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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2
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68 views

Dual space of Sobolev functions with homogenous neumann BC

Let $\Omega$ be a bounded Lipschitz-domain with outward normal vector $\nu$ and let us take a look at the Sobolev-spaces $H^1:=H^1(\Omega)$, $H^1_0:=H^1_0(\Omega)$ and ...
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61 views

Properties of the first eigenvalue of the $p$-Laplace operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $p\in (1,\infty)$, $p\neq 2$. Consider the usual Sobolev space $W_0^{1,p}(\Omega)$ and its dual $W^{-1,p'}(\Omega)$, where $1/p+1/p'=1$. Define ...
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423 views

Using the Sobolev-Nirenberg-Gagliardo inequality in a proof

If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u ...
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84 views

(Sobolev space) A star domain in $\Bbb R^n$

We know the theorem: If $\Omega$ is a star domain in $\Bbb R^n$, then $C^{\infty}(\overline{\Omega})$ is dense in $W^{k,p}(\Omega)$. It means that ...
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32 views

Time continuity of a solution to the generalized porous medium equation

Let $\Phi(x,s)$ be a smooth, non-negative function with $\Phi(\cdot,0)=0$ and $\Phi$ increasing in $s$ and $\partial_s \Phi(x,0)=0$ and $\Omega$ be a bounded smooth domain. Now assume there is a weak ...
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147 views

Variation on the Sobolev space $H^1_0$

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let $$ C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\}, $$ and let $C^1_c(\Omega)$ be the space of ...
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141 views

Don't understand proof of global solution to parabolic PDE

Let $u \in L^p(0,T;V)$ denote the solution to the parabolic PDE $$u_t + \Delta u = f\qquad\text{a.e $t \in [0,T]$}$$ where $u_t \in L^q(0,T;V^*).$ We have the usual assumptions on $V \subset H \subset ...
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47 views

PDEs on noncylindrical domains

Can someone give me some information about this area? PDEs on non-cylindrical domains I take to mean parabolic (let's fixed parabolic) PDEs on domains which evolve in time. What is the state of ...
2
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102 views

$L^\infty(0,T;L^\infty(\Omega))$ embedding into continuous functions?

For each $t$, let $u(t):\Omega \to \mathbb{R}$ be a function defined on a bounded domain $\Omega \subset \mathbb{R}^n$. We can think of this as $$u(t)(x) = u(t,x)$$ for $x \in \Omega.$ If $\nabla u ...
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73 views

Extending $W^{2,n}(\Omega)\cap C^2(\Omega)$-functions to $W^{2,n}(\Omega)$

I have the following theorem: Let $u\in C^2(\Omega)\cap W^{2,n}(\Omega)$ satisfy $Lu\geq f$ for $f\in L^n(\Omega)$. Then for any ball $B_{2R} = B_{2R}(y)\subset\Omega$ and $p>0$ where $u$ is ...
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69 views

weak derivative and the value of a integral

Let $0 < r < R$ and $p>1$ and consider the function $$u(x) = \displaystyle\frac{\displaystyle\int_{|x|}^{R} t^{-1 }dt}{\displaystyle\int_{r}^{R} t^{-1 }dt},$$ if $r < |x|< R$ , and ...
2
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327 views

Is the Laplace operator in Sobolev space $W^{2,1}$ sectorial?

Let $\Delta$ be the Laplace operator, that is $\Delta f = f''$. It is well known (see, e.g. this lecture notes, Chapter 2) that $\Delta\colon D(\Delta) \to L^1(0,1)$ with domain $D(\Delta) = ...
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108 views

Solving Dirichlet problem by means of potential theory

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and consider the Dirichlet problem with $f\in H^{-1}(\Omega)$ $$\tag{1}-\Delta u=f$$ Is there a way to solve this problem by using ...
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60 views

Multipliers in trace spaces

I need a reference for the following fact. Let $\Omega \subset \mathbb R^n$ be an open domain with $C^{1,1}$ boundary (maybe, less regularity is needed). Let $H^{1/2}(\Gamma)$ be the trace space of ...
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167 views

A bounded sequence

I have a question please : Let $f:[0,2\pi]\times \mathbb{R} \rightarrow \mathbb{R}$ a differential function satisfying : $\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq ...
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584 views

Stampacchia Theorem: $\nabla G(u)=G'(u)\nabla u$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $G:\mathbb{R}\to\mathbb{R}$ a Lipschitz function with $G(0)=0$. Stampacchia's Theorem states that if $u\in W_0^{1,p}(\Omega)$, then $G(u)\in ...
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113 views

$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product

I have ran across the following theorem but the given proof does not convince me. Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and ...
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44 views

“Compensated weak compactness” in $W^{1,1}$

In a problem I'm working on I want to show that a bounded sequence $\{u_n\} \subset W_0^{1,1}((0,1))$ converges weakly. Of course, since $W^{1,1}$ is not reflexive I don't get this for free. The ...
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100 views

A question on weakly convergence and norm convergence.

Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. ...
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126 views

A question on a bounded sequence in $H^1(\mathbb{R}^n)$.

Let $r>0$, and $2 \le q \le 2^*$. Suppose that $\{u_k\}_k$ is a bounded sequence in $H^1(\mathbb{R}^n)$ and $\lim_{k\to \infty} \sup_{y \in \mathbb{R}^n} \int_{B_r(y)}|u_k|^q dx \rightarrow 0.$ ...
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107 views

Sobolev spaces in $\mathbb R^n$ with functions having support on a closed set

I am interested in $H^s$ Sobolev spaces in $\mathbb R^n$ which have functions with support in a given closed set $K$ , denoted by $H^s_K$. Here $K$ is the complement of some bounded open set $\Omega$. ...
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106 views

Limit in norm of a Sobolev space

I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\}$ and I have to show that the function ...
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81 views

How to find a basis of $H^2(-N,N)$?

I need to find an orthonormal basis of $H^2(-N,N)$ where $N \in \mathbb{N}$ and $H^2$ denotes the Sobolev space $W^2_2$. I have no idea how to start. A hint to some literatur would be perfect.
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185 views

Why is this function space dense in this Bochner-type space? (Rogers and Renardy)

Let $\phi(t) = \sum_{i=1}^N\beta_i(t)\phi_i(x)$, where $\phi_i$ is basis for space $V$, and $\beta_i \in C_c^\infty(0,T).$ Renardy and Rogers says that 1) Functions of the above form are dense in ...
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74 views

Inequality involving Bessel potential.

I'm not able to prove the following inequality: Fix $s>0$ $$\|fg\|_{H^s}\lesssim \|fJ^sg\|_{L^2}+\|gJ^sf\|_{L^2},$$ where $\widehat{J^sf}(\xi)=(1+|\xi|^2)^{s/2}\hat{f}(\xi)$ (Bessel potential). ...
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79 views

Bound for a Sobolev function in an integral

For a compact bounded set $\Omega$, for the expression $$\int_\Omega \Delta u (\nabla u \cdot \nabla f)$$ where $u \in H^2$ and $f \in C^\infty$, is it possible to show that the expression is $\geq$ ...
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111 views

Convergence of Schwartz kernels implies convergence of operators

Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
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53 views

Unusual Compact Embeddings

Can anybody give a reference to the following two facts? The embeddings $$H_0^{1,2}(\mathbb R^n)\to L^2(\partial B_1(0))$$ and $$H^{1,2}(\partial B_1(0))\to H^{1/2,2}(\partial B_1(0))$$ are compact? ...
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98 views

Weak derivative and homeomorphisms commute

Suppose $\phi:H^1(S) \to H^2(R)$ is a linear homeomorphism, where $S$ and $R$ are compact surfaces in $\mathbb{R}^n$. Let $\varphi \in D(0,T;H^1(S))$ be a $H^1(S)$-valued $C_c^\infty(0,T)$ function. ...
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160 views

Sobolev-type inequality.

Let $0<\alpha<n$, $1<p<q<\infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$\left\|\int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha}} \right\|_{L^q(\mathbb{R}^n)} \leq ...
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75 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by $$W^{k,p}(\Sigma,M) = \{ u \in ...
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83 views

ODE with irregular coefficient

Here's a simple ODE \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} I want the solution $h(x)$ to be (at least) continuous with its first and second order derivative exist only ...
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201 views

An exercise about Sobolev Spaces

Let $\Omega\subset\mathbb{R}^n$ be an limited open set of class $C^1$ and $1\leq p<\infty$. Show that $$\bigcap_{m=0}^{\infty}W^{m,p}(\Omega)=C^{\infty}(\overline{\Omega}).$$
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134 views

How can we glue Sobolev functions?

