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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35 views

Evans PDE Chapt 5 problem 4, existenc of smooth functions form a partition of unity

I have to be honest that I am very lost on the same kind of problem about proving existence of smooth function. I have not done much topology, hence unfamiliar with the "covering" business. In the ...
2
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1answer
57 views

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

Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-periodic functions on the line with norm \begin{equation*} \| u ...
2
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1answer
59 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
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0answers
33 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ (Au)[v]=a(u,v)\quad ...
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1answer
27 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
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1answer
43 views

Are there $L^2$ functions whose Laplacian is in $L^2$ yet its gradient is not in $\mathbf{L}^2$?

Let $\Omega$ be a bounded smooth domain. Is it true that $H_\Delta^1(\Omega):=\{ u\in L^2(\Omega) : \Delta u\in L^2(\Omega)\}\subset H^1(\Omega):=\{ u\in L^2(\Omega) : \nabla u\in \mathbf{L}^2(\Omega ...
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1answer
36 views

show that $uv\in W^{1,r}(\Omega)$

Let $\Omega\subset\mathbb{R}^{n}$ is bounded,with $\partial\Omega\in C^{1},p,q\geq 1$,$u\in W^{1,p}(\Omega),v\in W^{1,q}(\Omega)$,show that $ uv\in W^{1,r}(\Omega)$.here ...
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1answer
41 views

Prove $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$ [duplicate]

Prove that for $n>1$,the non-bounded function $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$,Here $\Omega=B(0,1)\subset \mathbb{R}^{n}$ I think we have to prove that $$ ...
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3answers
61 views

prove that $u$ is equal a.e. to an absolutely continuous function

Prove that if $n=1$ and $u\in W^{1,p}(0,1) $ for some $1\leq p<\infty$, then $u$ is equal a.e. to an absolutely continuous function,and $u'$ (which exists a.e.) belongs to $L^{p}(0,1)$. My ...
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1answer
46 views

Sobolev space on a closed subset

Hi I have a question about Sobolev spaces. Let $U \subset \mathbb{R}^{d} $ be a bounded open subset and $dx$ be a Lebesgue measure on $U$ \begin{align} W^{1,2}(U):=\left\{u \in L^{2}(U;dx): ...
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0answers
26 views

Stuck in trace theorem

I am reading Sobolev space in the book Partial Differential Equation by Evan and I do not understand some point in the proof of the trace theorem. Let $U$ is open, bounded and $\partial U$ is $C^1$. ...
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1answer
30 views

An ineqaulity involving critical Sobolev exponent

This is related to my previous question An inequality involving Sobolev embedding with epsilon. There I wished to get that, for given a nice bounded domain $\Omega$ in $\mathbb{R}^n$, $\forall ...
2
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1answer
90 views

Show that $\frac{\int_\Omega|\nabla u|^2+\int_\Omega\alpha|u|^2}{\int_\Omega|u|^2}$ attains a minimum in $W_0^{1,2}(\Omega)$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $|\;\cdot\;|_p$ be the seminorm $$|u|_p^p:=\int_\Omega|\nabla u|^p\;d\lambda^n\;\;\;\text{for ...
3
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1answer
36 views

Is it possible, that the fist two weak eigenvalues of $-\Delta$ in a bounded domain are equal?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $\lambda_1$ be the first weak eigenvalue of $-\Delta$ in $\Omega$ $\varphi_1$ be the weak eigenfunction associated with $\lambda_1$ ...
2
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1answer
29 views

If $l_i$ is the first weak eigenvalue of $-\Delta$ in a domain $G_i$ and $G_1\subseteq G_2$, then $l_1\ge l_2$ and equality is possible

Let $\Omega_i\subseteq\mathbb{R}^n$ be a domain $\lambda_i$ be the first weak eigenvalue of $-\Delta$ in $\Omega_i$ It's easy to verify that $\Omega_1\subseteq\Omega_2$ implies $\lambda_1\ge ...
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1answer
37 views

An inequality involving Sobolev embedding with epsilon

Let $\Omega$ be a nice bounded domain in $\mathbb{R}^n$ and let $2^*=2n/(n-2)$ for $n>2$ or anything finite for $n\leq2$, i.e., the Sobolev exponent. Then, for any $p\leq 2^*$, one has $$ ...
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1answer
31 views

Is $C_0^\infty(\mathbb{R}_+)$ a dense subspace of $W_0^{1,2}(\mathbb{R}_+)$?

