3
votes
0answers
62 views

Equivalence of two definitions of weak solution (from a book, I don't understand something!!!!)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in ...
6
votes
1answer
109 views

Is a Sobolev function absolutely continuous with respect to a.e.segment of line?

Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes. It seems to me ...
4
votes
1answer
77 views

Example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t\notin L^\infty(0,T,H^1)$

Could someone give me an example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t$ exists (in the distributional sense), $u_t\in L^\infty(0,T,L^2)$ and $u_t\notin L^\infty(0,T,H^1)$? Thanks. ...
1
vote
0answers
25 views

Uniform continuity of weighted Sobolev functions.

I am trying to show an embedding result for a weighted Sobolev space and have come to the following problem: I have a function $f: (0,a] \rightarrow \mathbb{R} $ such that: $f$ is bounded and ...
3
votes
0answers
36 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
2
votes
1answer
96 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 $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
3
votes
0answers
52 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 ...
2
votes
0answers
62 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 ...
2
votes
1answer
49 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 ...
0
votes
1answer
32 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 ...
0
votes
1answer
18 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\{ ...
1
vote
1answer
28 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: ...
3
votes
0answers
44 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
0
votes
0answers
62 views

convergence sequence in L^1

Let an sequence $u_n$ such as $u_n$ converge to $u$ in $H^1_0(\Omega)$ weak, and $u_n$ converge to $u$ in $L^2(\Omega)$ strong and a.e $x \in \Omega$. Let $g_n(x,u_n)$ an Caratheodory function such ...
1
vote
1answer
68 views

convergence in L^{1} strong

I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies ...
6
votes
1answer
93 views

Proof of equivalence of Sobolev Space and Lipschitz functions

The attachment is a proof from Evans book "Measure Theory and Fine Properties of Functions" pg 132 Theorem 5. The statement of the theorem is: Let $f:U \rightarrow \mathbb{R}$. Then $f$ is locally ...
0
votes
1answer
34 views

Is $(H_0^1,\|\cdot\|_{L^2})$ a closed subspace of $L^2$?

Let $-\infty<a<b<\infty$ and $f\in L^2(a,b)$. Suppose $(f_n)$ is a sequence in $H_0^1(a,b)$ such that $\|f_n-f\|_{L^2}\overset{n\to\infty}{\longrightarrow}0$. Can we conclude that $f\in ...
0
votes
1answer
40 views

$\|f+g'\|_{L^2}=\|f'-g\|_{L^2}=0\Rightarrow f=g=0$ a.e?

Let $-\infty<a<b<\infty$ and $f,g\in H^1(a,b)$. So, $f,f',g,g'\in L^2(a,b)$. Suppose $$\int_a^b|f+g'|^2\mathrm dx=\int_a^b|f'-g|^2\mathrm dx=0.$$ Is it possible to conclude that $f=g=0$ ...
1
vote
1answer
41 views

If $(f_n)$ is Cauchy in the $L^2$-norm, then is $(f'_n)$ Cauchy in the $L^2$-norm?

Let $(f_n)$ be a sequence in $H^1(a,b)=\{f\in L^2(a,b);\;f'\in L^2(a,b)\}$, where $-\infty<a<b<+\infty$. If $(f_n)$ is a Cauchy sequence in the norm $\|\cdot\|_{L^2}$, is it possible to ...
1
vote
0answers
36 views

Aproximating a function on SO(3)

By $SO(3)$ I mean rotation matrices. Let $\cal{L}=\{f:[0,l]\to \rm{SO}(3)\} \cap L^1([0,l];\mathbb{R}^{3\times3})$. How to approximate funkctions from $\cal{L}$ with functions from ...
2
votes
1answer
100 views

Explanation on a “different” proof that $C_c(\Omega)$ is dense in $L^p(\Omega)$.

