Questions tagged [sobolev-spaces]
A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.
1,152 questions
1 vote
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63 views
About the first variation of total variation of BV functions
Let $u v\in W^{1,1}(\Omega)$ and consider the gradient functional $$ J(u) = \int_\Omega |\nabla u(x)| \, dx $$ and the perturbation $u_\varepsilon = u + \varepsilon v$, with $\varepsilon > 0$. On ...
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3 votes
1 answer
301 views
Is the restriction of a Sobolev function to some full-measure set continuous?
Motivation: As a consequence of this question, every function $u$ in $W^{1,2}(\mathbb{R}^n,\mathbb{R})$ has a representative $u^*$ such that there is a null set $E\subset \mathbb{R}^n$ with the ...
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6 votes
1 answer
413 views
Classical PDE theorems on manifolds with boundary
At first, let $M$ be a smooth, non-compact, complete (geodesic balls are precompact) and connected Riemannian manifold with boundary. We define the weak gradient of a function $u \in L^2_{\mathrm{loc}}...
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2 votes
0 answers
92 views
An interpolation scale for uniformly local Sobolev spaces?
Are there any known results about the interpolation of uniformly local Sobolev spaces on the real line? For $s \geq 0$, the space $H_{\mathrm{u,l}}^s(\mathbb{R})$ is defined as follows, $$ H^s_{\...
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4 votes
0 answers
103 views
Regularity of the gradient for elliptic equations set in a bounded open set
Let $\Omega$ be a bounded open set (no regularity assumptions), and let $f \in L^{\infty}(\Omega)$. It is well-known that there exists a unique solution $u \in H_0^1(\Omega)$ of $-\Delta u = f$. In ...
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1 vote
0 answers
108 views
What is the curl of inverse Dirichlet laplacian?
Let $\Omega$ be bounded smooth simply connected. Let $\Delta^{-1}_D$ be the inverse of the Dirichlet problem. I was wondering what ${\rm curl}(\Delta^{-1}_D({\rm grad}(p))$ is for $p\in L^2$. Firstly ...
- 181
2 votes
2 answers
230 views
Necessary and sufficient conditions for coefficients of elliptic operator to obtain interior regularity
In the comments section in this other MSE question concerning a certain calculation left to the reader in Evans's Partial Differential Equations, @peek-a-boo and I were discussing the requirements on ...
- 183
6 votes
1 answer
443 views
Functions with compactly supported Fourier transform
The following is a repeat from the following question on MathStack Exchange. I am just hoping to have more success here. This is a follow-up from that question. The question is this: I want to ...
1 vote
0 answers
149 views
A generalized comparison principle - is it true?
Consider $v_1,v_2\in H^1([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ (Bochner spaces) be two weak solutions of the following doubly-nonlinear parabolic problems $$\begin{cases}\dfrac{\partial v_2^2}{\...
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3 votes
0 answers
134 views
About weak solutions of PDE's - integral inequality
Working with weak solutions of PDE's we often can deduce a inequality like the following one: $$ \int_{\Omega}w(x)\phi(x)\ dx+\int_{\Omega}\xi(x)\cdot\nabla\phi(x)\ dx\geq 0,\quad \forall\ \phi\in H^1(...
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0 votes
1 answer
231 views
Generalize Sobolev bound to $\mathbb R^n$
I am trying to generalise the following inequality on $\mathbb R:$ From $u(x)-u(r)=\int_r^x u'(s) \ ds$ we have $$\left \lvert \frac{1}{2r} \int_{-r}^r u(x) \ dx-u(r) \right\rvert^2 = \frac{1}{2r} \...
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0 votes
1 answer
147 views
Growth of solutions to the Poisson equation in growing domains
Let $N \ge 3$. Let $T, \Omega \subset \mathbb{R}^N$ be open and bounded sets, and assume they contain the origin. For $\varepsilon > 0$, I am interested in the solution of the Poisson equation \...
0 votes
1 answer
139 views
Continuity about extensions of $L^1$ functions to Sobolev functions
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain. It's well known that the trace operator $\mathcal{T}: W^{1,1}(\Omega) \to L^1(\partial \Omega)$ is surjective. My question is, given a ...
