0
votes
2answers
22 views

$\|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $ holds?

I want to find the relation of $p$ and $k$ such that the inequality $$ \|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $$ holds when r.h.s $<\infty$. Here $f$ ...
4
votes
0answers
45 views

Higher interior regularity

From PDE by Evans, 2nd edition, pages 332-333. My question and work shown are at the bottom of this post. THEOREM 2 (Higher interior regularity). Let $m$ be a nonnegative integer, and assume ...
0
votes
0answers
26 views

Interior $H^2$ regularity - using a textbook's identity to show an important estimate

This is yet another continuation of my previous question, and concerns the same long textbook proof which I asked in those two questions as well. This is now from page 331 of PDE by Evans, 2nd ...
0
votes
2answers
26 views

Interior $H^2$ regularity - inequalities over regions $U$ and $W \subset U$

This question is a direct continuation of my previous question. However, this one is requesting only for a relatively simple explanation. Note: $W \subset U \subset \mathbb{R}^n$. On page 330 of PDE ...
2
votes
1answer
24 views

Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
1
vote
2answers
45 views

Show that for all $(\tau, \xi) \in \mathbb R^{n+1}$ we have $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$

Show that, for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$ This is the exercise 7.4 in the book by Francois Treves. It is just a fundamental ...
1
vote
1answer
34 views

Lp bounds of the Heat Kernel

These days, I am struggling with a problem which seems very straightforward (and I'm pretty sure it is straightforward) but it resists to my attempts to prove it. Here it is: Let $\mathcal H_t$ be ...
1
vote
1answer
50 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 \\ ...
2
votes
1answer
41 views

How to prove this integral inequality?

Here is a problem: Let $B_r=\{ (x_1,x_2,\cdots,x_n)\in \mathbb{R}^n: x_1^2+x_2^2+\cdots+x_n^2<r^2\}.$ Let $f$ be a $C^1$ real function on $B_2$. Prove that $$\inf_{a\in R}\int_{B_2} ...
0
votes
0answers
26 views

Proof of ellitpticity Monge Ampere Equation

I want to ask how to prove an inequality below: Let $A(x,z,p)\in R^n\times R^n$ be a symmetric matrix, $\Omega$ be a bounded domain and $\Omega^\delta=\{x|dist(x,\Omega)<\delta\}$ Given a ...
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 ...
1
vote
0answers
22 views

Proving an elementary inequality of real vectors related to the p-Laplacian [duplicate]

How would you prove the following inequality? $$ \left|\left|a\right|^{p-2}a-\left|b\right|^{p-2}b\right|\leq C_p\left|a-b\right|^{p-1} $$ where $a,b\in\mathbb{R}^{n}$ and $C_p>0$ is some constant ...
0
votes
0answers
16 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. ...
0
votes
0answers
23 views

About an inequality equivalence in Optimal system

currently I'm working on an optimal control problem and I have what I think is a really simple question about it, The variable that I'm looking to optimize is $\lambda$ and it lies in an $X^d$ space ...
0
votes
1answer
34 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
2
votes
0answers
25 views

Integral Inequality Question: Effect of Exponents

Is it true that, if $\alpha q < (1-\alpha)q$ then $\int|k(x,y)|^{\alpha q}dy \leq \int|k(x,y)|^{(1-\alpha)q}dy$, where $k(x,y)$ is a positive measurable function?
9
votes
0answers
166 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
2
votes
1answer
44 views

Need help filling in the details of proof of Jensen's Inequality

In a book on PDEs that I'm reading, I am trying to fill in the details of the proof of Jensen's Theorem, and am having a little trouble with the algebra. Here is the statement of Jensen's Theorem in ...
5
votes
1answer
109 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
41 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)| ...
2
votes
1answer
183 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 ...
2
votes
2answers
52 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
3
votes
2answers
151 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 ...
1
vote
1answer
160 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 ...
0
votes
1answer
58 views

Calculus Inequality

I am thinking about the following problem: Suppose $1 \le f(x) \le 2$ on $[0,1]$. Let $x_0$ and $x_1$ be fixed points such that $0< x_0 <x_1< 1$. Show that there exist positive constants ...
5
votes
2answers
179 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 ...
1
vote
3answers
98 views

Is there a lower bound for $\int_{B_{r}}f$ when $f$ is a positive function?

Let $f$ be a positive Lebesgue integrable function on a ball $B_{r}\in\mathbb{R}^n$. I'm looking for a positive constant $c$ depending only on $f$ that satisfies $$c\omega_nr^n\leq\int_{B_r}f$$ where ...
1
vote
1answer
68 views

Why is the case $1/2<\rho\leq 1$ trivial in proving the following inequality?

I'm studying Elliptic Partial Differential Equations by Q. Han and F. Lin. In Lemma 1.41 is given the elliptic equation $D_j(a_{ij}D_i u)=0$ where the coefficient matrix $(a_{ij})$ is constant ...
2
votes
1answer
47 views

Why does the following inequality holds for any weak solution $u\in C^1(B_1)$ of uniformly elliptic equation $D_i(a_{ij}D_ju)=0$?

