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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About Sobolev-Poincare inequality on compact manifolds

Let $2^* = \frac{2n}{n-2}$ where $n$ is the dimension of a compact closed manifold $M$. We get from the Sobolev/Poincare inequality the identity $$\lVert u \rVert_{2^*} \leq C\lVert \nabla u ...
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21 views

Morrey's inequality

From PDE Evans, 2nd edition, page 281: Now \begin{align} \int_0^s \int_{\partial B(0,1)} |Du(x+tw)| \, dS(w) dt &=\int_0^s \int_{\partial B(x,t)} \frac{|Du(y)|}{t^{n-1}} \, dS(y) dt \\ ...
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28 views

What domains of $\mathbb{R}^n$ have the property that $H^1(\Omega)=H^1_0(\Omega)$?

i wonder what are sufficient conditions on an unbounded domain of $R^n$ called $\Omega$ to get : $C_c^\infty (\Omega)$ dense in $H^1 (\Omega)$ ? where $C_c^\infty$ stands for the set of functions with ...
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17 views

Gagliardo-Nirenberg-Sobolev inequality

There is one step of the textbook's proof that I wish could be clarified. Pages 277-278 of PDE Evans, 2nd edition, says: Integrate this inequality with respect to $x_1$: \begin{align} ...
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37 views

Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
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21 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
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Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
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25 views

About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
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1answer
41 views

Trace-zero functions in $W^{1,p}$

This is an excerpt of a textbook's proof for a theorem (Trace-zero functions in $W^{1,p}$), from PDE Evans, 2nd edition, page 275. Next let $\zeta \in C^\infty(\mathbb{R}_+)$ satisfy $$\zeta ...
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40 views

Trace Theorem question

From PDE Evans, page 272. My question is towards the bootom of this post. THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator ...
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1answer
25 views

Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
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1answer
41 views

Extension Theorem

From PDE Evans, 2nd edition, pages 268-270. My question is at the bottom of this post. THEOREM 1 (Extension Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Select a bounded open set $V$ ...
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1answer
23 views

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
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57 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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1answer
12 views

About the weighted Sobolev norm

I'm wondering that the Sobolev norm with weight $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ is equivalent to the norm $$ ...
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1answer
47 views

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that ...
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25 views

Taylor theorem on Sobolev spaces

I am trying to understand the Taylor theorem for Sobolev spaces that appears in http://science.org.ge/moambe/5-2/5-10%20Boyarsky.pdf. I am not sure in what sense the aproximation is. I feel that it is ...
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27 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
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1answer
120 views

Is this proposition about $L^2$ functions correct?

Is this proposition correct? Will you please give a contour example if it is wrong? If $J \in L^2(\mathbb{R}) \cap C^1 (\mathbb{R})$, $f \in C^{\infty}(\mathbb{R})$, $|f'|\leq K$, where $K > 0$ is ...
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1answer
24 views

Difference in Definitions of Quasiconvexity

So I've seen a few different definitions of quasiconvexity of a function and I cannot, after a bit of working, figure out how all of them are related: Let $X$ be a convex subset of a real vector ...
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18 views

Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$. It follows that for almost all $t$, $u_n(t)$ is bounded in ...
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1answer
67 views

Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...
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25 views

Fractional order Sobolev space over manifold

I stuck with a problem: If $\Omega\subset\mathbb{R}^{n}$ is open and bounded with a sufficiently smooth boundary $\Gamma$. Then how to prove dual space of $H^{1/2}(\Gamma)$ is $H^{-1/2}(\Gamma)$ ? ...
2
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1answer
30 views

A diffeomorphism between manifolds (or surfaces) that preserves the mean value of functions

Let $M$ and $N$ be two Riemannian manifolds with $f:N \to M$ a diffeomorphism with the following properties: for all $u \in H^1(M)$, $\hat u := u\circ f$ satisfies $\hat u\in H^1(N)$ and furthermore ...
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1answer
32 views

Constant in Sobolev-Poincare inequality on compact manifold $M$; how does it depend on $M$?

Let $M$ be a smooth compact Riemannian manifold of dimension $n$. Let $p$ and $q$ be related by $\frac 1p = \frac 1q - \frac 1n$. There is a constant $C$ such that for all $u \in W^{1,q}(M)$ ...
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1answer
31 views

The range of the distributional laplacean, defined in $W_0^{1,1}(\Omega)$.

Let $\Omega\subset \mathbb{R}^N$ be a bounded, smooth domain. Assume that $u\in W_0^{1,1}(\Omega)$ and consider the distributional laplacean of $u$; $\Delta u$. My question is: when is $\Delta u\in ...
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1answer
15 views

Green identity for measures with compact support

Let $\Omega\subset\mathbb{R}^N$ be a bounded, smooth domain. Assume that $\mu \in \mathcal{M}(\Omega)$ has compact support in $\Omega.$ Let $u\in W_0^{1,1}(\Omega)$ be a solution of $$ \left\{ ...
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1answer
30 views

Proving that weak limit in $L^p$ and strong limit in $H^{-1}$ are the same

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. Let $p \geq 1$ and suppose that $u_n \rightharpoonup u$ in $L^p(\Omega)$ and $u_n \to v$ in $H^{-1}(\Omega)$. How to show that $u=v$? I can do ...
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1answer
42 views

A technical step in proving Hardy's inequality

A technical step in proving Hardy's inequality $$ \int_{B(0,r)}\frac{\mu^{2}}{|x|^{2}}dx\le C\int_{B(0,r)}(|D\mu|^{2}+\frac{\mu^{2}}{r^{2}})dx $$ where $n>3, r>0, \mu\in H^{1}(B(0,r))$ is to ...
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0answers
26 views

Want a dense subset of a Sobolev-Bochner space!!