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0 votes
1 answer
144 views
About the Sobolev spaces $W^{k,n/k}(U)$
Currently, I am reading chapter 5 of L.C. Evans's Partial Differential Equations, 2nd Ed. I am on section 5.6.3, General Sobolev Inequalities. Evans says the following: (here, $p\in [1,\infty)$.) ...
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1 vote
1 answer
165 views
Notation in 5.6.3: General Sobolev Inequalities in Evans PDE book
I am currently reading Evans's PDE book. I reached section 5.6.3, General Sobolev Inequalities, which goes as follows: THEOREM 6: (General Sobolev Inequalities). Let $U$ be a bounded open subsetof $\...
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3 votes
1 answer
140 views
Comparing Trace of two functions who agree a.e. except on a small set
Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ ...
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2 votes
0 answers
83 views
How does truncation affect energy functional of Neumann Problems?
I am working on an energy functional arising from an inhomogeneous Neumann Problem. The boundary term makes it difficult to compare the energies while truncating. Concretely, Suppose $U$ is an open ...
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1 vote
1 answer
222 views
Sobolev extension: "we can arrange for the support of $\bar u$ to lie within $V\supset\supset U$". How, exactly?
My questions: Is this sort of what Evans was referring to when he claims we can "arrange" for the support of $Eu$ to be compact? If not, what did he mean? How can I easily see that we can &...
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0 votes
1 answer
146 views
Can we strengthen the extension theorem for zero trace functions?
I am crossposting this from MSE because I received no answers there. I have posted an answer, because I thought I had solved the problem. But, after some further thought, I am doubtful if I can be ...
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0 votes
1 answer
150 views
Sobolev trace theorem on bounded Lipschitz domain
The region $\Omega \subset \mathbb{R}^3$ is said to satisfy the $W^{1,r}$-trace theorem provided the trace mapping has an extension $\gamma$ that is a continuous linear map of $W^{1,r}(\Omega)$ into $...
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9 votes
1 answer
524 views
What is the optimal constant for $\|f\|_{L^\infty}\leq C\|f\|_{L^2}^{1/2}\|f'\|_{L^2}^{1/2}$?
Let $f\in H^1(\mathbb R)$, then by the Gagliardo–Nirenberg interpolation inequality, there is a constant $C>0$ which is independent of $f\in H^1(\mathbb R)$ such that $$\|f\|_{L^\infty(\mathbb R)}\...
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1 vote
0 answers
91 views
Regularity of backwards linear parabolic PDEs
In [1] or [2] one can find a regularity theory for (quasi-)linear parabolic PDEs of with Cauchy data, on Euclidean domains. However, in each case the boundary data is "forward in time", ...
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0 votes
0 answers
52 views
Convergence Rate Truncates Besov-Type Sequence Norm
Let $x:=(x_{i,j})_{i\in \mathbb{N},\, j=0,\dots,2^i}$ be a real-sequence and consider the (small) Besov-type sequence spaces with quasi-norms for $0<q,p,\alpha< \infty$ $$ \|x\|_{\alpha,p,q} := \...
4 votes
1 answer
322 views
Bounds on Sobolev norms of mollified functions - interpolation
Let $a,b\in \mathbb{R}$ with $a< b$, $n\in \mathbb{N}$ with $n>0$, $1\le p\le \infty$, let $W^{a,p}(\mathbb{R}^n)$ be the corresponding Sobolev space, and let $(m_{\delta})_{\delta>0}$ be a ...
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2 votes
1 answer
317 views
On the weak convergence of measures and Sobolev spaces with negative index
I have a question about the weak convergence of measures. Let $\mathcal{B}(\mathbb{R}^d)$ denote the set of all borel measurable sets in $\mathbb{R}^d$, and let $m \colon \mathcal{B}(\mathbb{R}^d) \to ...
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2 votes
0 answers
109 views
The relation between capacity and the Poincaré inequality
Let $N > 2$ and let $\omega, \, \Omega \subset \mathbb{R}^N$ be domains containing the origin. Define $\varepsilon \omega := \{\varepsilon x : x \in \omega\}$ for $\varepsilon > 0$. I am ...
0 votes
0 answers
115 views
Differentiability of $H^1$ function on some hypersurfaces
Let $u\in H^1(B_1^+;\mathbb{R}^3)$ for $B_1^+:=B_1\cap\{x_3>0\} \subset\mathbb{R}^3$. Suppose that $u(x)\in S \subset \mathbb{R}^3$ for $x \in B_1 \cap \{x_3=0\}$ in the sense of trace. The ...
