User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:35:30Z http://mathoverflow.net/feeds/user/13289 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93263/c2-estimates-for-elliptic-equations C^{2} estimates for elliptic equations rrrq 2012-04-05T23:10:43Z 2012-07-26T19:26:01Z <p>I am curious about the following question:</p> <p>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 $\beta(x)$.</p> <p>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 $|\beta(x)|$?</p> <p>The condition I forgot to put: Suppose $u$ is convex and smooth...</p> http://mathoverflow.net/questions/90176/a-question-about-lp-norm-of-curvature-on-convex-curves a question about Lp norm of curvature on convex curves rrrq 2012-03-04T02:52:08Z 2012-03-04T11:48:54Z <p>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</p> http://mathoverflow.net/questions/70085/is-there-such-a-priori-estimates-for-mean-curvature-type-equation Is there such a priori estimates for mean curvature type equation? rrrq 2011-07-12T04:14:21Z 2011-07-12T04:14:21Z <p>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 $C_{3}$.</p> http://mathoverflow.net/questions/61153/a-question-about-the-number-of-intersections-of-lines-in-r3 A question about the number of intersections of lines in $R^{3}$ rrrq 2011-04-09T17:25:29Z 2011-04-10T02:17:41Z <p>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?</p> <p>The up bound will also be a up bound for Erdos's unit distance problem in $R^{2}$.</p> http://mathoverflow.net/questions/58718/degree-reduction-argument-in-guth-katzsproof-of-erdos-distinct-distance-problem Degree reduction argument in Guth-Katz'sproof of Erdos distinct distance problem in the plane rrrq 2011-03-17T03:20:21Z 2011-03-17T12:21:08Z <p>In the middle of page 9 of <a href="http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf</a>.</p> <p>They said " Now we select a random subset....choosing lines independently with probability $\frac{Q}{100}$. With positive probability....</p> <p>I can not see why there is positive probability...</p> <p>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?..</p> http://mathoverflow.net/questions/56876/any-similar-inequality-in-literature Any similar inequality in literature? rrrq 2011-02-28T05:42:26Z 2011-02-28T20:41:30Z <p>I got the following inequality:</p> <p>$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary. </p> <p>$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$, </p> <p>for all $u$ satisfies $\Delta^{2}u\leq 0$, $-\frac{\partial u}{\partial\gamma}\leq 1$, $\gamma$ is the outer nomal of $\partial B_{4}$.</p> <p>$S$ is the sharp constant which can be determined by letting $u$ solve $\Delta^{2}u=0$, </p> <p>$-\frac{\partial u}{\partial\gamma}=1$, $u=0$ on $\partial B_{4}$. </p> <p>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.</p> <p>Does anyone see similar inequality in the literature? What is the criteria for an inequality to be good? </p> <p>Any comments or refference will be appreciated..</p> http://mathoverflow.net/questions/93263/c2-estimates-for-elliptic-equations Comment by 2012-04-06T21:19:48Z 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 http://mathoverflow.net/questions/93263/c2-estimates-for-elliptic-equations Comment by 2012-04-06T21:12:15Z 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.. http://mathoverflow.net/questions/93263/c2-estimates-for-elliptic-equations Comment by 2012-04-06T20:32:31Z 2012-04-06T20:32:31Z and I am most interested in the 2-d case! http://mathoverflow.net/questions/93263/c2-estimates-for-elliptic-equations Comment by 2012-04-06T20:29:26Z 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 $|D^{2}u|\leq |\frac{f}{\beta}|$? http://mathoverflow.net/questions/93263/c2-estimates-for-elliptic-equations Comment by 2012-04-06T18:49:52Z 2012-04-06T18:49:52Z @Deane, my bad, I forgot to write the condition that $u$ is convex... http://mathoverflow.net/questions/93263/c2-estimates-for-elliptic-equations Comment by 2012-04-06T03:39:47Z 2012-04-06T03:39:47Z and of couse there is a simple estimate that $|D^{2}u|$ bounded by $|\frac{f}{\beta}|$ http://mathoverflow.net/questions/93263/c2-estimates-for-elliptic-equations Comment by 2012-04-06T03:36:24Z 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 $\beta$ http://mathoverflow.net/questions/90176/a-question-about-lp-norm-of-curvature-on-convex-curves/90194#90194 Comment by 2012-03-04T22:04:43Z 2012-03-04T22:04:43Z Thank you, sergei, thats promising! http://mathoverflow.net/questions/90176/a-question-about-lp-norm-of-curvature-on-convex-curves/90194#90194 Comment by 2012-03-04T18:53:43Z 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. http://mathoverflow.net/questions/61153/a-question-about-the-number-of-intersections-of-lines-in-r3/61178#61178 Comment by 2011-04-10T15:26:23Z 2011-04-10T15:26:23Z It looks like &quot;no five lines in a quadric&quot; 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... http://mathoverflow.net/questions/61153/a-question-about-the-number-of-intersections-of-lines-in-r3 Comment by 2011-04-10T04:14:20Z 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 <a href="http://www.cs.tau.ac.il/~michas/pst5.pdf" rel="nofollow">cs.tau.ac.il/~michas/pst5.pdf</a>. http://mathoverflow.net/questions/61153/a-question-about-the-number-of-intersections-of-lines-in-r3/61178#61178 Comment by 2011-04-10T03:46:54Z 2011-04-10T03:46:54Z That example can not satisfy the constions in my question, in fact &quot;for fixed 2 lines, no 3 lines intersect these 2 lines at the same time&quot; 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.... http://mathoverflow.net/questions/58718/degree-reduction-argument-in-guth-katzsproof-of-erdos-distinct-distance-problem/58742#58742 Comment by 2011-03-18T03:25:28Z 2011-03-18T03:25:28Z this can be done by using estimate in <a href="http://en.wikipedia.org/wiki/Binomial_distribution" rel="nofollow">en.wikipedia.org/wiki/Binomial_distribution</a>. 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. http://mathoverflow.net/questions/58718/degree-reduction-argument-in-guth-katzsproof-of-erdos-distinct-distance-problem/58742#58742 Comment by 2011-03-18T03:21:56Z 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 $1-e^{-N/100}$ http://mathoverflow.net/questions/58718/degree-reduction-argument-in-guth-katzsproof-of-erdos-distinct-distance-problem/58742#58742 Comment by 2011-03-18T01:39:25Z 2011-03-18T01:39:25Z in fact one can do estimate directly to see the probability is positive