, as the real interpolation space ...

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### Sobolev space notation for interpolation and Gagliardo norm best practice

I work on a compact manifold $M$. I'm using three different Sobolev spaces:
$A^s(M)$ defined set of functions $u$ with the Gagliardo norm.
$B^s(M)$ defined as interpolation space between different ...

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### How to modify a SBV convergence sequence to obtain uniform integrability?

Given $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Assume $(u_n)\subset SBV(\Omega)$ a sequence of functions such that $u_n\to u_0$ weakly in $SBV$ for some function $u_0\in ...

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### Composition operators on fractional-order (periodic) Sobolev spaces

(The question was originally posted on MSE.)
Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication ...

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### Sobolev chain rule on non-compact manifolds

Let $(M,g)$ be a non-compact Riemannian manifold (not of bounded geometry).
Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$ with $f'$ bounded and $f(0)=0$. Is the Sobolev chain rule valid for functions ...

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### Is it true that for dimension d=3, if $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ then exponent of v or -v is not summable?

I have the following Question:
1) Is it true that
if $\Omega\subset\mathbb R^3$, $\Omega$ - bounded, $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ implies $\int\limits_{\Omega}{e^vdx}=+\infty$ ...

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### A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

(The question was originally posted on Math StackExchange.)
Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ...

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### Differentiation of $(u(t),v(t))_{L^2(\Omega)}$ when $u, v \in H^1(I\times \Omega)$

Let $I=(0,\infty)$. Consider $u, v \in L^2(I;H^1(\Omega))$ with $u_t, v_t \in L^2(I;L^2(\Omega))$ where $\Omega$ is a bounded doamin.
Is it true that
$$\frac{d}{dt}(u(t),v(t))_{L^2(\Omega)} = (u'(t), ...

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### Operator on a Sobolev space [closed]

I'm studying Sobolev spaces using Evans' PDE book.
I can't figure out this simple fact.
Let $L$ be an operator in this form:
$$Lu= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}.$$
I can't understand why ...

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### About weak derivatives [closed]

I have a question about weak derivatives.
Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...

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### Does strong convergence in $W_p^1$ imply strong convergence of derivatives of absolute values in $L_p$?

Let $G$ be a Lipshitz domain and $u_n \to u$ in $W_p^1(G)$. Is it correct, that $\frac{\partial |u_n|}{\partial x_m} \to \frac{\partial |u|}{\partial x_m}$ in $L_p(G)$?
I know, that $\frac{\partial ...

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### An inequality with critical Sobolev exponent

Let $\Omega\subset \mathbb{R}^n, n\geq 3$ be a nice bounded domain and $2^*=2n/(n-2)$ the critical Sobolev exponent. One may expect that $\forall \epsilon>0$, $\exists C_\epsilon<\infty$ such ...

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### Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ...

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### Elliptic pde with bilaplacian; boundary conditions.

I am interested in the solvability of
$$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$ with $ \partial_\nu u = \Delta u=0$ on $ \partial \Omega$ where $ f(x)$ is some smooth bounded function on $ ...

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### How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
...

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### Generalization of maximum principle to other norms

Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...