The sobolev-spaces tag has no usage guidance.

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406 views

### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...

**6**

votes

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76 views

### Best constant for a trace inequality

Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality
$$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla ...

**5**

votes

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92 views

### 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 $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

**5**

votes

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270 views

### Which functions can be approximated by piecewise constant functions?

Let $\Omega \subset \mathbb{R}^d$ be a connected and bounded domain. We call a function $f\in L^\infty(\Omega)$ nice if for each $\epsilon>0$ there exist $n\in \mathbb{N}, a_1,\dots,a_n \in ...

**5**

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173 views

### Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: ...

**5**

votes

**0**answers

171 views

### A finely open set, not open up to polar set?

I already asked this on M.SE, but get no answers.
Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...

**4**

votes

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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 ...

**4**

votes

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101 views

### Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts:
Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...

**4**

votes

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214 views

### Alternative representations of Sobolev space

Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...

**4**

votes

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270 views

### Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is
\begin{equation}
\omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})}
\end{equation}
where ...

**3**

votes

**0**answers

45 views

### 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 $W^{1,\infty}$ ?

**3**

votes

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72 views

### Chain rule for weakly differentiable functions

Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ...

**3**

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57 views

### Pointwise (a.e) evaluation of $\sum_{n \geq 0}(u,w_n)_{L^2}w_n$ and equalities in $L^2$

Let $w_n$ be a orthonormal basis of $L^2(\Omega)$. Given $u \in L^2$ we can write $$u=\sum_{n \geq 0}(u,w_n)_{L^2}w_n.$$
Suppose $w_n$ are the eigenfunctions of the Neumann Laplacian. We can write
...

**3**

votes

**0**answers

120 views

### Lipschitz continuity of a composition operator

Let $M$ be a compact Riemannian submanifold of $\mathbb{R}^K$, $U\subset \mathbb{R}^K$ an open neighboorhood of $M$ such that the shortest point Projection $P_M\colon U\rightarrow M$ is well-defined ...

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149 views

### Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space
$$X = \{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} ...

**3**

votes

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133 views

### Bounded functions dense in Sobolev Spaces

Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...

**3**

votes

**0**answers

153 views

### Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation
$$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$
$$u(0) = u_0$$
$$u|_{\partial\Omega} =0$$
for all $v \in ...

**3**

votes

**0**answers

173 views

### Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t ...

**3**

votes

**0**answers

147 views

### Linear interpolation in weighted Sobolev spaces

I am reading a paper regarding the pricing of certain financial derivatives making use of the finite element method. In this paper the following weighted Sobolev spaces are introduced:
$W_{0}$ = $ \{ ...

**3**

votes

**0**answers

135 views

### convergence of $e^{it\Delta}f$

I heard of a conjecture that
$e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$
but couldn't find a proper reference.

**3**

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97 views

### Existence and smoothness of convolutions of distributions in Sobolev spaces

Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative.
It is easy to show that $f *g$ is defined pointwise when ...

**3**

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181 views

### Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required.
Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy
$$0 < a \leq g(x,t) \leq b < ...

**3**

votes

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184 views

### Trace Theorem for $p=\infty$

I am considering the Sobolev space $W^{1,\infty}(\Omega)$ on a bounded Lipschitz domain $\Omega \subseteq \mathbb{R}^2$. I am wondering whether the trace theorem holds in this case with constant one ...

**3**

votes

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189 views

### Degree of a function in $H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$

We can define the degree of a function $f \in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$ as
$$\mathrm{deg} \hspace{1mm} f = \frac{1}{2\pi i} \int_{\mathbb{S}^1} f^{-1} \frac{\partial f}{\partial ...

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200 views

### $\tilde{u}=0,\ a.e.\ x\in\Gamma$?

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 ...

**3**

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498 views

### Open sets and Poincaré's inequality

In many references, Poincaré inequality is presented in the following way :
Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...

**3**

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216 views

### Where can I find interpolation inequalities for derivatives of the following form?

Here they are:
$$||f||_{\infty} \leq C ||f||_q^{\frac {qk} {n+kq}} \left( \sum_{|\mu|=k} ||D^\mu f||_{BMO} \right)^{ \frac n {n+kq}}$$
and
$$||f||_{Lip_\alpha} \leq C ||f||_q^{\frac {qk} {n+kq} \frac ...

**2**

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60 views

### 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 ...

**2**

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56 views

### 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 $ ...

**2**

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**0**answers

39 views

### 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 ...

**2**

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**0**answers

13 views

### 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 ...

**2**

votes

**0**answers

47 views

### 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 $u \in W^{1, 1} (\Omega \setminus \{a\})$, ...

**2**

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**0**answers

178 views

### 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 = ...

**2**

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**0**answers

179 views

### 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 ...

**2**

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**0**answers

65 views

### Sobolev embedding on warped product

Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote ...

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49 views

### Finding an explicit constant in finite element error estimates

Background: In a finite element approximation to the solution of a linear PDEs, estimates on the order of convergence of the approximation to the solution rely on a theorem of Bramble and Hilbert ...

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110 views

### Sobolev trace theorem

Set $Q:=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$,
where $\Omega$ is knows as a bounded domain
with smooth boundary $\partial D$.
We choose any subdomain $D\subset Q$
with smooth boundary $\partial ...

**2**

votes

**0**answers

85 views

### Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)

Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve
$$\int_0^\infty\int_\Omega \nabla v\nabla ...

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137 views

### Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...

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165 views

### Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)

Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$.
I am ...

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96 views

### Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...

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**0**answers

92 views

### The definition of $W_0^{1,\infty}$

I know how to define $W_0^{1,p}(\Omega)$, $\Omega\subset R^N$ open bounded smooth boundary, for any $1\leq p<\infty$. However, for definition of $W_0^{1,\infty}(\Omega)$, I always confused. ...

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131 views

### Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point.
I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...

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62 views

### Approximating a superharmonic function, by smooth superharmonic functions

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$, $u\ge 0$ and $-\Delta u\ge 0$ in the sense of distributions ($u$ is superharmonic).
The standard ...

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**0**answers

82 views

### Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...

**2**

votes

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137 views

### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...

**2**

votes

**0**answers

98 views

### Almost a Green formula

Let $\Omega$ be the half-space $\mathbb{R}^{n-1}\times \{ x_n>0 \}$, let $v \in L^2(\Omega)$ and $\phi\in \mathcal{C}^{\infty}(\overline{\Omega})$ with compact support in $\overline{\Omega}$. What ...

**2**

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**0**answers

139 views

### compact embedding in duals of weighted Sobolev spaces

On the whole space $\mathbb{R}^d$ consider the weight $\omega(x)=\sqrt{1+|x|}$. Under which conditions on $k,q$ is the embedding
$$
L^p(\mathbb{R}^d,\omega(x)dx)\subset\subset ...

**1**

vote

**0**answers

31 views

### 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 ...

**1**

vote

**0**answers

35 views

### 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) : ...