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Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm that controls both a function and its weak derivatives in some Lebesgue space.

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Any function in Sobolev $H^{s}(\mathbb{R}^n)$ space is continuous and bounded if $s>n/2$

I'm studying this book M.E. Taylor, and M.E. Taylor, Partial Differential Equations. 1: Basic Theory, Corr. 2. print (Springer, New York Heidelberg, 1997), more specifically, the proposition 1.3. ...
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Convexity of Fisher Information [duplicate]

I came across with this functional called Fisher Information: $$ \int_0^1 \int_{\Omega}|\nabla \log (\bar{\rho})|^2 d \rho, $$ where $\bar\rho$ is the density $\frac{d \rho}{d \mu}=\bar{\rho}$ with ...
math95's user avatar
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Intuition for fractional Sobolev norm

If $k$ is a nonnegative integer, then the Sobolev space $W^{k,p}$ is defined as the set of functions whose first $k$ weak derivatives are in $L^p$. To define a fractional Sobolev space on $\mathbb{R}^...
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Fractional regularity and the quotient $\frac{f(x)-f(y)}{|x-y|^\alpha}$

Holder, Sobolev, and Besov Spaces are often used to measure regularity, and in particular, fractional regularity. On one hand, the relation between their respective norms and regularity/...
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what is called this sub space $ H^{1}_{0,p} $ [closed]

What's the name of this sub sobolev space $ H^{1}_{0,p}= \left\{ u\in AC\left( \left[0,+\infty \right), \mathbb{R} \right) : u\left( 0 \right) = u\left( +\infty \right) =0 , \sqrt{p} \acute{u} \in ...
Daking Diss's user avatar
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Convexity of Dirichlet Integral Functional

The following is not homework, it is just personal study. Let $u \in H_0^1(\Omega)$ and consider the map $u \mapsto \int_{\Omega}|\nabla u|^2 dx.$ I want to show that this map is convex. I considered ...
Mud's user avatar
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Understanding spaces of negative regularity

Let $C^k(\mathbb{R}^n$) be the space of functions with $k$ continuous derivatives, and $H^s(\mathbb{R}^n)$ the Sobolev space $W^{2,s}$. Their dual spaces are commonly denoted as $C^{-k}$ or $H^{-s}$. ...
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Bochner-Sobolev spaces with second time derivative and embeddings

In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these spaces as $$W^{1,p,q}(a,b,X,Y) = \left\{v, \ \...
Maths_GEES 's user avatar
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sub space $ H^{1}_{0,p}$ [closed]

What's the name of this sub sobolev space $ H^{1}_{0,p}= \left\{ u\in AC\left( \left[0,+\infty \right), \mathbb{R} \right) : u\left( 0 \right) = u\left( +\infty \right) =0 , \sqrt{p} \acute{u} \in ...
Daking Diss's user avatar
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Elliptic Equation on sphere

The literature I read recently says that the function $$\phi_1(\theta)=(\theta\cdot e_d)_+$$ defined on the sphere solves the equation $$ -\Delta _{\mathbb{S}}\phi _1=\left( d-1 \right) \phi _1\qquad \...
zik2019's user avatar
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Regularity of continuous approximation on time discretization

I have been trying to understand how to deal with continuous approximations when dealing with time discretizations, consider the abstract PDE $$ \partial_t u + A(u) = f(u) $$ and then instead ...
Daniel Moraes's user avatar
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Sobolev embedding theorem on the torus

In his lecture notes at https://www.math.ucla.edu/~tao/254a.1.01w/notes2.dvi, Terence Tao proves the following (non-endpoint) Sobolev embedding theorem: Let $1 \le p < q \le \infty$ and $\frac{1}{p}...
efontana's user avatar
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Reflexitivity of sobolev spaces

Let $\Omega \subset \mathbb{R}^n$ open I'm interested to understand the reflexivity property of Sobolev spaces $W^{1, p}(\Omega)$ for $p \in [1, +\infty]$, using the fact that $L^p(\Omega)$ are ...
Manuel Bonanno's user avatar
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How to understand the Sobolev space defined by completion.