Theorem: Let $\Omega\subset \mathbb{R}^n$ be an open set and $1\leq p < \infty$. The space $C_c(\Omega)$ is dense in $L^p(\Omega)$. Haim Brezis has a French book called "Analyse fonctionnelle: ...
1
vote
0answers
22 views

construction of a sequence of smooth function [duplicate]

Consider $r>0$ , $1<p< \infty $. Consider $K $ a compact set in $R^n$ with $K \subset B(x_0 , r)$. Define $$ B = \{ u \in C^{\infty}_{0}(B(x_0 , r)) ; u=1 \ on \ K , 0 \leq u \leq 1\}$$ ...
1
vote
1answer
52 views

aproximation in Sobolev Spaces

consider $r>0 , p>1$ and $K \subset B(x_0 , 2r) \subset R^n$ . $K$ compact. Define the sets : $$A = \{ u \in C^{\infty}_{0} (B(x_0 , 2r)); \textit{ such that } \ u=1 \textit{ in a open ...
1
vote
2answers
68 views

About smooth approximation in a Sobolev space

I want to prove the following fact : Consider $\Omega \subset R^n$ a bounded and open set. Let $v \in H^{1}_{0}(\Omega)$ a nonnegative function. Then exists a sequence $v_m$ in ...
4
votes
1answer
355 views

Some doubts about Trace Theorem (for Sobolev functions).

Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and consider the Sobolec space $W^{1,p}(\Omega)$ for $p\in (1,\infty)$. I have some doubts with th trace theorem. Roughly speaking, the ...
5
votes
2answers
139 views

Sobolev spaces - about smooth aproximation

Consider $\Omega $ a open and bounded set of $R^n$ . Let $u \in H^{1}(\Omega)$ a bounded function. I know that exists a sequence $u_m \in C^{\infty} (\Omega) $ where $u_m \rightarrow u$ in ...
4
votes
0answers
75 views

$u\in W^{1,p}(B)$ implies $u\in W^{1,p}(\partial B_t)$ for almost $t\in (0,1]$?

Suppose that $u\in W^{1,p}(B)$ where $B=B(0,1)\subset\mathbb{R}^N$ and $p\geq 1$. It was showed on this post (as an application of Fubini's theorem) that for almost $t\in (0,1]$ $$u_{|\partial ...
4
votes
1answer
50 views

Existence of variation

Let $I[w] =\int_U L(Dw,w,x) dx$. Let $1<q<\infty$, and there exist constants $\alpha>0$,$\beta\ge0$ such that $$L(p,z,x)\ge \alpha |p|^q - \beta$$ This implies that if $I[w]$ exists, $$I[w] ...
1
vote
2answers
61 views

Approximation for $L_{\text{loc}}^{\infty}(U)$ is this proof correct?

Let $U\subset\mathbb{R}^n$ be open and bounded. I am trying to extend Evans' proof (in his PDE book) for approximating functions in $L_{\text{loc}}^{p}(U)$ for the case that $p=\infty$ using ...
3
votes
1answer
162 views

Gauss–Ostrogradsky formula for Distributions

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
2
votes
1answer
174 views

dominated convergence theorem

I am studying the proof of a theorem and in a part of the proof I have the following situation: Let $u : \Omega \rightarrow R$ a nonnegative measurable function, with $\Omega$ open and bounded. ...
2
votes
2answers
96 views

Sobolev Spaces and Measure Theory

Suppose $u\in W^{1,p}_0(\Omega)$, where $\Omega\subset\mathbb{R}^n$ is a bounded domain (open and connected) and $p\geq 1$. Let $a\in\mathbb{R}$ and suppose the set $\Omega_a=\{x\in\Omega:\ u(x)=a\}$ ...
10
votes
1answer
492 views

Understanding a theorem concerning Sobolev spaces

I have two doubts in the proof of the theorem below. If you want the detaIls can be found here. Theorem A Let $q$ a given real number $q>p$. Let $u \in W^{1,p}$ be a solution to (E2). Assume ...
2
votes
1answer
178 views

Poincaré Inequality - Product Of Measures

I'm given two euclidean spaces $ \mathbb{R}_1 , \mathbb{R}_2 $ , with probability measures on them , that satisfy the Poincaré's inequality: $ \lambda^2 \int_{\mathbb{R}^k} |f - \int_{\mathbb{R}^k} f ...
3
votes
1answer
104 views

Evaluation of $L^p$ function

Functions in $L^p$ are only defined $µ$-almost everywhere, so for a given evaluation point $x$, $F(x)$, $f\in L^p$ can be changed to any value, so in general it would not be well-definied to just ...
3
votes
3answers
331 views

Total variation of (weakly) differentiable functions

the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as $$ \sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert ...