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
18 views

can I get weak convergence in sobolev spaces from convergence of distributions

my question is the following. Given a sequence $(f_k)_k$ in $W^{1,q}(\Omega)$ with $q \in (1,\infty)$ and $\Omega \subseteq \mathbb{R}^n$ open and bounded. If I want to show $f_k$ converges in $W^{1,...
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46 views

Hadamard counterexample Dirichlet-problem

Good afternoon, I try to understand the follwoing counterexample of Hadamard: The classical solution of $ \begin{cases} \Delta u = 0 ~~\text{ in }\Omega\\ u=g \text{ auf }~~ \partial \Omega \end{...
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1answer
15 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 ...
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1answer
39 views

Scalar property of $ C(\Omega)=\sum_{|\alpha|\leq m}\color{blue}{\big|\Omega\big|^{\dfrac{2|\alpha|-n}{n}}} \int_{\Omega}|D^\alpha f|^2\ dx $

This is closely related to a previous question: Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$ This question focuses on the direct calculation (by change of ...
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1answer
35 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 \mathbb{...
2
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1answer
58 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 ...
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0answers
28 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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20 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 "...
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0answers
40 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 x_n>0,|x|<R\}...
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1answer
121 views

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 ...
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1answer
29 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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2answers
48 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 ...
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1answer
50 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 ...
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0answers
7 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 $u \geq m$. ...
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1answer
65 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 $...
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1answer
19 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 L^...
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2answers
47 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 ...
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1answer
41 views

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

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
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1answer
46 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 ...
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1answer
20 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 ...
2
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2answers
116 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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1answer
21 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 \mathbb{...
0
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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,q^{\star}...
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0answers
33 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: $$-...
2
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2answers
54 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 $C$ ...
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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 ...
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0answers
17 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 L^2(\...
2
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0answers
26 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 ...
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1answer
28 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 $C_0>0$ ...
0
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1answer
22 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 ...
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1answer
51 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 f=...
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1answer
17 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 \...
2
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1answer
47 views

Passing from Classical Formulation to Weak Forulation on PDEs. Integration by parts in $n$ dimensions?

I am reading about the Variational Method for solving PDEs and ODEs. The process to pass from a classical solution to a weak solution is pretty clear when we are dealing with ODEs, by using ...
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0answers
18 views

Given an operator $Q$ between a Hilbert space $U$ and $L^2(ℝ^d;ℝ^d)$, is it possible to make sense of $U∋u↦(Qu)(x)$ for a fixed $x∈ℝ^d$?

Let $U$ be a Hilbert space $H:=L^2(\mathbb R^d;\mathbb R^d)$ for some $d\in\left\{2,3\right\}$ $Q$ be a Hilbert-Schmidt operator from $U$ to $H$. I want that $\tilde Q(x)$, where $$\tilde Q(x):=...
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1answer
23 views

Show that $||u||_{L^{1}(I)} \le \frac{1}{k} ||f||_{L^{1}(I)}$

Suppose there exists a unique $u \in H_0^{1}(I)$ satisfying $$\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$$ Show that $$||u||_{L^{1}(I)} \le \frac{1}{k} ||f||_{L^{1}(I)}$$ In the ...
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1answer
19 views

prove that there exists a unique $u \in H_0^{1}(I)$ satisfying $\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$

Let $I=(0,1)$ and fix a constant $k \gt 0$. Given $f \in L^{1}(I)$ prove that there exists a unique $u \in H_0^{1}(I)$ satisfying $$\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$$ For ...
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0answers
54 views

Help understanding the proof of Trace Theorem given in Evans

I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997): Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there ...
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2answers
38 views

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with ...
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0answers
25 views

Representation of the delta distribution as an element of the dual of $H^1$

I'm working with some Sobolev spaces and I just wanted to consider the elements of $H^{-1}$ as elements on $H^1$ (Riez Theorem). Since the delta function $\delta(f) = f(0)$ is an element of the dual ...
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0answers
23 views

Show that the following are equivalent for $u \in C^{1}(0,1)$

Let $u \in C^{1}(0,1)$. Prove that the following are equivalent: (a) $u \in W^{1,1}(0,1)$ (b) $u' \in L^{1}(0,1)$ (where $u'$ means the derivative in the usual sense) (c) $u \in BV(0,1)$ My try: ...
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1answer
36 views

On the trace theory and restrictions of Sobolev space functions

Consider a connected open bounded subset $D\subset \mathbb{R}^d$ with smooth boundary. It is easier to state for the disc so I will do so. I don't think it would change the argument (except for in our ...
1
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1answer
18 views

Show that the following is true for a bounded sequence in $W^{1,p}(I)$

Let $I=(0,1)$. Assume that $u_n$ is a bounded sequence in $W^{1,p}(I)$ with $1 \lt p \le \infty$. Show that there exist a subsequence $(u_{n_k})$ and some $u$ in $W^{1,p}(I)$ such that $||u_{n_k}-u|...
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0answers
8 views

Prove the Poincare's inequality on $B^{0}(0,1)$. [duplicate]

Fix $\alpha >0$. Let $U=B^{0}(0,1)$. Show that there exists a constant $C$, depending only on $n$ and $\alpha$ such that $\int_{U} u^{2}\mathrm{d}x\leq C\int_{U} |Du|^{2}\mathrm{d}x,$ provided $|\{...
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0answers
28 views

Question about the proof of $W^{1,p}_0(\Omega) \Rightarrow u=0 $ on $\partial \Omega$

I am reading the proof of Theorem 9.17 in the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. The theorem says: Suppose that $\Omega$ is of class $C^1$...
4
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1answer
43 views

Prove that following are true for $\phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$

Fix a function $ \phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$ and set $u_n(x)=\phi(x+n)$. Let $1 \le p \le \infty$. Then Check that $u_n$ is bounded in $W^{1,p}(\mathbb{R})$ Prove that there ...
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0answers
15 views

Fractional Sobolev space on an interval

Consider a fractional Sobolev space $H_p^t(I)$ defined on an interval $I\subset \mathbb{R}$. When $I=\mathbb{R}$ the space can be defined via Fourier transform. Is it possible to do it when $I=(-1,1)...
2
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0answers
9 views

Sobolev spaces on non-compact manifolds — independence on charts

Are there some standard references where basic facts about fractional-order (or at least integer-order) Sobolev spaces on non-compact manifolds are treated? More precisely I would like to be able to ...
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0answers
28 views

Importance of Sobolev Spaces

Why Sobolev spaces are so important in study of partial differential equations? What could have light up the mind of researchers to use these spaces to analyze PDEs?
2
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0answers
49 views
2
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1answer
34 views

An example of open set $\Omega$ in $\mathbb{R^n}$ for which $C^{\infty}_c(\Omega)$ is dense in $W^{l,p}(\Omega)$

We know that the statement $C^{\infty}_c(\mathbb{R^n})$ is dense in $W^{l,p}(\mathbb{R^n})$ is always true for any $l\in \mathbb{N}$ and $p\geq \infty$, $p\neq \infty$. My professor told me that it ...