All Questions
15 questions
10
votes
1
answer
1k
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 ...
- 2,041
5
votes
0
answers
277
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
$$\int_{B_r} (Ae(u)...
- 109
5
votes
0
answers
488
views
Any similar inequality in literature?
I got the following inequality:
$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary.
$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,
...
- 215
3
votes
1
answer
334
views
Improved Hardy Inequality when orthogonal to radial functions?
A formulation of Hardy's Inequality on $\mathbb{R}^d$ states that when $0\le s<d$ and $1< p<\tfrac{d}{s}$ that
$$ \|\frac{u}{|x|^s}\|_{L^p(\mathbb{R}^d)}\lesssim_{s,p,d} \||\nabla|^s u\|_{L^...
- 141
2
votes
1
answer
286
views
An inequality for abstract Cauchy problem
Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...
- 97
2
votes
1
answer
267
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 ...
- 83
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
0
answers
73
views
Question about Gidas-Ni-Nirenberg result
Background:
So I know that the Euler Lagrange equation associated with the Sobolev inequality takes the following form,
$$-\Delta u = u^p$$
where $p=2^*-1$ and here we assume that $u>0$ on $\mathbb{...
- 537
1
vote
1
answer
529
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 ...
- 199
1
vote
1
answer
158
views
How do I integrate this inequality that appears in a paper of Rabinowitz?
Sorry if this is too easy for MO, but I found it in a research paper, so I thought that it was worth posting here.
I was reading a paper by Rabinowitz(this one to be more precise) and I came across ...
- 113
1
vote
1
answer
146
views
Extremizers of the Sobolev inequality
Background:
I am reading the paper: Best constant in Sobolev inequality by Talenti (see here) and I am trying to understand the following step.
On p. 365, the author is arguing that the solutions to ...
- 537
1
vote
2
answers
345
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
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...
- 1,649
1
vote
1
answer
113
views
Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$
Let $\rho_1, \rho_2 \in L^1(\Omega;\mathbb R_+)$ such that $\int \rho_i|\ln \rho_i| < \infty$. Is it true that there exists a constant $C>0$ such that
\begin{align*}
\int_\Omega \left(\rho_{1} \...
user140746
1
vote
0
answers
84
views
How to show that $\Delta W \leq −2(n − 4)V$?
I am reading a preprint and trying to understand the proof of Lemma 3.5. On Pg. 19 above eqn (3.49) the authors claim that $\Delta W \leq −2(n − 4)V$ where the functions $W$ and $V$ are defined below,
...
- 537
0
votes
1
answer
125
views
Bounding integral expression with Sobolev norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844