I read that in some lecture notes that the space of $C^\infty$ funtions compactly supported on the positive real line is a dense subspace of the Sobolev space $W_0^{1,2}(\mathbb{R}_+)$. How can one ...
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0answers
20 views

Develop a concept of weak solvability for $-\langle\nabla,A\nabla u\rangle=f$

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable with $A(x)$ is symmetric, for all $x\in\Omega$ $\exists c_1,c_2>0$ with ...
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1answer
104 views

Is $T$ a compact mapping from $W_{0}^{1,2}\left(\Omega\right)$ into itself? [closed]

Let $\Omega$ be an open bounded subset in $\mathbb{R}^{6}$ and $f$ be in $L^{8}\left(\Omega\right)$. For any $w$ in $W_{0}^{1,2}\left(\Omega\right)$, define $T\left(w\right)$ be in ...
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0answers
44 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
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0answers
23 views

If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...
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2answers
44 views

Example of an $H^{-1}$ function that isn't $L^2$

I'm going back over some PDE and Sobolev space theory, and the following is puzzling to me. Consider a nice domain $\Omega$ and the space $H^1_0(\Omega)$ of functions with $L^2$ first derivatives, ...
2
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1answer
47 views

Stuck in the proof of Extension theorem

I am reading the extension theorem in sobolev spaces in the book '' Partial Differential Equation'' by Evan and I get stuck at one point. Let $U\subset\mathbb{R}^n$ is open and bounded, and $\partial ...
0
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1answer
25 views

Absolute Continuity defined by Necas

I read a definition of the absolute continuity in Necas' book "Direct Methods in the Theory of Elliptic Equations": Let $\Omega$ be a domain in $\mathbb{R}^n$ , $P$ a line verifying ...
2
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1answer
46 views

Conditions under which a function vanishing on the boundary belongs to $H_0^1$

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u ...
2
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0answers
63 views

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 ...
4
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1answer
94 views

Prove that a series converges in $L^2(\Omega)$

Let $\Omega$ be a bounded smooth domain and let $\varphi_k$ be eigenfunctions of the Neumann Laplacian with eigenvalues $\lambda_k$. Let $u \in H^1(\Omega)$. I want to show that for all $y \in ...
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1answer
39 views

If the weak derivative $\nabla u$ of $u\in L^2(\Omega)$ exists, then $\int_\Omega|\nabla u|^2=\int_\Omega|\nabla u^+|^2+\int_\Omega|\nabla u^-|^2$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded $u\in \mathcal{L}^2(\Omega)$ be weakly differentiable, i.e. $$\int_\Omega u\nabla\psi\;d\lambda^n=-\int\psi\nabla u\;d\lambda^n\;\;\;\text{for all ...
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1answer
54 views

Reference about Sobolev spaces

I'm looking some book from which I could learn more about Sobolev spaces. I'm interested rather in abstract theory: some topics which I would like understand in detail include: general construction ...
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0answers
54 views

Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$

Assume $\Omega$ is open bounded domain in $\mathbb R^n$ Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$ with inner product ...
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1answer
16 views

About Chebyshev inequality for integrals

Let $u \in H^1(\Omega) \cap C(\Omega)$ ($\Omega \subset R^n$ a smooth and bounded domain) a nonnegative function. Let $B(x,R) \subset \overline{ B(x,R) } \subset \Omega $ a ball. Suppose that ...
2
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1answer
38 views

A classical solution of Poisson's equation is also a weak solution

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be a solution of $$\left\{\begin{matrix}-\Delta u&=&f&&\text{in }\Omega\\ ...
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1answer
27 views

About equi-integrability

Suppose $\Omega\subset \mathbb R^N$ is bounded and lipschitz boundary. Suppose $u_n, u\in H^1(\Omega)$ such that $u_n\to u$ weakly in $H^1$. Then can I conclude that $$ ...
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1answer
37 views

Dual space of Sobolev Space

I'm studying pde's and I'm on the topic of Sobolev Spaces and I have a small question regarding the dual of a Sobolev Space. I'm trying to understand a proof about the characterization of the dual of ...
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1answer
25 views

Is function $u$ nice when all $\Delta^k u$ are nice?

