For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

learn more… | top users | synonyms

2
votes
1answer
64 views

basic exercise distribution theory

Consider $f \in L^{2}(R^n)$ with $\Delta f \in L^{2}(R^n) $. Show that ${\partial}^{|\alpha| } , (|\alpha| \leq 2 )f \in L^{2}(R^n)$. (the derivatives is in the distribution sense). My book gives ...
1
vote
1answer
53 views

Behavior of $u\in W_0^{1,p}(\Omega)$ near the boundary.

Assume that $\Omega\subset\mathbb{R}^N$ is a bounded regular domain. Let $1<p<\infty$ and take $u\in W_0^{1,p}(\Omega)$. Is true that given $\epsilon>0$ there is a neighbourhood $V$ of ...
1
vote
0answers
36 views

An weak type sobolev multiplication theorem

Suppose that $f_i\rightharpoonup f$ in $W^{1,p}$ and $g_i\rightharpoonup g$ in $W^{1,q}$ with $q=\frac{np}{n-p}$, what the best $r$, for which we have $f_ig_i\rightharpoonup fg$ in $L^r$ (maybe ...
1
vote
0answers
23 views

norm of $x \in \mathbb R^d$ is in Sobolev space

For which values of $\alpha, k,p,d$ is $$ \|x\|^\alpha \in \textrm{W}^{k,p} (B(0,1)) \quad ? $$ where $\displaystyle{ \textrm{B}(0,1) = \{x \in \mathbb R^d : \|x\|<1\}}$ This is an ...
1
vote
3answers
514 views

Weak convergence and strong convergence

Suppose $f_i$ is uniformly bounded in $W^{1,p}$ for some $+\infty>p>1$, then by passing to a sub-sequence, we can suppose $f_i$ is weakly convergent to $f$ in $W^{1,p}$. Assume furthermore that ...
2
votes
1answer
55 views

Does $H_0^1(\Omega)$ embed into $H_0^1(R^d)$?

Given a domain $\Omega$ in $\mathbb{R}^d$ and a function $f\in H_0^1(\Omega)$, the closure of the test functions on $\Omega$, does the extension of f by 0 to all of $\mathbb{R}^d$ necessarily lie in ...
3
votes
0answers
78 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
3
votes
1answer
138 views

Gradient of weak solutions of elliptic PDE at interfaces

Let $\Omega$ be a $C^1$-domain partitioned into two open sets $\Omega_1$ and $\Omega_2$ with interface $I=\partial\Omega_1 \cap \partial\Omega_2$. Now, say $u \in H^1(\Omega)$ is a weak solution of a ...
1
vote
0answers
20 views

Weak convergence version of sobolev multiplication theorem

Suppose that $1/n-1/p+1/q=0$, and $f_i$ weakly convergent to $f$ in $W^{1,p}$, $g_i$ weakly convergent to $g$ in $W^{1,q}$, can we conclude that $f_i*g_i$ weakly convergent to $f*g$ in $W^{1,p}$?
0
votes
0answers
18 views

Using Sobolev-Gagliardo-Nirenberg [duplicate]

I am currently studying a proof of a General Sobolev Inequality. I have the following question: Consider the Sobolev Space $W^{k,p}(U)$. With the added assumption that $k > \frac{n}{p}$. Let $l = ...
3
votes
0answers
60 views

$W^2_p$ regularity of solutions of linear elasticity

I want to prove the following statement: Let $\Omega$ be bounded, polygonal, and convex, then for the solution of the linear elasticity (elliptic) equation with inhomogenous Dirichlet boundary ...
2
votes
1answer
185 views

Evans PDE p.308 Exercise 16 (2nd ed)

Here is the statement of the problem (Evans PDE 2nd Ed., p.308, exercise 16) Show that for $n \geq 3$ there exists a constant $C$ so that $$ \int_{\mathbb {R}^n} \frac{u^2}{\vert x ...
0
votes
0answers
46 views

A Sobolev norm inequality

Assume $\mu\in H^{1}$ is compactly supported in $\Omega$ and $\Delta_{g}\mu\in H^{m}$ for some $m\ge 0$. Then if $K\subset \Omega$, there is a constant $C=C(K,m)$ such that $$|\mu|_{H^{m+2}(K)}\le ...
1
vote
1answer
174 views

Definition of weak solution in $W^{1,2}(\Omega)$.

