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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1answer
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Smooth extensions from boundary of convex domains

Given a convex domain $\Omega$ with piecewise $C^1$ boundary data $g \in C(\partial \Omega)$ and $g \in C^1(\Gamma_i)$ with $\bigcup_{i} \Gamma_i = \partial \Omega$. Now, I want to know if there ...
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1answer
67 views

Compact embedding into boundary

I was looking for a result: $W^{1,2}(Q)$ is compactly embedded in $L^{2}(\partial Q)$ ; where Q $\subset \mathbb R^{2}$ is a bounded domain & $Q \in C^{1,1}$." (which is mentioned in the ...
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1answer
66 views

Why does the mollified function converge uniformly to the original $W^{1,\infty}$ function

I'm studying Evans' PDE book and on page 294 it states that if we have $u \in W^{1, \infty } (\mathbb R^n)$ and has compact support, and take $\eta _\epsilon$ as the standard mollifier, then $\eta ...
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0answers
20 views

Discretization of $W^{1,\infty} (\mathbb R^N, \mathbb R^N)$

I am working on a shape differentiation problem. I have a theoretical result that I want to implement numerically. Does anyone know of a reference for a discretization (finite sequence of subspaces ...
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0answers
38 views

Question about the interpret of Picone inequality for non-regular functions.

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open. 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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0answers
43 views

How do I show that a distribution is a function?

While reading Grigor'yan's book on the heat kernel I have encountered the following definition of a Sobolev space on a Riemannian manifold $M$: $W^2 (M) = \{ u \in W^1 (M) : \Delta u \in L^2 (M) ...
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1answer
62 views

Fractional Sobolev spaces definition

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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2answers
64 views

show that a certain bounded linear operator does not exist

I'm trying to do problem 8 from section 5.10 on Evans' PDE book. Basically the problem asks if $U$ is a bounded open subset of $\mathbb R^n$ with $C^1$ boundary and $u \in L^p (U)$, show that ...
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1answer
62 views

infimum of a functional in $W^{1,p}((0,1))$

Consider the functional $$\mathcal{F}(u)=\int_{0}^{1}x^{\alpha}|u'(x)|^pdx,\ \ u\in W^{1,p}((0,1)),$$ where $\alpha\ge 0$ and $1<p<\infty$. Given $a<b$, find the value of ...
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1answer
43 views

How to define Biharmonic operator for second order Sobolev spaces and show that it is continuous?

I am studying an article where the authors assume that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Somewhere in the paper we have $$ \Delta^2 (\cdot) : W^{2,2}(\Omega) \to W^{-2,2}(\Omega) $$ (This ...
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0answers
20 views

Infimum of a function on Sobolev spaces

Let $\alpha>0$ and $1<p<\infty$ such that $p>\alpha +1$. For $u\in W^{1,p}((0,1))$ such that $u(0)=a, u(1)=b, a<b$, using the Holder inequality, we get ...
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0answers
31 views

Find the infimum value of a functional.

Consider the functional $$F(u)=\int_{0}^{1}x^{\alpha}|u'(x)|^pdx,\ \ \ u\in W^{1,p}(0,1)$$ where $\alpha\ge 0$ and $1<p<\infty$. Given $a<b$, find the value of $$\inf\{F(u): u\in ...
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0answers
22 views

Understanding the embedding of $W^{\infty, 2}$

I am trying to understand Sobolev's embedding theorem, more precisely to understand when a Sobolev generalized function of infinite order is smooth of some order. Consider the following statement ...
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0answers
14 views

Sobolev space for classic function approximation?

Hi guys could be any convenience in using a sobolev space instead of square integrable space for function approximation? I know that sobolev space are mostly used for PDE, but i was wandering if ...
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0answers
44 views

Sobolev spaces and symmetric operators

I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following: Suppose we are ...
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0answers
29 views

Bessel potential action on a product of function

Let's consider the Bessel potential $$J^s:=(I-\Delta)^{\frac{s}{2}}$$ Does it exist some kind of Leibniz rule for $$J^s(fg)$$?
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1answer
18 views

Is Jacobi weight power type or general type Sobolev weight?

Motivated by Shuhao Cao's answer in Weighted Poincare Inequality, I checked out Kufner's book weighted Sobolev spaces. Question Is the Jacobi weight either a power-type weight or a general weight in ...
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2answers
68 views

Mistake in reasoning about Sobolev spaces

I am new to Sobolev spaces and, while trying to construct a proof, I make some subtle mistake that I cannot detect. The setting: let $C \subset \Bbb R^n$ be a closed, measure-$0$ set. Let $U = \Bbb R ...
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1answer
36 views

Given $u\in L^2(\mathbb{R}^N)$ and $\nabla u \in L^\infty(\mathbb{R}^N)$, is $u\in L^\infty(\mathbb{R}^N)$?

