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C^{2} estimates for elliptic equations

2012-04-05T23:10:43Z

I am curious about the following question:

suppose $u$ is a solution to the uniformly elliptic equation $\sum_{i,j=1}^{n}a_{ij}(x)u_{ij}=f(x)$ in $\Omega$ and $u=0$ on$\partial \Omega$, where $\Omega$ is a bounded convex domain and for simplicity it is close to a unit ball in hausdorff distance, $a_{i,j}$ and $f(x)$ are smooth. $a_{ij}$ has largest eigenvalue $\alpha(x)=1$, and smallest eigenvalue $\beta(x)$.

is it possible to prove a $C^{2}$ estimate: $|D^{2}u|\leq C$ in the compact subdomain $\Omega'$ of $\Omega$, where $C$ depends on $|f|_{L^\infty}$ and the distance between $\partial \Omega$ and $\partial \Omega'$, but doesnt depend on the lower bound of $|\beta(x)|$?

The condition I forgot to put: Suppose $u$ is convex and smooth...

a question about Lp norm of curvature on convex curves

2012-03-04T02:52:08Z

Suppose we have two strictly convex closed curves $C_{1}$ and $C_{2}$, $C_{1}$ contains $C_{2}$, then can we conclude $\int_{C_{1}} \kappa_{1}^{p} ds\geq \int_{C_{2}} \kappa_{2}^{p} ds$, $\kappa_{1}$ and $\kappa_{2}$ are corresponding curvatures of $C_{1}$ and $C_{2}$, $p$ is between 0 and 1

Is there such a priori estimates for mean curvature type equation?

2011-07-12T04:14:21Z

I am dealing with a mean curvature type equation as following: $\displaystyle{\sum_{i,j=1}^{2}}(\delta_{ij}-\frac{u_{i}u_{j}}{1+|Du|^{2}})u_{ij}=(1+|Du|^{2})^{\frac{1}{2}-\frac{1}{2\alpha}}$, where $\alpha>1$ fixed. $u$ is convex and defined on the entire $R^{2}$ suppose when $|x|$ is large, $C_{1}|x|^{\alpha}\leq|Du(x)|\leq C_{2}|x|^{\alpha}$, where $C_{1}$ and $C_{2}$ are fixed positive constants. Then is there such estimate that: when $|x|$ is large $|D^{2}u|\leq C_{3}|x|^{\alpha-1}$ for some fixed constant $C_{3}$.

A question about the number of intersections of lines in $R^{3}$

2011-04-09T17:25:29Z

Suppose I have n lines in $R^{3}$ with the conditions that: no 3 lines in one plane, no 3 lines intersect at one point, for fixed 2 lines, no 3 lines intersect these 2 lines at the same time. what is the up bound of the number of intersections? The up bound $n^{\frac{3}{2}}$ is a simple corollary of Guth-Katz's paper or one can prove it directly by algebraic method. Is it possible to establish the up bound like $n^{\frac{4}{3}}$ or some better one?

The up bound will also be a up bound for Erdos's unit distance problem in $R^{2}$.

Degree reduction argument in Guth-Katz'sproof of Erdos distinct distance problem in the plane

2011-03-17T03:20:21Z

In the middle of page 9 of http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf.

They said " Now we select a random subset....choosing lines independently with probability $\frac{Q}{100}$. With positive probability....

I can not see why there is positive probability...

Could any one explain a bit about what is going on there? I feel they are applying large number law, but I can not see it clearly, for example what is the probability measure space, what is the random variables, how the law is used?..

Any similar inequality in literature?

2011-02-28T05:42:26Z

I got the following inequality:

$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary.

$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,

for all $u$ satisfies $\Delta^{2}u\leq 0$, $-\frac{\partial u}{\partial\gamma}\leq 1$, $\gamma$ is the outer nomal of $\partial B_{4}$.

$S$ is the sharp constant which can be determined by letting $u$ solve $\Delta^{2}u=0$,

$-\frac{\partial u}{\partial\gamma}=1$, $u=0$ on $\partial B_{4}$.