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0 votes
1 answer
154 views
Homogeneous Besov space and norm estimate
Let $\sigma>0$ and $f\in S(\mathbb R^d)$ be a Schwartz function and define $$||f||_{\overset{.}{H}^\sigma}^2:=\frac{1}{(2\pi)^N}\int_{\mathbb R^d}|\xi|^{2\sigma}|\hat{f}(\xi)|^2d\xi,$$ where $\hat ...
0 votes
1 answer
153 views
Embedding of Sobolev spaces with the weight $|x|^\tau$
Let $\Omega \subset \mathbb{R}^N$ a bounded domain. For $\tau \in \mathbb{R}$, $m \in \mathbb{N}$ and $1 \leq p < \infty$ define $$ L^p(\Omega, |x|^\tau) = \left\{u : \Omega \to \mathbb{R}: u \text{...
- 239
3 votes
1 answer
285 views
Energy functionals with disconnected sublevel sets in the weak topology
Since the first question of A double problem on compact and connected sets in the weak topology has been answered, I reformulate here the second question alone: Let $g:{\bf R}\to {\bf R}$ be a ...
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2 votes
1 answer
301 views
Logarithm of a Sobolev function has a weak gradient?
Let $w\in W^{1,p}(\Omega)$ for some $p>1$ and $\Omega\subset\mathbb{R}^N$ be an open, bounded and connected Lipschitz domain. If $w>0$ a.e. on $\Omega$ is it true that $\nabla\left (\log w\right ...
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4 votes
1 answer
299 views
A mollification in the $L^2$ space with measure $dx/|x|$
Define the norm $$ \Vert f \Vert_{-\frac{1}{2}}^2 = \int_\mathbb{R} |f(x)|^2 \frac{dx}{|x|} $$ and the Hilbert space $$ L^2(\mathbb{R}, dx/|x|) = \left\{ f:\mathbb{R} \to \mathbb{R}: \Vert f \Vert_{-\...
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1 vote
0 answers
151 views
Product estimate in fractional-order Sobolev spaces
In Products of functions in fractional-order Sobolev spaces Bazin mentions for $f,g\in H^s({\mathbb R}^n)$, $s>n/2$, the estimate $$ \|fg\|_{H^s({\mathbb R}^n)}\leq c_n(\|f\|_{H^s({\mathbb R}^n)}\|...
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2 votes
0 answers
103 views
Does the Leray projection map $H ^1 (\Omega)^3$ to $H_0^1 (\Omega)^3 $? [closed]
Most of the information I’m using here come from Boyer and Fabrie, “Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models”. I’ll use them to formulate my ...
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4 votes
2 answers
293 views
Sobolev inequality on $\mathbb{R}^d \setminus B_R$
Does the following inequality hold $$\left(\int_{\mathbb{R}^d \setminus B_R} |u|^{\frac{2d}{d-2}}dx\right)^{\frac{d-2}{2d}} \leq S \left(\int_{\mathbb{R}^d \setminus B_R} |\nabla u|^2 dx\right)^{1/2}$$...
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1 vote
0 answers
134 views
Real interpolation of negative Sobolev spaces $(L^p,W^{-k,p})_{\theta,p}$ for $1\le p\le\infty$
For an open set $U\subset\mathbb R^n$, $k\ge1$ and $1\le p\le\infty$, the space $W^{-k,p}(U)$ consists of all distributions of the form $\sum_{|\alpha|\le k}\partial^\alpha g_\alpha$ where $g_\alpha\...
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1 vote
0 answers
64 views
Concluding a radial weak solution with radial test functions is a weak solution with all test functions
Let $B \subset \mathbb{R}^N$ be the unit ball and $E = \{u \in H^1_0(B) : u \text{ is radial}\}$. Define the functional $I : E \to \mathbb{R}$ by $$ I(u) = \int_{B} |\nabla u(x)|^2 dx - \frac{1}{p}\...
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4 votes
2 answers
290 views
Is the graph of a $W^{1,2}$-function path-connected?