In page 16 of this book, the author state: For $1\leq p <\infty$, consider the normed space of all smooth functions $\phi \in \mathbb{R}^n$ such that $$ \|\phi\|_{1,p} = \|\phi\|_p + \|\nabla \phi\...
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A counterexample that $C^\infty(\mathbb{R}^n)$ is not complete.

Can anyone give an example which can directly indicate that the following norm space is not complete: $$ C^{\infty}(\mathbb{R}^n)\cap W^{1,p}(\mathbb{R}^n) = \{f\in C^{\infty}(\mathbb{R}^n): f \in L^p(...
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Sobolev space on the circle

I'm styudying Sobolev space on the circle. The Sobolev space is $H^{s,p}(\mathbb{S}^1)=\left\{u\in L^p(\mathbb{S}^1)\colon \left\|\mathcal{F}^{-1}\left((1+k^2)^{s/2}\widehat{u}(k)\right)\right\|_{L^p(\...
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convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero? Let $U$ and $E$ ...
Alucard-o Ming's user avatar
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integration by parts in Hilbert space

I want to prove that $u,v∈H^1 (R^n )$ we have : $∫_{R^n} \frac{∂u}{∂x_i} vdx=-∫_{R^n} \frac{∂v}{∂x_i} udx$ We know that $C_c^∞$ $(R^n )$ is dense in $H^1$ $(R^n)$, this means that $∃ u_n∈C_c^∞ (R^n )$...
Alucard-o Ming's user avatar
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1 answer
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Determining weak solution for Dirichlet problem

Let $D$ be the unit disk in the plane and let $\Omega= D\setminus\{0\}$. The Dirichlet problem \begin{cases} Δu = 1 & \text{in } \Omega \newline u=0 & \text{on } \partial \Omega \end{cases} ...
john_psl1298's user avatar
3 votes
1 answer
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Eliminating Neumann boundary condition for elliptic PDE

In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated: I am now wondering if this also works with Neumann boundary ...
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Proof of Theorem 3 (More Calculus) from 5.9.2 Evans PDE

From Evans PDE, the text presents the following Theorem 3 (More calculus): Suppose $\mathbf{u} \in L^2\left(0, T ; H_0^1(U)\right)$, with $\mathbf{u}^{\prime} \in$ > $L^2\left(0, T ; H^{-1}(U)\...
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$f=0$ on $\partial\Omega$ implies $f\in H_0^1(\Omega)$

Let $\Omega\subset\mathbb{R}^n$ be an open set. Let $f\in C(\bar{\Omega})\cap H^1(\Omega)$ with $f=0$ on $\partial\Omega$. Claim: Then $f\in H_0^1(\Omega)$ holds. Since $H_0^1(\Omega)$ is the closure ...
MaxwellDgt's user avatar
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If $\Omega \subseteq \Omega_1$, $u \in H^1(\Omega_1)$, and $u = 0$ on $\Omega_1 \setminus \overline{\Omega}$, do we have $u \in H^1_0(\Omega)$?

Let $ \Omega \subseteq \overline{\Omega} \subseteq \Omega_1$ be two open, bounded domains in $\mathbb{R}^n$ with Lipschitz boundary. Suppose $u$ belongs to the Sobolev space $H^1(\Omega)$ in such ...
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Significance of Poincare's inequality

I am self-learning the Evans' PDE book. In Chapter 5.8, it states the Poincare's inequality as Let $U$ be a bounded, connected, open subset of $\mathbb{R}^n$, with a $C^1$ boundary > $\partial U$. ...
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If $\Delta u \in L^2(\Omega)$ then $u\in H_0^1(\Omega)$?