Let $U \subset \mathbb{R}^n$ be a bounded domain. Let $$W=\{ u \in L^2(0,T;H^1(U))\cap L^\infty(0,T;L^\infty(U)) : u' \in L^2(0,T;H^{-1}(U))\}$$ and let $$D=\{u \in L^2(0,T;H^1(U)) \cap ...
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Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
3
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1answer
26 views

Sobolev's inequality from Reed & Simon vol. II

In Reed & Simon vol. II, an inequality called Sobolev's inequality is stated in Eq. (IX.19): Let $0<\lambda<n$ and suppose that $f\in L^p(R^n)$, $h \in L^r(R^n)$ with $p^{-1} + r^{-1} + ...
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1answer
48 views

If $Du=0$ a.e. , does $u=c$ a.e.?

Let $W^{1,p}(U)$ be the Sobolev space, where $U$ is a connected bounded domain in $\mathbb{R}^n$ and $u\in W^{1,p}(U)$ satisfying $Du=0$ a.e. in $U$. Then $u$ is constant a.e. in $U$. I don't ...
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1answer
28 views

The Nash inequality on a compact manifold without a boundary.

Let $M$ be a compact manifold without boundary. Does the Nash inequality $$\lVert u \rVert^{1+\frac 2n}_{L^2} \leq C\lVert u \rVert^{\frac 2n}_{L^1} \lVert \nabla u \rVert_{L^2}$$ or something ...
2
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1answer
75 views

Sobolev Spaces and Derivative

I need help on the problem 8.9 at page 238 of the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis. Set $I=(0,1)$. Let $u \in W^{2,p}(I)$ with ...
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Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
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66 views

Weak Derivative Heaviside function

I have to prove that the Heaviside function $$ H(x):=\begin{cases} 1 &\mbox{if } x \in [0,+\infty) \\ 0 &\mbox{otherwise}\end{cases} $$ doesn't admit weak derivative in ...
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1answer
25 views

A priori estimates for functions in $C_0^\infty (\overline{\Omega})$.

Let $u\in C_0^\infty (\overline{\Omega})$, where $\Omega\subset \mathbb{R}^N$ is a bounded domain. Fix some $a\in \Omega$ and choose $r>0$ such that $\overline{\Omega}\subset B(a,r)$. Define ...
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1answer
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If $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;H^1)$ and $u$ and $u_t$ are bounded, is $u \in C([0,T];L^\infty$)?

Let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$ so $$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$ holds for all $\varphi \in C_c^\infty(0,T)$. Suppose we know ...
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1answer
59 views

Characterization of weak solution

5 Nonlinear elliptic variational inequalities Preliminaries In order to explain the importance of elliptic variational inequalities, first consider the weak solution of the linear ...
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Sobolev space: $\vert u\vert^{p-2}u $ belongs to $u\in W^{1,p'}(\Omega)$

Show that if $p\geq2$ and $u\in W^{1,p}(\Omega)$ then the function $\vert u\vert^{p-2}u $ belongs to $u\in W^{1,p'}(\Omega)$
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If $u_n \to u$ in $C([0,T];H^{-1})$ and $\lVert u_n\rVert_{L^\infty} \leq C$ then $u_n(t) \rightharpoonup u(t)$ in $L^1(\Omega)$?

Let $u_n \to u$ strongly in $C([0,T];H^{-1}(\Omega))$ and suppose that $u_n$ is uniformly bounded in $L^\infty((0,T)\times\Omega)$. Then $$u_n(t) \to u(t)$$ weakly in $L^1(\Omega)$ for each $t$? Can ...
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1answer
18 views

Spherical rearrangement

Let $u\colon\Omega\subset\mathbb{R}^N\to\mathbb{R}$ be a non negative measurable function, and $\Omega$ open and bounded. Consider $u^*$ the spherical rearrangement $$ u^*(x)=\sup\{t\geq0 : \mu\{x: ...
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39 views

Regularity of a Weak Solution

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
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1answer
43 views

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases} ...
2
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1answer
35 views

What is the dual of $A\cap B$

I encountered with some elliptic problem which admits a variational formulation in terms of space $X$ and I need to understand its dual. Suppose that $2<p<\infty$, $\Omega\subset {\mathbb R}^d$ ...
2
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1answer
35 views

Convergence in $L^p$ plus bounded gradient implies that the limit belongs to $W^{1,p}$?

I have a question with this problem I have found in the latest edition of the book Functional analysis, Sobolev Spaces author Haim Brezis pag 264 Remark 4 Let $(u_n) \subset W^{1,p} $ such that $u_n ...
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1answer
28 views

Equivalent Definitions of Negative Order Sobolev Spaces

Ignoring fractional sobolev spaces, if we restrict ourselves to $k>0$ when $k$ is an integer, then the Sobolev space of order $k$, for $W^{k,p}(\mathbb{R})$ is the space of functions $f$ such that ...
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2answers
66 views

Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$ \Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd ...
0
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1answer
43 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...