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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0
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2answers
114 views

A modified version of Poincare inequality

We know the general version of Poincare inequality: $$ \int_\Omega |u-u_\Omega|^2dx\leq C\int_\Omega|\nabla u|^2dx,\quad \forall u\in W^{1,2}(\Omega), $$ where $u_\Omega$ is the average of $u$ over ...
0
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1answer
22 views

Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset ...
2
votes
2answers
107 views

In what conditions a weak solution is a classical solution?

I'm studying elliptic equations in divergence form $$-D_{j}(a_{ij}(x)D_{i}u) + c(x)u = f(x) \text { in a domain } \Omega \subset \mathbb{R}^{n}$$ I call a function $u \in H^{1}(\Omega)$ a weak ...
0
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0answers
17 views

Aproximating a Sobolev function by $p$-subharmonic functions.

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain of class $C^2$. Take $u\in W_0^{1,p}(\Omega)$ for $p\in (1,\infty)$ and assume that $$\frac{\partial u}{\partial \nu}(x)<0,\ \forall\ x\in ...
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1answer
60 views
+100

Existence of operator

I want to show that for $ s> \frac{1}{2} $ there is a bounded linear operator $ T: H^s(\mathbb{R}^n) \to H^{s-\frac{1}{2}}(\mathbb{R}^{n-1})$ following the below steps: Consider that $ u \in ...
1
vote
2answers
424 views

Poincaré inequality using $H^1$ seminorm

Does this inequality holds for Poincaré Inequality? $$||v||_{L^2} \leqslant C_p |v|_{H^1}$$ and $$ |v|_{H^1} = ||v'||_{L^2} $$ where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm. I'm ...
2
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1answer
44 views

Generalized Poincaré Inequality on H1 proof.

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
0
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1answer
27 views

Question regarding Evan's proof of Global Approximation by $C^∞(\overline{U})$ functions

The page where the proof is is on Google Books. A picture of the statement and the proof. I reproduce the statement of the result: Suppose $U$ is bounded with $C^1$ boundary, and $u∈ W^{k,p}(U)$. ...
0
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0answers
16 views

For manifolds $M,N$ show that $W^{1,p}(M,N)$ is path-connected iff $C^0(M,N)$ ist path-connected.

I'm asked to show that for compact, smooth Riemmanian manifolds $M,N$ we have that $W^{1,p}(M,N)$ is path-connected if and only if $C^0(M,N)$ is path-connected. The theorem (0.1) is taken from ...
1
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1answer
26 views

Can I write $H^1$ as $H^1_0 \oplus H^1_{\perp}$?

Let $\Omega\subset \mathbb{R}^d$, with $d\in \{1,2,3\}$ be an open bounded, simply connected domain. Define $H_0^1$ as the subspace of $H^1$ whose member functions have vanishing trace on the ...
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0answers
38 views

Can $f\in L^2(\Omega)$ imply $\nabla f\in [(H^1(\Omega))^*]^n$?

This is closely related to a previous question of mine. The only difference is the definition of $H^{-1}(\Omega).$ Suppose $f\in L^2(G_R)$ where $$ G_R=\{x\in\mathbb{R}^n\mid ...
1
vote
1answer
64 views

Can $f\in L^2(\Omega)$ imply $\nabla f\in H^{-1}(\Omega):=(H_0^1(\Omega))^*$?

I'm reading a regularity proof in a monograph of PDE. A step of the proof may or may not be implied by the following statement (I don't know if it is true or not): Suppose $f\in L^2(G_R)$ where ...
2
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1answer
138 views

Question regarding characterization of the dual space of $H_0^1(\Omega)$ in Evans's Partial Differential Equations

Given $\Omega\in R^N$ open bounded with smooth boundary. We define $H^{-1}$ to be the dual space of $H_0^1(\Omega)$ and from Evan's PDE book, chapter $5$, we know that for any $f\in H^{-1}$, there ...
4
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1answer
476 views

how can I show that $H_0^k(\Omega)=\{u\in H^k(M):\text{supp }u\subset\overline{\Omega}\}?$

Let $\bar{\Omega}$ be a smooth, compact manifold with boundary; we denote the interior by $\Omega$. We can suppose $\bar{\Omega}$ is contained in a compact, smooth manifold $M$, with $\partial\Omega$ ...
1
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0answers
41 views

Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$

I learned the following from Constantin and Foias's Navier-Stokes Equations (Chapter 4): We say that a function of a bounded open set $\Omega\subset\mathbb{R}^n$, $c(\Omega)$, is scale invariant ...
2
votes
2answers
111 views