Now I'm studying "Elliptic Partial Differential Equations" by Q.Han and F. Lin. Throughout the section 5 of the chapter 1, $u\in C^1(B_1)$ is a weak solution of $$D_i(a_{ij}D_j u)=0$$ where ...
2
votes
0answers
128 views

Gagliardo Nirenberg Sobolev inequality

Assume that $f$ satisfies the equality in the Gagliardo Nirenberg Sobolev inequality for the best constant. What can be said about $f$?
2
votes
2answers
66 views

Showing that this Coercivity condition implies uniform boundedness of a minimising sequence.

The following problem is in Dacorogna's book "Introduction to the Calculus of Variations": Let $\Omega\subset\mathbb{R}^n$ be open and bounded with a Lipschitz boundary. Let $f\in C(\mathbb{R}^n\times ...
5
votes
1answer
133 views

Estimation on elliptic operator

Assume the strongly elliptic property, i.e. $$\sum_{|\alpha|= m}a_\alpha \xi^\alpha\neq0,\ \forall \xi \in \mathbb{R}^d\backslash\{0\},$$ and $$\sum_{|\alpha|\leq m}a_\alpha ...
3
votes
1answer
97 views

How to deduce the uniform ellipticity condition from an integral condition

Let $\Omega$ be a bounded open set, and let $A$ be a $N\times N$ symmetric matrix with entries in $L^\infty(\Omega, \mathbb{R})$, such that for some positive $\lambda$ and $\Lambda$ the following ...
1
vote
1answer
63 views

An inequality from PDEs: $|v|^\alpha v -|w|^\alpha w \leq C(|v|^\alpha + |w|^\alpha)|v-w|$

In using Stritcharz estimates to prove well-posedness of the nonlinear Schrödinger equation $$i\partial_tu = \Delta u - \lambda|u|^\alpha u$$ where $\alpha>0$, one requires the following inequality ...
3
votes
1answer
159 views

Inequality between Neumann and Dirichlet eigenvalues

Let $\Omega$ be a fixed, smooth and bounded domain in $\mathbb{R}^n$. If we denote with $\{\lambda_n\}_{n \ge 1}$ the nondecreasing sequence of eigenvalues of the Dirichlet problem $$\left\{ ...
1
vote
1answer
57 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 ...
2
votes
1answer
72 views

Proving the equality in weak maximum principle of elliptic problems

This one is probably simple, but I just can't prove the result. Suppose that $\mathop {\max }\limits_{x \in \overline \Omega } u\left( x \right) \leqslant \mathop {\max }\limits_{x \in \partial ...
2
votes
1answer
85 views

Bounding the solution of a wave equation in 3 dimensions

Let $u:{\mathbb{R}^ + } \times {\mathbb{R}^3} \to \mathbb{R}$ be a solution of the Cauchy problem $\left\{ \begin{gathered} {u_{tt}} - \Delta u = 0 \\ u\left( {0,x} \right) = {u_0}\left( x ...
4
votes
1answer
493 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
273 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 ...
4
votes
1answer
358 views

Kato's Inequality with Laplacian

Let $u \in C^2(\mathbb R^n)$ be complex-valued. I am trying to prove the inequality $$\Delta |u| \geq \Re\left(\frac{\bar u}{u} \Delta u\right),$$ in the distributional sense. I tried calculating ...
2
votes
1answer
583 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
0answers
42 views

Show inequality for modified bilinear form

Let $\Omega_h$ denote to the domain that is bounded by a polygon, and $V_h$ to the space of all $c\in C^0(\Omega)$ such that $v_{|T}$ is linear on any (curved) triangle T and $v=0$ in the vertices of ...
0
votes
1answer
95 views

Why this inequality yields at most exponential growth?

Let $\Omega=\mathbf{R}^{n-j}\times\omega$, where $\omega\subset\mathbf{R}^j$ is a smooth bounded domain. Consider a function $u:\overline\Omega\rightarrow\mathbf{R}$ that satisfies $$u(x,y)+k\leq ...
3
votes
1answer
197 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 ...
7
votes
2answers
542 views

Why is Korn's inequality not trivial?

Let $\Omega \subset \subset \mathbb{R}^N$ have smooth boundary, $N \geqslant 2$ and $$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} ( v) \mathrm{d} x := ...
2
votes
2answers
261 views

An inequality about the gradient of a harmonic function

Let $G$ a open and connected set. Consider a function $z=2R^{-\alpha}v-v^2$ with $R$ that will be chosen suitably small, where $v$ is a harmonic function in $G$, and satisfies $$|x|^\alpha\leqslant ...
1
vote
0answers
126 views

Harnack Inequality…

Consider the eigenfunction $\varphi_R>0$ $$L\varphi_R=\lambda_R\varphi_R, \ \ \ in \ \ B_R,$$ and $$\varphi_R=0, \ \ \ in \ \ \partial B_R,$$ where $L$ is a elliptic operator and $\lambda_R$ is the ...
5
votes
2answers
146 views

$ W_0^{1,p}$ norm bounded by norm of Laplacian

Let $f\in W_0^{1,p}(U)$, for $U$ a bounded domain and $p < n/(n-1)$. I am trying to prove that there is an inequality of the form $$\|f\|_{W^{1,p}} \leq C \int_{\Omega} |\Delta f| $$ where ...