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. We consider $$\left\{\begin{matrix} \Delta u-\lambda u =f \ \rm in \ \Omega & \\ u\mid_{\partial {\Omega}} =0 ...
1
vote
1answer
115 views

Using Sobolev-Nirenberg-Gagliardo

I am currently studying a proof of a General Sobolev Inequality. I have the following question: Consider the Sobolev Space $W^{k,p}(U)$. With the added assumption that $k > \frac{n}{p}$. Let $l = ...
1
vote
1answer
331 views

Gagliardo-Nirenberg interpolation inequalities on a bounded domain

In the original paper of Nirenberg, the following inequality is proven : if $\frac{1}{p} = \frac{j}{n} + \left( \frac{1}{r} - \frac{m}{n} \right) \alpha + \frac{1 - \alpha}{q}$ and $\frac{j}{m} \leq ...
4
votes
1answer
159 views

Is $L^2(\Omega)$ dense in $H^{-1}(\Omega)$?

Is it true that $L^2(\Omega)$, identified with its own dual, is dense in $H^{-1}(\Omega)$? $H^{-1}(\Omega)$ is the dual of $H^1_0(\Omega)$ and $H^1_0(\Omega)$ is the $H^1$-closure of smooth functions ...
2
votes
0answers
136 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 ...
2
votes
0answers
75 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 ...
1
vote
1answer
440 views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
0
votes
1answer
35 views

Showing that function are equal almost everywhere in Sobolev Spaces

Consider the Holder space $C^{0,1-\frac{n}{p}}(\mathbb{R}^{n})$ and the Sobolev Space $W^{1,p}(\mathbb{R}^{n})$. Take $u_{m} \in C_{c}^{\infty}(\mathbb{R}^{n})$ such that Morrey's Inequality we have ...
2
votes
0answers
25 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 ...
4
votes
1answer
122 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
2
votes
1answer
57 views

Range of the Dirac operator on the real line closed?

This is quite a short question: consider the Dirac operator $-i \tfrac{d}{dx}\colon H^1(R) \to L^2(R)$, where $H^1(R)$ denotes the Sobolev space of square-integrable functions with square-integrable ...
0
votes
0answers
29 views

Prove $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{-m} \cdot \|{v}\|_{L^{q}(T)}$

My professor asked me to derive this inverse estimate: $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{l-m} \cdot \|{v}\|_{W^{l,q}(T)}$, for $l \le m$ So I divided the problem into 2 steps: ...
1
vote
0answers
33 views

Trigonometric functions are dense in Sobolev Spaces

So I am trying to prove that the following functions $\{f_n=\frac{1}{\sqrt{2 \pi}}e^{inx}\}_{n=-\infty}^{n=\infty}$ are dense in the space $H^{n}[0,2\pi]$. For the proof it would be safe to assume ...
2
votes
1answer
52 views

How to show that $W^{2,\infty}(B_1)=C^{1,1}(\bar B_1)$?

Suppose that $B_1$ is the open unit ball in $\mathbb R^n$, denote $W^{2,\infty}(B_1)$ be the sobolev spaces and $C^{1,1}(\bar B_1)$ is the Holder spaces. It seems the equality ...
4
votes
1answer
92 views

Representing the dirac distribution in $H^1(\mathbb R)$ through the scalar product

Since in dimension $1$, $H^1$ is continuously embedded in $C_0$, we know that the Dirac distribution $\delta_0 \in H^1(\mathbb R)'$. Then by Riesz representation theorem we know that there exists a ...
2
votes
1answer
90 views

Explanation on a “different” proof that $C_c(\Omega)$ is dense in $L^p(\Omega)$.

Theorem: Let $\Omega\subset \mathbb{R}^n$ be an open set and $1\leq p < \infty$. The space $C_c(\Omega)$ is dense in $L^p(\Omega)$. Haim Brezis has a French book called "Analyse fonctionnelle: ...
3
votes
2answers
155 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
1
vote
2answers
79 views

Book searching in Elliptic Equation

I am learning a course with the subject of Elliptic Equations. If you know about it, please recommend me a book on Elliptic Equations. And if that's possible, someone post these books/author/...that ...
4
votes
1answer
55 views

Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
2
votes
2answers
297 views

Proof of Sobolev Inequality Theroem

I have a short question about the proof of Theorem 2 below. I have included Theorem 1's statement since it is used in the proof of Theorem 2. Definition: If $1 \leq p < n$, the Sobolev Conjugate ...
1
vote
2answers
98 views

1 in Sobolev spaces H^s

Following Rauch's Partial Differential Equations, he defines the generalization of the Sobolev Spaces for any $s\in\mathbb{R} $ as $$H^s(\mathbb{R}^n)=\{u\in S' | (1 + |x|^2)^{s/2}\widehat{u}\in ...
4
votes
1answer
92 views

Can $u\in W_0^{1,p}\cap L^\infty$ be approximated by a sequence $u_k\in C_0^\infty $ with $\|u_k\|_\infty$ bounded?

