1
vote
0answers
22 views

Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
2
votes
1answer
26 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
-1
votes
0answers
14 views

$\big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} \le C_{>0}\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$ holds? [duplicate]

I want to know that whether the following inequality holds or not for complex-valued functions $f_1$, $f_2$, $f_3$ on $\mathbb R$: $$ \big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} ...
1
vote
1answer
54 views

Hardy's inequality

Hardy's inequality (for integrals, I think) presented in Evans' PDE book (pages 296-297) contains a formula whose notation is substantially different than the conventional estimate presentation of ...
2
votes
1answer
34 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 \\ ...
5
votes
0answers
45 views

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 ...
1
vote
0answers
57 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
0
votes
0answers
17 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
2
votes
1answer
72 views

Poincaré inequality for a subspace of $H^2(\Omega)$

Suppose that $\Omega\subset\mathbb{R}^d$ is a smooth, bounded, and connected domain. Let \begin{equation} H=\{u\in H^2(\Omega):\int_\Omega u(x) dx=0 ~\text{and}~ \nabla u\cdot v=0~ ...
0
votes
0answers
31 views

inequality for linear functions in sobolev space

Ist the following Statement true for $f$ and $g$ linear? $\vert fg \vert_{H^2} \leq C \Vert f \Vert_{H^1} \Vert g \Vert_{H^1}$, where $\vert \cdot \vert_{H^2}$ denotes the seminorm. My Idea: It is ...
2
votes
2answers
82 views

Directional derivative in a Sobolev-like inequality

I am trying to do the following problem: Let $\Omega \subset \subset \overline{\mathbb{R}_{+}^{n+1}} = \{ (x, x_{n+1}); \, x_{n+1}\geq 0\}$ (i.e., $\Omega$ is bounded inside the closed upper ...
1
vote
1answer
113 views

Gagliardo Nirenberg Sobolev inequality for n >= 2

I have a quick question regarding the Gagliardo-Nirenberg-Sobolev inequality. It states that: Assume $1 \leq p < n$. There exists a constant $C$, depending only on $p$ and $n$, such that ...
0
votes
1answer
63 views

Using Cauchy Schwarz for functions in Sobolev Space

Hi I just want to confirm something simple and check that the following is allowed: The Cauchy-Schwarz inequality states if $A = ((a_{ij}))$ is a symmetric, non-negative $n \times n$ matrix then ...
5
votes
1answer
112 views

The second Friedrichs' inequalities?

In paper On the Validity of Friedrichs' Inequalities,$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ \tag{1}\qquad \|\mathbf{u}\|_{1,\Omega} \le ...
2
votes
1answer
150 views

Friedrichs's inequality?

Friedrichs's second inequality is stated as follows(see www.win.tue.nl/~drenth/Phd/friedrichs.ps): For all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n}\cdot\mathbf{u} = 0$ or ...
1
vote
2answers
129 views

Inequality involving Lp space, Holder Space, Sobolev Space

Do we have the following inequality or some variation of this: $||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)}$ for $n < p \leq \infty$ for $u \in W^{1,p}(U)$ where $U$ ...
4
votes
2answers
80 views

Using Results in Sobolev Spaces

I have two questions about using results in Sobolev Spaces and a last one on an inequality involving Holder spaces: 1.The Gagliardo-Nirenberg-Sobolev Inequality states the following: Assume $1 \leq p ...
2
votes
1answer
42 views

estimate on $| \nabla (u |u|^2) - \nabla(w|w|^2)|$ for $u,w \in H^1$

suppose $u, w \in H^1 (R^2)$. I'd like to know where does the following inequality come from (it appears in a proof I've been reading and I can't figure it out) $$ | \nabla (u |u|^2) - \nabla(w|w|^2)| ...
3
votes
1answer
84 views

Similar to Poincare inequality on Sobolev spaces

The following looks quite similar to Poincare's inequality: Let $\displaystyle{ 1 \leq p < \infty}$ and $\displaystyle{ U \subset \mathbb R^n}$ open and such that $\displaystyle{ U \subset ...
4
votes
2answers
131 views

Using the Extension Operator Theorem for Sobolev Spaces

I want to know if certain conditions hold after applying the Sobolev Extension Theorem: Assume $U$ is a bounded open subset of $\mathbb{R}^{n}$ and $\partial U$ is $C^{1}$. Suppose $1 \leq p < n$. ...
2
votes
1answer
190 views

Evans PDE p.308 Exercise 16 (2nd ed)

Here is the statement of the problem (Evans PDE 2nd Ed., p.308, exercise 16) Show that for $n \geq 3$ there exists a constant $C$ so that $$ \int_{\mathbb {R}^n} \frac{u^2}{\vert x ...
1
vote
1answer
116 views

Using Sobolev-Nirenberg-Gagliardo

I am currently studying a proof of a General Sobolev Inequality. I have the following question: Consider the Sobolev Space $W^{k,p}(U)$. With the added assumption that $k > \frac{n}{p}$. Let $l = ...
2
votes
0answers
142 views

Using the Sobolev-Nirenberg-Gagliardo inequality in a proof

If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u ...
1
vote
1answer
462 views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
3
votes
2answers
160 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
4
votes
1answer
56 views

Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
1
vote
1answer
166 views

Poincaré inequality for $W_0^{1,\infty}$

In the book A first course in Sobolev spaces by Leoni, the following Poincaré inequality for $W_0^{1,p}(\Omega)$ is stated: Suppose $\Omega\subset \mathbb{R}^n$ has finite width (lies between two ...
6
votes
2answers
189 views

An inequality of J. Necas

The above inequality belongs to J. Necas. Sur les normes equivalentes dans $W^k_p$ et sur la coercivite des formellement positives. Sem. math. sup. Universite Montreal, 102-128 (1966). But I can't ...
2
votes
1answer
52 views

$H^1$ norm estimation of an affine function

Let $v(x)=\alpha +(\beta-\alpha)x$ a function in $H^1(\Omega)$ with $\Omega=[0,1]$ and $\alpha$ and $\beta$ are constants. How do we prove that there exist an constant $M >0$ such that ...
3
votes
1answer
115 views

How is the inner product in $H^{-1/2}$ defined?