Let $u:A\cup B\to R$ be a function (where $A$ and $B$ are disjoint connected sets and $A\cup B$ is connected) such that $u$ restrict to $A$ and to $B$ are in $W^{1, p}$. Which result guarantees me ...
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373 views

Weak convergence and limit of this sequence

Let $f_n$ be bounded uniformly in the $H^1$ norm, so we have (weak convergence) $$f_n \rightharpoonup f \qquad \text{in} \qquad H^1(\Omega\times [0,T]).$$ Then by compact embedding, we have strong ...
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101 views

Compactness of an embedding between weighted spaces

I read somewhere that if: $N\geq 2$ is an integer, $p\in ]1,N[$, $r>N/p$, $m\in L^r(0,a)$ (with $a>0$) and $m>0$ a.e. in $(0,a)$, then the weighted Sobolev space $W^{1,p^\prime} ...
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80 views

Difference between $C^\infty (U)$ and $C^\infty (\overline U)$

I am learning Sobolev spaces. There seem to be a difference while approximating a function in $W^{k,P}(\Omega)$ by smooth function $C^\infty (\Omega)$ and $C^\infty ( \overline \Omega)$, where we use ...
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89 views

Extension of Uncertainty Relations to a specific potential in Schrödinger Equation

Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle ...
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321 views

Adjoint of multiplication operator on Sobolev space

This is sort of an idle question, and I'll admit I didn't think very hard about it. Let $H^1 = H^1(\mathbb{R}^n)$ be the Sobolev space with norm $||f||_{H^1}^2 = ||f||_{L^2(\mathbb{R}^n)}^2 + ...
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31 views

Uniqueness of solution to Helmholtz-style equation

Let $\Omega\subset \mathbb{R}^n$ be open, bounded and connected with $C^1$ boundary. Suppose $q\in L^\infty(\Omega)$ and that $q\geq C>0$ almost everywhere. Consider the following PDE problem: ...
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15 views

Is the charateristic function $\chi _{\Omega }$ in the Sobolev space $W^{1,2}_{0}(\Omega)$?

Given $\Omega$ is a bounded, $C^1$ domain in $\mathbb{R}^n$. $\chi _{\Omega }(x)$ is the characteristic function of $\Omega$. I have done the followings: We can get $\chi _{\Omega }(x) \in ...
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16 views

Given an operator $Q$ between a Hilbert space $U$ and $L^2(ℝ^d;ℝ^d)$, is it possible to make sense of $U∋u↦(Qu)(x)$ for a fixed $x∈ℝ^d$?

Let $U$ be a Hilbert space $H:=L^2(\mathbb R^d;\mathbb R^d)$ for some $d\in\left\{2,3\right\}$ $Q$ be a Hilbert-Schmidt operator from $U$ to $H$. I want that $\tilde Q(x)$, where $$\tilde ...
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22 views

Show that the following are equivalent for $u \in C^{1}(0,1)$

Let $u \in C^{1}(0,1)$. Prove that the following are equivalent: (a) $u \in W^{1,1}(0,1)$ (b) $u' \in L^{1}(0,1)$ (where $u'$ means the derivative in the usual sense) (c) $u \in BV(0,1)$ My try: ...
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26 views

Question about the proof of $W^{1,p}_0(\Omega) \Rightarrow u=0 $ on $\partial \Omega$

I am reading the proof of Theorem 9.17 in the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. The theorem says: Suppose that $\Omega$ is of class ...
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0answers
27 views

Importance of Sobolev Spaces

Why Sobolev spaces are so important in study of partial differential equations? What could have light up the mind of researchers to use these spaces to analyze PDEs?
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14 views

Heat Equation stationary convergence

Consider the heat equation: $$u_t -\Delta u=0 \quad \text{in} \quad Q_T=\Omega \times(0,T) $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) \quad \text{in} \quad \Omega $$ a weak formulation is: find $u \in ...
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11 views

Given $u \in W^{1,p}(\Omega)$ then $\overline{\alpha u} \in W^{1,p}(\Omega)$, with $\alpha \in C^1 \cap L^{\infty},\nabla \alpha \in L^{\infty}$

Given $u \in W^{1,p}(\Omega)$ and $\alpha$ a function which verifies $\alpha \in C^1(\Omega)$, $\alpha \in L^{\infty}(\mathbb{R}^n)$, $\nabla \alpha \in L^{\infty}(\mathbb{R}^n)^n $ y $supp(\alpha) ...
1
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0answers
17 views

Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad ...
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0answers
50 views

Product of two $W^{1,p}_0$ functions

I have a fairly simple question. Suppose that $p>n$ and $u,v\in W^{1,p}_0$, how can I prove that the product is also in $W^{1,p}_0$ ? Of course, we have to employ Morrey's inequality. My idea was: ...
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0answers
23 views

Help showing compactness of the support of a function in the Sobolev Space $W^{1,p}$

In Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, in the proof of Theorem 8.12, it is needed to show that the support of a function is compact. The function ...