Let $u:\mathbb{R}^n\to\mathbb{R}$ be a function in $W^{1,2}$ and let $u^*(x)=\lim_{r\to 0} \frac{1}{\omega_n r^n} \int_{B_r(x)} u(y) dy$ be the fine representative of $u$. From Evans-Gariepy Theorem 4....
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2 votes
1 answer
196 views
Commuting weak derivatives in time with space integral
Consider $Ω$ bounded and for simplicity of notation take $f'$ to be the weak time derivative. I’m not sure if the following claim is true: $$f \in W^{1,1} ((0,T);L^p (\Omega));1 \leq p \leq \infty \...
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5 votes
0 answers
152 views
Is the Nemytskii map $u \mapsto \max(0,u)$ Hölder continuous from $H^1_0(\Omega)$ into $H^1_0(\Omega)$?
I recently read that there is a chance that the positive part function $f(x) := \max(0,x)$ is such that the associated Nemytskii map $f\colon H^1_0(\Omega) \to H^1_0(\Omega)$ is $1/2$-Hölder ...
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2 votes
1 answer
135 views
An endpoint Besov space being reflexive?
I would like to know: is it known in the case of the Torus, provided for $s\in\mathbb{R}$, $q\in(1,\infty)$, that $$({\mathrm{B}}^{-s}_{\infty,q'}(\mathbb{T}^n))' = {\mathrm{B}}^{s}_{1,q}(\mathbb{T}^n)...
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5 votes
1 answer
224 views
Whether a function induced by the path integral is $L^p$ integrable?
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain (i.e. open connected subset) and $\alpha\in (0,1]$. Fix arbitrary $z_0\in \Omega$. Define the function induced by the path integral \begin{align*} ...
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1 vote
1 answer
180 views
dual of $H^1_0$ is not a distribution space?
Let $Ω⊂R^d$ be a smooth open bounded set with boundary $Γ$. Define $D(Ω)$ the space of test function in $Ω$ $U=(D(Ω))^d∩Ker(div)=ϕ∈D(Ω)×…×D(Ω)$ and $div (ϕ)=0$ $V=$ the closure of $U$ in $H^1$ so ...
- 181
0 votes
1 answer
153 views
The Sobolev inequality
The Sobolev inequality is valid in $ \mathbb{R}^N $: $$\|u\|_{L^{p^*}(\mathbb{R}^N)} \leq C \|\nabla u\|_{L^p(\mathbb{R}^N)}, \quad \forall u \in W^{1,p}(\mathbb{R}^N)?$$ My question is: does this ...
- 461
9 votes
2 answers
508 views
Convergence to a Lipschitz function
Let $f \in W^{1,1}(\mathbb R^n)$. Suppose for each $k \in \mathbb Z_+$, $f_k \in W^{1, k} (\mathbb R^n)$, $\|f_k\|_{W^{1, k}(\mathbb R^n)} \leq 1$ and further $f_k \to f$ uniformly. Question: Is it ...
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1 vote
0 answers
87 views
How to get a curve condition from Hölder continuous
I have read a paper (https://afm.journal.fi/article/download/134517/83089), and I have one question about the proof of necessity of Theorem 2.2 of this paper. Maybe this is an easy question, but I am ...
- 75
0 votes
0 answers
83 views
Moser-Trudinger inequality for surfaces of non-negative Gaussian curvature
Let $(\Sigma,g)$ be a compact Riemannian surface without boundary. The classical Moser Trudinger inequality implies that there exists constants $\beta,C$ depending on $M,g$ such that \begin{equation} \...
- 1
1 vote
0 answers
127 views
Fubini property of Sobolev space on domains
Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. The standard Sobolev space $W^{k,p}(\Omega)$ is the space of all $f\in L^p(\Omega)$ such that $\|f\|_{W^{k,p}}^p=\sum_{|\alpha|\le k}\int_\...
- 1,481
4 votes
1 answer
174 views
Tangential Sobolev spaces
Let $\Omega \subset \mathbf R^n$ be a smooth domain and define $U_s=\{x\in\Omega \mid d(x,\partial \Omega)<s\}$; let $f\in W^{1,p}(\Omega)∩W_{\mathrm{loc}} ^{2,p}(\Omega)$; let $v$ be the unit ...
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0 votes
1 answer
156 views
If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality \begin{equation*} \int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
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