Dual space of a closed subspace of a Hilbert space

I'm reading Girault and Raviart's book concerning Finite Element Methods for Navier-Stokes equations, and they use in the proof of one result, the following argument: As $V=\{v\in H_0^1(\Omega)^N; ...
1
vote
1answer
47 views

How is this inequality called? (And how to improve this process)

I am reading a book and it mentions the following: Let $u \in H^1_0(G)$; then $$\lVert u\rVert ^2_{L^\infty(G)} \le C \lVert u \rVert_{L^2(G)}\lVert u'\rVert_{L^2(G)}$$ Note: Here $G = (a,b) \subset ...
6
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0answers
69 views

What is a distribution in $H^{-1}(\Omega)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...
0
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0answers
4 views

About test functions for supersolutions

Let $B_{1}$ the unit open ball in $\mathbb{R}^{n}$ and $u \in H^{1}(B_{1})$. For $k,m >0$, let $\bar{u} = u^{+} + k$ and $\bar{u}_{m} = \bar{u}$ if $u < m$ and $\bar{u}_{m} = k+m$ if $u \geq m$. ...
6
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2answers
605 views

characterization of the dual space of the Sobolev space $H_0^1$

I am slightly confused about the properties of the dual space of the Sobolev space $H_0^1$ as outlined on page 299 in Evans. In particular, following the notation in the book, item 3 says that ...
9
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1answer
1k views

How is the acting of $H^{-1}$ on $H^1_0$ defined?

I have a question about the Sobolev Space $H^1_0(U)$, where $U$ is a open subset of $\mathbb{R}^n$. Let us denote with $H^{-1}(U)$ the dual space of $H^1_0$. How is the acting of $H^{-1}$ and ...
0
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0answers
37 views

properties of the dual of $H^1_0(\mathbb{R}^2;\mathbb{R}^2)$

Consider $f(x_1,x_2)=\chi_{B_1(0,0)}(x_1,x_2)$. 1) Is $\nabla f\in (H^1_0(\mathbb{R}^2;\mathbb{R}^2))^*$? 2) $\langle \nabla f , u \rangle= ?$ with $u\in H_0^1(\mathbb{R^2})$? For the first ...
0
votes
1answer
80 views

A proof about the characterization of the dual of $H_0^{-1}$

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 ...
14
votes
2answers
2k views

Dual space of $H^1$

It holds that $W^{1,2}=H^1 \subset L^2 \subset H^{-1}$. This is clear since for every $v \in H^1(U)$, $u \mapsto (u,v)_{H^1}$ is an element of $H^{-1}$. Moreover for every $v \in L^2(U)$, $u \mapsto ...
6
votes
3answers
849 views

dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
0
votes
1answer
50 views

convolutions and the mollification of functions in $L^1_{\hbox{loc}}(\Omega)$

The following is from the appendix in Leonin's First Course in Sobolev Spaces: Here is my question: Could anybody explain what is the meaning of "$u_\varepsilon$ is well-defined"? What is ...
0
votes
1answer
18 views

Concerning the proof of Sobolev Embedding $W^{m,p}(\mathbb{R}^n) \subset L^{\infty}(\mathbb{R}^n)$

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. Corolary 9.13 says (Among other results), that for $m \geq 1$: $$ W^{m,p}(\mathbb{R}^n) \subset ...
1
vote
1answer
48 views

Can weak convergence in $V$ imply strong convergence in $H$?

In the proof of the existence of strong solutions of the stationary NSE in the setting of Hilbert spaces, the following argument is made in Constantin and Foias's Navier-Stokes Equations (p60): ...
1
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2answers
40 views

$u$ continuous and the weak derivative $Du$ continuous $\Rightarrow$ $u \in C^1$?

Supose we have $u \in W^{1,p}$ (i.e $u$ has weak partial derivatives, which we denote by $Du$), and that both $u$ and $Du$ are continuous (More precisely, there is a continuous representative in the ...
0
votes
1answer
34 views

Weak solution in $\mathbb{R}^{N}$

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
0
votes
1answer
16 views

Does Green's (first) identity hold for Weak Derivatives?

