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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20 views

$\left\|f\right\|_{L^{\infty}(\mathbb R^d)} \leq K \left\|f\right\|_{H^{s}(\mathbb R^d)}$

The Swchartz, $\mathcal S(\mathbb R^d)=\left\{f\in C^{\infty}(\mathbb r^d): \sup_{x\in \mathbb R^d}(1+|x|^{2})^{\frac{k}{2}}\sum_{|\alpha\leq l|}|D^{\alpha}f(x)|< \infty\right\}$, for all $k, l \in ...
2
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74 views

Traces of Sobolev functions in an unbounded domain

I have a doubt concerning the trace of Sobolev functions. Let $C=\Omega\times(0,\infty)$ an infinite cylinder of basis a smooth domain $\Omega$ of $\mathbb{R}^{N}$ and consider the classical Sobolev ...
2
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64 views

Alternative derivation of Poincaré inequality

I've been trying to prove the Poincaré inequality via a representation formula for Sobolev functions $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p < n$, wlog with compact support. The setting is ...
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166 views

Can we characterize homogeneous Sobolev spaces by means of Sobolev embedding?

For $\alpha>0^{[1]}$, let $D^\alpha=(-\Delta)^{\frac{\alpha}{2}}$ be the operator defined on all Schwartz class functions $f$ as follows: $$ \widehat{D^\alpha f}(\xi)=\lvert \xi\rvert^\alpha ...
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72 views

Gagliardo-Nirenberg-Sobolev inequality

There is one step of the textbook's proof that I wish could be clarified. Pages 277-278 of PDE Evans, 2nd edition, says: Integrate this inequality with respect to $x_1$: \begin{align} ...
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67 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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52 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
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78 views

Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$. It follows that for almost all $t$, $u_n(t)$ is bounded in ...
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62 views

Want a dense subset of a Sobolev-Bochner space!!

Let $U \subset \mathbb{R}^n$ be a bounded domain. Let $$W=\{ u \in L^2(0,T;H^1(U))\cap L^\infty(0,T;L^\infty(U)) : u' \in L^2(0,T;H^{-1}(U))\}$$ and let $$D=\{u \in L^2(0,T;H^1(U)) \cap ...
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44 views

Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
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70 views

Why is this bilinear form elliptic on $H^1_0$?

I have the following bilinear form: $$ a(v,w) = \int_\Omega \nabla v(x) \cdot \nabla w(x)dx + \int_\Omega \left( \sum_{i=1}^n \beta_i v_{x_i}(x) \right) w(x) dx $$ where $\Omega \subset ...
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57 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
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48 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
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117 views

Sobolev Spaces separable

How do I demonstrate that the Sobolev spaces $W^{1,\infty}$ is not separable? PS: I know that space $L^{1,\infty}$ is not separable but was unable to use this information.
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41 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
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50 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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53 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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309 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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82 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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30 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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132 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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135 views

Generalized chainrule for Sobolev functions with a cut-off

let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $f\in C^1(\bar\Omega \times (0,g(x)),[0,1])$ and $f(x,\cdot)$ increasing and $g(x)\in\mathbb{R}$ continuous (maybe better, Lipschitz?). I want ...
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113 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 ...
2
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38 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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95 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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63 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 ...
2
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65 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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247 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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95 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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57 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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161 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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350 views

Showing that smoothing operators are compact

Suppose I have a bounded, linear map $T: H^1(X) \to H^1(X)$ such that $T(H^1(X)) \subset C^\infty(X)$. Is $T$ a compact operator? I'm guessing this depends on whether or not $X$ is (pre)compact, and ...
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427 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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97 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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96 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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121 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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91 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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95 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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75 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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67 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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75 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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100 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) ...
2
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93 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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142 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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69 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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81 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 ...
2
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192 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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130 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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325 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 ...
2
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0answers
97 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} ...