Suppose that $u\in L^2(\Omega)$ and $\Delta u\in L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a set. Also, suppose that $u$ vanishes at the boundary(For example, the trace of $u$ at $\partial \...
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2 votes
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Equivalence of Sobolev spaces Definitions on Riemannian manifold via $k$-Covariant Derivative and $k$-Gradient Derivative

I am studying Sobolev spaces on Riemannian manifolds and have encountered two different definitions in the literature. In the references: E. Hebey. Nonlinear analysis on manifolds: Sobolev spaces and ...
Raoní Cabral Ponciano's user avatar
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A substitution in an integral of Bochner functions

I have given functions $f \in L^2(0,T;L^2(U))$ and $g \in L^2(0,T;H^1(U))$ and a function $H \in C^1([0,T]\times U)$ such that $$1 \leq H(t,x) \leq 2$$ for a.e. $(t,x)$. Here, $U=\partial\Omega$ is ...
BBB's user avatar
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Regularity of an elliptical problem

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with boundary of class $C^{\infty}$, with $N \geq 3$. Moreover, consider the uniformly elliptic operator $$\mathcal{L}v := -\Delta v - \lambda v.$$...
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2 votes
1 answer
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Schwartz theorem

Can i relax the conditions on f of the Schwartz theorem in real analysis? That is: if $f: \Omega \to R^2$, s.t $f \in C^2(\Omega)$, then i can change the order of derivatives and the result is the ...
Lucio Rosi's user avatar
4 votes
1 answer
46 views

Continuous representative of Sobolev function on triangle set

According to Morrey`s inequality $$||u||_{C^{0,\gamma}(U)}\leq C\cdot ||u||_{W^{1,p}(U)}$$there is a continuous representative for every $u\in W^{1,p}(U)$. $U$ is supposed to be a bounded domain with ...
hannah2002's user avatar
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1 answer
35 views

Lower bound for mean value of Sobolev function

Assume that for a Sobolev function $f\in W^{2,p}(E)$, for some $p>1$ and $E\subset \mathbb{R}^m$, that the weak derivative $\nabla f$ is essentially bounded in $E$ and that we already know that $|\...
HelloEveryone's user avatar
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Function that don't belong to any Sobolev space

Let $u:B(0,1)\subseteq \mathbb{R}^N$, with $N\geq 2$ such that $$ u(x) = \begin{cases} 1 & \text{if } x_N \geq 0,\\\\ 0 & \text{if } x_N <0. \end{cases} $$ I want to show that $u\not\in W^{...
matdlara's user avatar
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Auxiliar inequality for Rellich-Kondrachov theorem

To prove the Rellich-Kondrachov Theorem it is used the following statement If $u\in W^{1,1}(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$, then $||\...
Shiva's user avatar
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1 vote
1 answer
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Proving a linear functional belongs to $L^{4/3}(0,T; H^{-1})$

Let $H^{-1}$ be the dual to the Sobolev space $H^1$ and $T > 0$. Let $u: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $u \in L^\infty(0,T; L^2) \cap L^2(0,T; H^1)$. It can be shown that $$(u \...
CBBAM's user avatar
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Inverse of the Dirichlet Laplacian?

I have read somewhere that $(-\Delta)^{-1}u$ for $\Delta :H_0^1 \rightarrow H^{-1}$ is defined as the unique weak solution to $-\Delta v= u$ with $v=0$ on the boundary (assuming $U$ is some sufficient ...
Perelman's user avatar
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Upper bound on infinity norm from Sobolev W^1,2 norm in >1 D? [duplicate]

Let $\Omega:=[0,1]^d, d\in\mathbb{N}$. Let $u\in W_0^{1,2}(\Omega)$, the 0 subscript meaning $u$ is 0 on $\partial\Omega$. Assume also $$ \int_{\boldsymbol x\in\Omega}\nabla u(\boldsymbol x)\cdot\...
Leon Avery's user avatar
1 vote
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31 views

Chain rule with Sobolev functions

Let $f: \mathbb{R} \to \mathbb{R} $ be Lipschitz and let $u\in W^{1,p}(\Omega)$, with $1\le p <\infty$ and $\Omega\subset \mathbb{R}^N$ open. Assuming that $f(0)=0$ I have to show that $f\circ u\in ...
Shiva's user avatar
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1 vote
1 answer
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Confusion on Definition of Weighted Sobolev space

I have been reading a little bit about weighted Sobolev spaces, and I have found myself with the following quandary: The norm of the unweighted sobolev space, $W^{1,p}$ is defined as$$ \left(\int |f(x)...
APP's user avatar
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4 votes
1 answer
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Sobolev Embedding Theorem: counter example cone condition

