3
votes
1answer
86 views

Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...
2
votes
0answers
78 views

Variant form of the gronwall inequality

I know the following statement for gronwall inequality: Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have, $f' \leq \phi f$ and $f(0)=0$ then $f=0$ Now is ...
4
votes
0answers
104 views

Carleman estimates on monotonicity formulas

I am trying to derive a monotonicity formula for a certain Dirichlet critical point (or even maybe a minimizer) of an energy of the type, say for simplicity, an energy of the from $$\int_{B_r} ...
0
votes
1answer
138 views

A Poincaré-type inequality with logarithmic function

For any function $f(x)$ we denote $\bar{f}:=\frac{1}{\Omega}\int_\Omega f(x)\,dx$. Let $\Omega\subset \mathbb{R}^n$ be a bounded smooth domain and $u(x)> 0$ be a smooth function defined on ...
3
votes
1answer
116 views

An interpolation type inequality

Let $u(x),x\in R_+$ be a non-negative decreasing smooth function with compact support $[0,L]$, I want to know the following inequality is true? $a\in (0,1)$ $$\int_0^\infty \frac{1}{1+x}u^{1+a}dx \le ...
18
votes
1answer
895 views

What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities: If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...
3
votes
1answer
257 views

on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
1
vote
2answers
266 views

how to solve a singular integral equation involving the kernel $1/x$

Dear all, Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that $$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$ ...
7
votes
2answers
748 views

(sharp)Garding's inequality and inequality with lower bounds

The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let $$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$ with principal part ...
1
vote
1answer
309 views

sobolev embedding theorem in the smooth metric measure space

we know the sobolev embedding theorem of Saloff-Coste $\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu $ wtih $Ric\ge-(n-1)K$, for ...
7
votes
1answer
421 views

What would the best treatment of Gehring's lemma look like?

In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...
7
votes
1answer
486 views

Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$: $\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...
5
votes
0answers
445 views

Any similar inequality in literature?

I got the following inequality: $B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary. $(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$, ...