Assume that $\Omega\subset\mathbb{R}^N$ is a bounded Lipschitz domain and let $p\in [1,\infty)$. Suppose that $u\in W_0^{1,p}(\Omega)\cap L^\infty (\Omega)$. Is it possible to approximate $u$ by a ...
0
votes
1answer
40 views

$|\nabla f|^2, |\nabla g|^2 \in W^{1,2}_{\mathrm{loc}}(M)$, then $\langle \nabla f ,\nabla g \rangle \in W^{1,2}_{\mathrm{loc}}(M)$?

M is a Riemannian manifold, suppose $|\nabla f|^2, |\nabla g|^2 \in W^{1,2}_{\mathrm{loc}}(M)$, then $\langle \nabla f ,\nabla g \rangle \in W^{1,2}_{\mathrm{loc}}(M)$?
2
votes
0answers
64 views

The Sobolev space $H^s(\mathbb R^n)$ is closed under multiplication when $s>n/2$ [duplicate]

Show that if $u,v\in H^s(\mathbb{R}^n)$ for $s>{n\over 2}$, then $uv \in H^s(\mathbb{R}^n)$ and $$ \|uv\|_{H^s(\mathbb{R}^n)} \le C\|u\|_{H^s(\mathbb{R}^n)} \|v\|_{H^s(\mathbb{R}^n)}, $$ the ...
1
vote
1answer
161 views

Poincaré inequality for $W_0^{1,\infty}$

In the book A first course in Sobolev spaces by Leoni, the following Poincaré inequality for $W_0^{1,p}(\Omega)$ is stated: Suppose $\Omega\subset \mathbb{R}^n$ has finite width (lies between two ...
2
votes
2answers
234 views

Inf-sup condition only for existence (Necas theorem)?

So existence and uniqueness follows given (c1) and (c2). Is it possible that existence holds if (c1) only holds? I saw a similar result in other books (Showalter's Monotone Operators in Banach ...
4
votes
1answer
162 views

About the trace of Sobolev functions

Let $\Omega\subset\mathbb{R}^d$ be a Lipschitz domain with compact boundary and $1<p<\infty$.$$\gamma_0:W^{1,p}(\Omega)\mapsto W^{1-1/p,p}(\partial\Omega)$$ is the trace operator. ...
5
votes
1answer
103 views

A differentiable function not in $H^1$

Let $n\geq2$, $0<R<\infty$ and $\Omega=B(0,R)\subset\mathbb{R}^n$. I am looking for a function $u\in C^1(\Omega\setminus\{0\})$ such that $u\in L^2(\Omega)$, $\nabla u\in L^2(\Omega)$ (where ...
2
votes
2answers
111 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
2
votes
1answer
112 views

Approximation in Sobolev Spaces

Consider the following proof in Lawrence Evans book 'Partial Differential Equations': How does it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in ...
3
votes
1answer
88 views

Find a function in $H^{\frac{1}{2}}$ that is not in $L^{\infty}$.

Les $\mathcal{S}$ be the Schwartz class and $\mathcal{S}'$ be its dual (also known as the set of tempered distributions). For a function $u$ let $\hat{u}$ denote de Frourier transform of $u$. Given a ...
1
vote
0answers
70 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 ...
0
votes
1answer
24 views

Does for $u\in L^1(\Omega)$ and every $t$ also hold $\nabla u \cdot 1_{\{u=t\}}=0$ a.e.?

The result is known if $u$ is more regular e.g. $u \in W^{1,1}(\Omega)$. Is it also possible to extend such an result to mere integrable or even just measurable functions? Unfortunately the result ...
0
votes
1answer
40 views

Relation between $p$-superharmonic functions and concave functions

If I understood correctly what I read. In the one-dimensional situation the $p$ -superharmonic functions are exactly the concave functions and in several dimensions, the concave functions are ...
2
votes
1answer
79 views

$C_0^\infty(0,T)\cdot V$ dense in the Bochner space $L^2(0,T;V)$

Let $V$ be a Banach space and $(0,T)$ a time interval. Consider the space $C_0^\infty(0,T)$ of infinitely often differentiable functions with values in $\mathbb R$ and compact support in $(0,T)$ and ...
0
votes
1answer
36 views

Could the functions in larger space than $L^2$ be approximated by finite element basis functions?

Let $u \in V:=\{v \in L^{1+\alpha}(\Omega): \nabla v \in \mathbf{L}^2(\Omega)\}$, where $0<\alpha<1$. Clearly, $H^1(\Omega):=\{v \in L^2(\Omega): \nabla v \in \mathbf{L}^2(\Omega)\} \subset ...
0
votes
2answers
72 views

A question about a trace operator (is this right?)

Suppose I have proven that for $u \in H^1(\Omega) \cap C^1(\bar \Omega)$ that $$|u|_{L^2(\partial\Omega)} \leq f|u|_{H^1(\Omega)}$$ for some constant $f$. Let $T:H^1(\Omega) \to L^2(\partial\Omega)$ ...