Let be $p\in[1,\infty)$ and $\Omega\subset\mathbb{R}^n$ an open subset that satisfies the cone condition. According to Sobolev's theorem, the following continuous embedding holds for $k>\frac{n}{p}$...
Oscar210899's user avatar
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1 answer
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Regarding $\nabla \times (u \times v)$ and Sobolev Spaces

Greetings fellow members, Let $\Omega$ be a bounded domain (open+connected) of $\mathbb{R}^3$. Let's consider $S(\Omega)$, a subspace of Sobolev $H^1(\Omega)$ defined by : \begin{align*} S(\Omega)=\{...
Aubium's user avatar
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Equivalence of norms in the trace space

I have the following question about trace spaces: Let $\Omega$ be a Lipschitz domain with Lipschitz boundary $\Gamma$. Consider the space of traces $$ H^{1/2}(\Gamma) = \{g \in L^2(\Gamma) | \exists ...
mathology's user avatar
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Euclidean space is a Lipschitz domain?

I'm reading the concepto of Lipschitz domain. For example, in Question. Is $\mathbb{R}^n$ a Lipschitz domain? i read that, the boundary of $\mathbb{R}^n$ is the empty set. Is empty set a Lipchitz ...
eraldcoil's user avatar
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Weak closure of $W_0^{1,p}(\Omega)$

I should prove the weak-closure of $W_0^{1,p}(\Omega)$, for $1\le p< \infty$. One way is proving that it’s strongly closed using the definition of $W_0^{1,p}$ as the closure of $C_c^{\infty}(\Omega)...
Shiva's user avatar
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0 answers
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Extension of a $W^{1,p}_0(\Omega)$ function

Let $u \in W^{1,p}_0(\Omega)$ and let $\tilde{u}$ the function defined extending $u$ to zero on $\mathbb{R}^N \setminus \Omega$. I should prove that $\tilde{u}\in W^{1,p}(\mathbb{R}^N)$. The exercise ...
Shiva's user avatar
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0 answers
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Sign of the first eigenfunction of the Laplacian

I am trying to prove that the first eigenfunction of the Laplacian operator in an open domain $\Omega$ does not change sign and that the first eigenvalue $\lambda_1$ is simple (with Dirichlet-boundary ...
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Is $u(x)=\frac{1}{|x|^{\alpha}}$ in $W^{1,p}(B_1(0))$?

Consider a function $$ u(x)=\frac{1}{|x|^{\alpha}} \quad x\in B_1(0) \subset \mathbb{R}^N. $$ I should find condition about $p, N, \alpha$ for $u$ to be in $W^{1,p}(B_1(0))$. Following different books ...
Shiva's user avatar
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find the explicit Solution $u$ and check that if $u\in H^2(I)

we considere $I=]a,b[$ and $$V=\{ u\in H^1(I) : u(\frac{1}{2})=0\}$$ $1)$ Prove that $V$ is an Hilbert space and $v\to \|u'\|$ is equivalente to norm of $H^1(I)$ $2)$ Show there exists a unique $v\in ...
kebiri5's user avatar
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1 vote
0 answers
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Lipschitz Domain and Sobolev Spaces

In theory of PDEs, we usually impose some smoothness assumptions on the domain $\Omega$, so that the standard Theorems involving Sobolev Spaces hold true, like continuous/compact embedding theorems, ...
cct's user avatar
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0 answers
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Integral convergence from weak convergence

Let there be a sequence $(u_n)_{n \in \mathbb{N}}$ in $H^1(\mathbb{R}^N)$ such that $u_n \rightharpoonup u$ in $H^1(\mathbb{R}^N)$. If $h \in L^{\frac{2}{2-q}}(\mathbb{R}^N)$, with $1 < q < 2$, ...
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0 answers
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Regularity of the Neumann trace

Let $\Omega$ be a Lipschitz domain with boundary $\partial \Omega$. The normal component of smooth functions on $\Omega$ is typically generalized as follows: Define the space $H(div; \Omega) = \{v \in ...
mathology's user avatar
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