Recall Green's First Identity: $$\int_{\Omega}v \Delta u \ (d\Omega) =\int_{\partial \Omega}v (\nabla u )\vec{n} \ d (\partial \Omega) - \int_{\Omega} \nabla u \nabla v \ (d \Omega)$$ Which ...
0
votes
1answer
40 views

Confused about the notation $||\nabla u||_{L^p(\Omega)}$

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, and also Evans' PDE. I am confused about what $||\nabla u||_{W^{1,p}(\Omega)}$ precisely means. In ...
0
votes
1answer
19 views

Showing regularity $(u \in C^2(\overline{\Omega}))$ for the Laplacian Problem.

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset ...
0
votes
2answers
138 views

finite elements-exercice

We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$ Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), ...
0
votes
1answer
13 views

Application of Holder and Poincare inequality

Let $p,q >1$ and $u \in W^{1,p}_{0}(\Omega)$ and $v \in W^{1,q}_{0}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ with smooth boundary. Suppose that $p,q \in (1,N)$, $q^{'} \in ...
1
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0answers
28 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: ...
1
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2answers
135 views

proof of the Meyers-Serrin Theorem in Evans's “Partial Differential Equations”

The following is a theorem and its proof in Evans's "Partial Differential Equations": Could anyone explain where (for which $x\in U$) is the convolution in step 2 defined and how to get (3) ...
15
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1answer
422 views

Suppose that there exist a set $\Gamma$ of positive measure such that $\nabla u=0, a.e.\ x\in\Gamma$.

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in ...
1
vote
1answer
49 views

Proving surjectivity of Laplacian for the $L^{p}$ case, $1<p<\infty$

For $1<p<\infty$ and $\lambda>0$ I want to show that $\lambda-\Delta:W^{2,p}(\mathbb{R}^{n})\to L^{p}(\mathbb{R}^{n})$ is bijective. Injectivity is obvious since if we have $\lambda-\Delta ...
2
votes
2answers
50 views

The Poincaré inequality for $H_0^1(\Omega)$

The following is the well known Poincaré inequality for $H_0^1(\Omega)$: Suppose that $\Omega$ is an open set in $\mathbb{R}^n$ that is bounded in some direction. Then there is a constant $C$ ...
1
vote
1answer
22 views

Closedness of first order differential operator on $L^2(\Omega)$

I am considering the when the following first order differential operator is a closed operator $$Au=b(x)\dfrac{\partial u}{\partial x_i},$$ on $L^2(\Omega)$ with the domain $D(A)=H^1(\Omega)$. Here I ...
0
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1answer
26 views

Bound first order derivative by $L^2$ norm of elliptic elliptic operator

Consider an symmetric 2nd order differential operator on a bounded domain with smooth boundary $$A=-\sum_{i,j=1}^n \partial_j (a^{ij}(x)\partial_i)$$ be uniformly elliptic if there exists $C_0>0$ ...
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0answers
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 ...
0
votes
1answer
2k views

proof on Poincare's inequality.

This might be a silly question. So basically, I have proved the Poincare's inequality for $p=1$ case. That is, for $u\in W^{1,1}(\Omega)$, I have $||u-\bar{u}||_{L^1}\leq C||\nabla u||_{L^1}$. Here ...
2
votes
0answers
25 views

How fast can Sobolev functions grow?

It is a simple fact that $L^p$-functions cannot grow arbitrarily fast. More precisely, one has for every $\ell>0$ $$ |\{f\geq\ell\}|\leq \frac{\|f\|_{L^p}^p}{\ell^p} $$ for every $f\in L^p$. My ...
1
vote
1answer
70 views

Trace operator and $W^{1,p}_0$

Let $W^{1,p}$ be the Sobolev space of $L^p$ functions with $L^p$ first derivatives. Let $W^{1,p}_0$ be the closure of the test functions in $W^{1,p}$. I am not explicitly writing the domain of the ...
0
votes
1answer
21 views

How to show a piecewise quadratic interpolant is $H^1$

I am preparing for a final exam and came across this question: Suppose that $\Omega\subset\mathbb{R}^2$ is an open bounded domain with triangulation $\mathscr{T}$. Suppose that $v_h$ is a ...
0
votes
1answer
15 views

Show a function is Sobolev

Let $T_{h}$ be a subdivision of a domain $\Omega \subset \mathbb{R}^d$ into elements $K$ with boundary $\delta K$so that the Gauss divergence theorem holds. If for a function $f$ it holds that $f ...
1
vote
1answer
17 views

Problem in passage in proof from Willem's book: is that $h$ in $L^{p'}$? How else can I use Dominated Convergence if not?

Here is my problem: Lebesgue's dominated convergence theorem implies that $$\begin{align}\left|\int_{\Omega_+} n\,\eta'(n\,x_N)w\, u\, ...