Since $H^{1/2}$ is a Hilbert space, $H^{-1/2}$ must also be a Hilbert space by the isomorphism of Riesz representation theorem. How is the inner product defined there? We know there is a nice ...
-1
votes
1answer
38 views

An inequality : $ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$

Let $u=u(x)$ be a real-valued function defined on $\mathbb R$. How does this inequality hold? $$ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$$ ...
2
votes
1answer
72 views

Laplacian inequality in $L^\infty$

Let $\Omega$ be a bounded domain of $R^n$ and let $y\in H^2(\Omega)\cap H_0^1(\Omega)$ such that the set $[x\in \Omega/ y(x)\ne 0]$ has non nul measure and $ \; \frac{\Delta y}{y} 1_{\{x\in \Omega/ ...
3
votes
1answer
513 views

Trace inequality

Could you please give me a hint on how to prove the following inequality $$\|u\|_{L^2(\Gamma)}\le C\|u\|^{\frac12}_{L^2(\Omega)}\|u\|^{\frac12}_{H^1}, \quad \forall u\in H^1.$$,
2
votes
1answer
70 views

Inequality in Sobolev Space

Given $\Omega \subset \mathbb{R}^3$, prove $\forall u, v, w \in H^{1,2} (\Omega)$ it holds that $ | \int_{\Omega} u \frac{\partial v}{ \partial x} w dx | \leq \| u \|_{1,2,\Omega}\|v \|_{1,2,\Omega}\| ...
4
votes
1answer
294 views

Poincaré inequality and Rellich Theorem in one dimensional weighted Sobolev space

Consider the weighted Sobolev space $W^{1,2}\big((0,R),r^{N-1}\big)$, $N=2,3,\ldots$ and its subspace $W_0^{1,2}\big((0,R),r^{N-1}\big)$. Anyone knows if the Poincaré inequality is true in this case? ...
2
votes
1answer
48 views

Regularity and the Varitational Inequality

Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
1
vote
1answer
58 views

How to establish the estimate?

I consider the inequality as follows: let $a>0,b>0$, and satisfy $$2a^2-b^2\leq C(1+a).\tag{1}$$ If we assume that $b\leq a$, then there exist a constant $C_1>0$ such that $$a\leq ...
3
votes
1answer
110 views

About a counterexample of an inequality?

I have known how to use the compactive argument to prove the inequality (1), i.e. $1\leqslant p<n$, $\Omega\subset R^n$ is a bounded domain,$\forall \varepsilon>0$, there is ...
4
votes
1answer
520 views

Sobolev embedding for $W^{1,\infty}$?

From the Evans'PDE book I learned $W^{1,\infty}(U)$ coincide with Lipschitz continuous $C^{0,1}(U)$ with $U\in\Bbb{R}^n(n\geqslant1)$ is bounded and $\partial U\in C^1$. I wonder the counterpart ...
2
votes
1answer
278 views

How to prove a version of Poincare inequality?

I want to use the contradiction argument and compact argument to prove the inequality below $\forall\epsilon>0$,there exists $C_\epsilon>0$,$\forall u\in W^{1,p}(U)$,we have ...
2
votes
1answer
596 views

Poincare Inequality

In page 290 of this book, Evans prove the Poincare inequality (Theorem 1) arguing by contradiction. Is there a direct proof of this theorem (Theorem 1) without arguing by contradiction?
1
vote
1answer
211 views

Poincaré inequality using $H^1$ seminorm

Does this inequality holds for Poincaré Inequality? $$||v||_{L^2} \leqslant C_p |v|_{H^1}$$ and $$ |v|_{H^1} = ||v'||_{L^2} $$ where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm. I'm ...
3
votes
1answer
202 views

Question on proof in Evans PDE

This is on page 542 of Evans PDE book. The last inequality states that $$\int_{U}{C(|Du|+1)|u|dx} \leq \frac{1}{2}\int_{U}|Du|^2dx + C\int_{U}{|u|^2+1 \ dx}$$ Where is this coming from? I think ...
1
vote
1answer
172 views

Infimum of the $L^2$ norm of the derivative of functions of $H^1_0$ of $L^2$ norm $1$

Let $I=(a,b)$ with $a<b$ real numbers. Define $\lambda_0$ by $$\lambda_0=\inf\{\|u'\|_{L^2(I)}^2:\ u\in H_0^1(I),\ \|u\|_{L^2(I)}=1 \}.$$ How can I prove that ...
4
votes
1answer
133 views

Prove this inequality from functional analysis

I want to prove this equality used in out lecture notes: Let $D=(0,r)^2 \subset \mathbb{R}^2, r\geqslant 0$. Then, for any $u \in H^1(D)$, there holds $$\lVert u\rVert \leqslant \frac 1 r ...
2
votes
1answer
177 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 ...
2
votes
1answer
148 views

Sufficient conditions to hold the following inequality.

If I have an inequality: $\lVert u\rVert_{L^p(R^n)} \le C\lVert\nabla u\rVert_{L^q(R^n)}$ , where $C \in (0,\infty)$ and $u \in C_c^1(R)$, is there a relation between $p, q, n$ such that the ...
1
vote
1answer
487 views

Poincaré inequality in unbounded domain

Help me please, how can I to show that Poincaré inequality in unbounded domain doesn't holds? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...