Let $\Omega \subset\mathbb{R}^d$ has smooth boundary and $$\Delta^k u \in W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega) \qquad k\geq0$$ Show that $u\in W^{n,2}(\Omega)$ for all $n\in \mathbb{N}$. This ...
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1answer
16 views

Intuitive question about the boundary values of a Sobolev function

Let $B_R$ a ball of radius $R$ in $R^n.$ Let $u \in H^1(B_R)$ and let $u^{1} \in H^1(B_R)$ with $u^1 - u^{+} \in H^1_{0}(B_R)$. ($u^{+}$ denotes the positive part of the function $u$). Let $v(y) : = ...
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1answer
24 views

Reference for denseness of testfunctions in sobolevspace

for my thesis I need a reference for a proof that $C_0^\infty(\mathbb R)$ is dense in $W^{2,2}(\mathbb R)$ in respect to the Sobolev-$\| \cdot \|_{W^{2,2}}$-Norm. I have tried Google but I can't find ...
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1answer
13 views

How can I conclude this “gluing property” for these Sobolev functions?

Let $u \in W^{1,2}(\Omega) \cap C(\Omega)$, where $\Omega$ is an open bounded domain in $R^n$ with smooth boundary.Let $B(x,R) \subset \overline{B(x,R)} \subset \Omega $ a ball. Consider $u^{\star} ...
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2answers
47 views

Significance of Sobolev spaces for numerical analysis & PDEs?

I never had an option to take a Functional Analysis module. I am tied up with other work for the next two months so I won't get a chance to self-study it until September. So one thing I was wondering ...
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0answers
21 views

Simple question about Sobolev Spaces

I'm studying Sobolev Spaces and my lecturer told us that $H^1_0(U)=H^1(U)$. Is this true for all $U\subseteq \mathbb{R}^n$? Even $U=\mathbb{R}^n$? And why exactly is this true?
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1answer
53 views

Multivariable calculus chain rule for weak derivatives

Let $g:(0,1) \rightarrow \mathbb R^n$ be absolutely continuous, $F \in W^{1,2} (\mathbb R^n).$ Is it true that a.e. it holds $$ \dfrac{dF(g(t))}{dt} = \nabla F(g(t)) \cdot g'(t) \quad ? $$ What I ...
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1answer
41 views

Counterexample to Rellich-Kondrachov Compactness Theorem, case $q=p^*$

I was searching some counterexample for I was searching some counterexample for Rellich-Kondrachov Compactness Theorem (You can see: PDE, Evans, chapter 5), for the case $q=p^*$ and I found this ...
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0answers
44 views

The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
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0answers
23 views

Functions equal in Sobolev spaces

Consider the Sobolev space $H^k(\mathbb{R}^n)$, where $k, n \geq 1$ are integers. Also, consider $u, v \in H^k(\mathbb{R}^n)$ such that $u$ and $v$ are equal almost everywhere in the Lebesgue measure ...
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0answers
28 views

About the compact embedding $W^{1,p}(U)\subset\subset L^p(U)$

How can I get this compact embedding $W^{1,p}(U)\subset\subset L^p(U)$ for a open, bounded and $C^1$ subset $U$ of $\mathbb R^n$? Basicly This is the Evans' Remark on pg. 289 2nd edition. But I ...
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1answer
30 views

Can I conclude that this Sobolev function is Lipschitz?

Let $u \in H^{1}(\Omega)$ ($\Omega \subset R^n$ a bounded domain with smooth boundary). Suppose that there is a constant $C>0$ such that $$ |u(x) - u(y)| \leq C |x-y|,$$ for every Lebesgue point ...
0
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1answer
50 views

Interpolation between derivatives

I am trying to prove the following: Let $u \in W^{2,2}(\mathbb R)$. Then $\| u^\prime \|_{L^2}^2 \leq \| u \|_{L^2} \| u^{\prime \prime} \|_{L^2}$ holds (these are meant to be weak derivatives). ...
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0answers
28 views

Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...
1
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1answer
24 views

Showing that there is a unique solution for the following equation

Let $I = (0, 1)$, $a : H_0^2(I) \times H_0^2(I) \to \mathbb{R}$ a continuous bilinear form defined by $$a(u, v) = \int\limits_I u'' (x) v'' (x) dx.$$ Show that for every $f \in L^2(I)$ there is a ...
5
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0answers
79 views

Sobolev type embedding

Consider a compact manifold $M$ and a point $q \in M$. Let us say that that the following inequality holds: $$ \Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in ...