The above inequality is also conformally invariant in the sense that it is invariant if one replace $u$ by the function $u\circ \tau+\frac{1}{4}ln(J_{\tau})$, where $\tau$ is the conformal map from $B_{4}$ to itself, $J_{\tau}$ is its Jacobian.

Does anyone see similar inequality in the literature? What is the criteria for an inequality to be good?

Any comments or refference will be appreciated..

Comment by

2012-04-06T21:19:48Z

One of the reasons I asked the equation is that for the bordline case there is an important example: Ω is a unit disc, $|Du|div(\frac{Du}{|Du|})=1$ has solution $u=\frac{1}{2}|x|2$, and the largest eigenvalue of the coefficients is 1, smallest one is 0

Comment by

2012-04-06T21:12:15Z

@Deane, that is a close example, but the bad thing is that it dosent satisfy the boundary condition, at the point $(1, \frac{1}{2})$ it doesnt equal to 0..

Comment by

2012-04-06T20:32:31Z

and I am most interested in the 2-d case!

Comment by

2012-04-06T20:29:26Z

1-d case, in my question, the condition largest eigenvalue=1 namely a=1... which is trivial. The point is can we get some estimate which is stronger than the estimate $|D^{2}u|\leq |\frac{f}{\beta}|$?

Comment by

2012-04-06T18:49:52Z

@Deane, my bad, I forgot to write the condition that $u$ is convex...

Comment by

2012-04-06T03:39:47Z

and of couse there is a simple estimate that $|D^{2}u|$ bounded by $|\frac{f}{\beta}|$

Comment by

2012-04-06T03:36:24Z

Schauder estimates requires $C$ depends on $C^{\alpha}$ modular of the coefficients and the lower bound of $|\beta(x)|$, which is not enough for my question. the key point is that I need some estimate which is independent of the ratio between $\alpha$ and $\beta$

Comment by

2012-03-04T22:04:43Z

Thank you, sergei, thats promising!

Comment by

2012-03-04T18:53:43Z

I think alvarezpaiya's 1st comments make sense. for Sergei's example, when p=0 the inequality in my question obviously right, and in fact it is a strict inequality. Then notice that the curves are strictly convex, so at least for when p very colose to 0, for sergei's example, the inequality still holds.

Comment by

2011-04-10T15:26:23Z

It looks like "no five lines in a quadric" but not exactly same. n lines in a (singly) ruled surface of degree $n^{\frac{1}{2}}$ is a situation appeared if one try to prove the up bound $n^{\frac{3}{2}}$, but still the full strength of that condition will not be used...

Comment by

2011-04-10T04:14:20Z

The best summary of Guth-Katz paper I can think is the link in JSE's answer below, for unit distance problem, one can find reference in the reference of cs.tau.ac.il/~michas/pst5.pdf.

Comment by

2011-04-10T03:46:54Z

That example can not satisfy the constions in my question, in fact "for fixed 2 lines, no 3 lines intersect these 2 lines at the same time" is the most important condition in the question, it is not easy for me to think some example with many intersections but satify that condition....

Comment by

2011-03-18T03:25:28Z

this can be done by using estimate in en.wikipedia.org/wiki/Binomial_distribution. then the probability that every line in $L_{2}$ intersects with at least $\frac{N}{2}$ lines in $L_{3}$ is bigger than $(1-e^{-N/100})^{N^{2}}$, which goes to 1 when $N$ very large.

Comment by

2011-03-18T03:21:56Z

one can do this as following Claim: suppose both $L_{1}$ and $L_{2}$ has $O(N^{2})$ lines, each line in $L_{1}$ intersects with at least almost $QN$ lines in $L_{2}$, now choosing line in $L_{2}$ independently with probability $\frac{1}{Q}$, the resulting subset of $L_{2}$ is denoted by $L_{3}$ then one has similar statement in 2: with probability bigger than 1/2 $L_{3}$ contains $\frac{O(N^{2})}{Q}$ lines for 3, the probability that a line in $L_{1}$ intersects more than $\frac{N}{2}$ lines in $L_{3}$ is bigger than $1-e^{-N/100}$

Comment by

2011-03-18T01:39:25Z

in fact one can do estimate directly to see the probability is positive