Given $u \in L^2(\mathbb{R}^N)$ and $\nabla u \in L^\infty(\mathbb{R}^N)$, is $u \in L^\infty(\mathbb{R}^N)$? Can I use Morrey's inequality? $$|u(x) - u(y)| \leq C_N \|\nabla u\|_\infty |x-y| ...
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1answer
40 views

$\||u|\|_{H^s(\mathbb R^n)} \le C \| u \|_{H^s(\mathbb R^n)}$ holds for even if $s$ is not an integer?

Let $u: \mathbb R^n \ni x\mapsto u(x) \in \mathbb C.$ I would like to know that the inequality $$ \||u|\|_{\dot H^s(\mathbb R^n)} \le C \| u \|_{\dot H^s(\mathbb R^n)} $$ or $$ \||u|\|_{H^s(\mathbb ...
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1answer
57 views

Elementary Sobolev space problem

For which $k$ does the following function belong to Sobolev space $H^k(-1,1)$: $$f(x) = \begin{cases} x e^{- \frac{1}{x} } & x > 0\\0 & x \leq 0 \end{cases}$$
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1answer
41 views

Does fundamental theorem of calculus hold for weakly differentiable function?

Does fundamental theorem of calculus hold for weakly differentiable function? That is $\int^b_a$$f'$=$f(b)-f(b)$ for $f$ being weakly differentiable??
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1answer
125 views

Absolute value of functions in Sobolev space

I'm just trying to show that if $u\in W^{1,p}(U)$ (with $1\leq p<\infty$) then $|u|\in W^{1,p}(U)$. Here $U$ is bounded. I originally took smooth functions $a_n$, equal to $|x|$ whenever $|x|\geq ...
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1answer
114 views

Prove that $\|u\|_{L^\infty(\mathbb R)} \le C\|u\|_{H^1(\mathbb R)}$

Use the Fourier transform to prove that if $u \in H^s(\mathbb R^n)$ for $s > \frac n2$, then $u \in L^\infty (\mathbb R^n)$, with the bound $$\|u\|_{L^\infty(\mathbb R^n)} \le ...
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1answer
45 views

New function defined by the trace of a $H^1$ function

Let $\Omega:=B(0,1)\subset \mathbb R^N$ the unite ball and $N\geq 2$. Given $u\in H^1(\Omega)$. Then the trace $T[u]$ is well defined over $\partial \Omega$. (by $H^1$ I mean $W^{1,2}$ space) Now ...
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1answer
81 views

Homogeneous Sobolev space is a Hilbert space

I am reading a book and I have some questions about the proof. The book wants to show $H^s(\mathbb{R}^d)$ is a Hilbert space iff $s<\frac{d}{2}$. $H^s(\mathbb{R}^d)$ is the homogeneous Sobolev ...
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0answers
84 views

Is the space $\mathcal{H}=\{v\in H^1(\Omega):\Delta v\in H^1(\Omega)^*\}$ a Banach space?

Let $\Omega$ be a Lipschitz domain in $\Bbb R^n$, is the space $$\mathcal{H}=\{v\in H^1(\Omega):\Delta v\in H^1(\Omega)^*\}$$ a Hilbert space when endowed with the norm $\|\cdot\|_\mathcal{H} = ...
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3answers
77 views

Divergence theorem in $H^1(\Omega)$.

Let $u,v\in H^1(\Omega)$, where $\Omega$ is a Lipschitz domain in $\Bbb R^n$. It is my understanding that the divergence theorem tells us $$\int_\Omega\nabla u\cdot\nabla ...
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1answer
51 views

Estimate the integral of a function on the boundary using the integral of its gradient

Show that for a sufficiently smooth boundary of $\omega$ and any $\epsilon > 0$, there exists $C$ such that $$\int_{\partial \omega} u^2\ ds \leq C\int_{\omega} u^2\ dx + \epsilon \int_{\omega} ...
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0answers
28 views

Can one explicitely construct a sequence of functions of compact support approximating $u\in W_{0}^{1,p}(\Omega)$?

We define $W^{1,p}_{0}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$ in the $W^{1,p}$-norm (or equivalently as the closure of the $W^{1,p}$-functions with compact support). Given $u\in ...
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1answer
49 views

Duals of Sobolev Spaces vanishing on parts of the boundary

I am revising for a Finite Elements course and have the following question about the definition of $H^{-1}$. Let $D\subseteq \mathbb{R}^2$ be bounded Lipschitz domain and let $\Gamma_0 \subseteq ...
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83 views

Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
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1answer
79 views

Double integral definition of periodic Sobolev spaces

Preliminaries: For $s \geq 0$ define the periodic Sobolev spaces $H^s(\mathbb T)$ with norm $$ \| f \|_{H^s(\mathbb T)}^2 = \sum_{k \in \mathbb Z} \big(1 + |k|^2 \big)^s \, \big|\widehat{f}_k \big|^2, ...
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1answer
53 views

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} ...
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1answer
43 views

Simple Inequality for Proving Equivalent Besov Seminorms

For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty$, and $h\in\mathbb{R}^{n}$, define the quantity $$I_{p}(h):=\left(\int_{\mathbb{R}^{n}}\left|f(x+h)-f(x)\right|^{p}dx\right)^{1/p}$$ and define ...
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2answers
71 views

Continuous function not sobolev

Let $I=(a,b)$ an open bounded interval. It is well known that $W^{1,p}(I)\subset C(I)$. It easy to see that there are $f\in C(I)$ such that $f\notin W^{1,p}(I)$ It is enough to take $I=(0,1)$ and ...
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0answers
57 views

A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...
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1answer
64 views

Is the regularity of $u$ necessary to deduce this result? (Evans PDE)

One of the exercises in Evans book on PDEs (at the end of chapter 7) is given as follows: Assume $$u_k\rightharpoonup u\quad\mbox{in}\quad L^2(0,T;H^1_0(U)),$$ $$u_k'\rightharpoonup ...
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36 views

About the dual of Sobolev spaces

I'm reading a paper about PDE, in which the author mentioned the space $W^{-1,1}(\Omega)$, does this mean the dual space $W^{1,\infty}(\Omega)^{*}$ ? I hope not. I only know the Sobolev dual space ...
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1answer
66 views

Decomposition of measures acting on sobolev spaces

This is a follow-up question to Decomposition of functionals on sobolev spaces. Let $\Omega \subset \mathbb{R}^n$ be a bounded, open set and $\mu \in H^{-1}(\Omega) = H_0^1(\Omega)^*$. Moreover, let ...
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39 views

Sobolev embedding theorem

I am supposed to prove the following: if $U\subset\mathbb{R}^2$ is open, $g\in H^1(U)$ with $\Delta g\in L^2(U)$ and $K\subset U$ is compact, then $||g|_K||_{C^0(K)}\leqslant C ||g||_{H^2(U)}$. The ...
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74 views

Comparison of Sobolev spaces on an open or closed interval

As noted in my previous question, I am currently working through some books on Sobolev spaces. I am struggling to determine whether, given an interval $I=(0,a)$,the Sobolev spaces $W^{m,p}(I)$ and ...
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0answers
43 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in ...
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1answer
47 views

Is this space a Banach space? 2

Consider the set of functions $$\mathcal{B}=\{v\in L^2(0,T;H^1_0(\Omega)): \partial_tv\in L^2(0,T;H^{-1}(\Omega))\},$$ equipped with the norm ...
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1answer
54 views

$f\mapsto \frac{df}{dx} - \frac{x}{\sqrt{1+x^2}}f $ has closed image and $1$-dimensional cokernel

Let $X$ be the completion of the space of smooth, compactly supported real-valued functions on $\mathbb R$ under the norm $$\|f\|_X^2=\int_{\mathbb R} \left(\frac{df}{dx}\right)^2 + f^2.$$ Let ...
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1answer
27 views

Prolongement 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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1answer
73 views

Extending Sobolev functions by zero

I believe that if you have a sufficiently regular (say Lipschitz) bounded domain $\Omega\subset\Bbb R^n$, then you can extend a function $u\in H^1_0(\Omega)$ by zero, and the extension lies in ...
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0answers
113 views

Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
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1answer
45 views

Show for the Hamilton's operator $H$ that $\overline{(H, C_0^{\infty}(\mathbb{R}))} = (H, W_2^2(\mathbb{R}))$ using Fourier transform

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable real-valued function defined on $\mathbb{R}$ bounded with its first derivative. Consider the Hamilton's operator $H$ such that: ...
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1answer
27 views

$C_0^0((a,b)) \subset W_0^{1,2}((a,b))$?

One can easily show that $W_0^{1,2}((a,b)) \subset C^0((a,b))$ for any finite interval $(a,b)$. Intuitively $W_0^{1,2}((a,b))$ should contain more functions than $C_0^0((a,b))$, but how to prove that? ...