User shaoming guo - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:59:55Z http://mathoverflow.net/feeds/user/5896 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89044/integrability-of-ground-state-solution-for-elliptic-equation Integrability of ground state solution for elliptic equation Shaoming Guo 2012-02-20T20:30:48Z 2012-02-20T20:30:48Z <p>For the solution of semi-linear elliptic equation, for example I'm considering the 2D cubic nonlinear Schroedinger equation, the correspongding elliptic equation is $\Delta u+u^3=u$, with $u>0$. By variational method, we can prove the existence of the solution(the minimizer of some functional), and moreover the uniqueness is also proved by MK Kwong, "Uniqueness of positive solutions of ...". </p> <p>Now I want to ask if there's any result about the integrability of the solution, like if the solution lies in $H^s$ or even Schwartz? because what we know about this kind of solution is radially decay with respect to some given point. </p> <p>For the 1D case, the solution can be written explicitely, $u(x)=\frac{2^{-1/2}}{cosh(x)}$, which is Schwartz function, maybe this could make my question more reasonable?</p> http://mathoverflow.net/questions/72966/are-there-soliton-solutions-for-euler-and-navier-stokes-equation are there soliton solutions for Euler and Navier-Stokes Equation Shaoming Guo 2011-08-16T08:20:40Z 2011-08-16T10:04:06Z <p>I'm now reading papers about the the well-posedness of Euler and Navier-Stokes Equation, so I wonder if we have soliton solutions for this two equations just like for KdV equation. I'm interested in this because if soliton solutions exist, then we can try larger space for initial data, which includes the soliton, to work in for the well-posedness, and also we can consider the stability for the soliton solutions.</p> <p>I searched in google, but haven't got any positive result.</p> http://mathoverflow.net/questions/72340/does-this-inequality-hold does this inequality hold? Shaoming Guo 2011-08-08T09:35:32Z 2011-08-08T18:05:53Z <p>this turned out to be wrong, see the comment of fedja below for the counter example.</p> <p>$\sum_{m>0,n>0,k>0,m+n>k}\frac{1}{(m+n-k)^{1/2}\dot (m+n)^{1/2}}a_m a_n b_k$$\le$ C$\|A\|_{l^2}^2$*$\|B\|_{l^2}$.</p> <p>where A is the sery ${a_m},m=1,2,3...$, B is the sery $b_k, k=1,2,3...$. C is a constant which is independent of A and B. $\| \|_{l^2}$ denotes the $l^2$ Hilbert norm of the sequence.</p> http://mathoverflow.net/questions/65935/what-is-the-lp-norm-of-the-uncentered-hardy-littlewood-maximal-function/65937#65937 Answer by Shaoming Guo for What is the $L^p$-norm of the (uncentered) Hardy-Littlewood maximal function? Shaoming Guo 2011-05-25T09:18:47Z 2011-05-25T09:18:47Z <p>I just searched in google, "best constants for uncentered maximal functions", by Grafakos and Smith. "the best constants for the centered H-L maximal inequality", by AD Melas.(1-D, weak type (1,1)). maybe you can find more results following this two.</p> http://mathoverflow.net/questions/60427/convergence-of-fourier-series-in-l-infty-norm/60432#60432 Answer by Shaoming Guo for Convergence of Fourier series in L^{\infty}-norm Shaoming Guo 2011-04-03T12:58:21Z 2011-04-03T12:58:21Z <p>I think you can find the answer in P191 "Classical Fourier Analysis",second edition, Loukas Grafakos</p> http://mathoverflow.net/questions/43059/relation-between-inclusion-and-embedding relation between inclusion and embedding Shaoming Guo 2010-10-21T15:36:50Z 2011-03-31T23:35:51Z <p>Assume that $X$ and $Y$ are two Banach spaces, now we have that $X$ is included in $Y$, in the sense that $\forall a\in X$, we have $a\in Y$. Then can we get that $X$ is embedded in $Y$, namely, $\forall b\in Y, \Vert b \Vert_Y \le C\cdot \Vert b \Vert_X$?</p> <p>I think there is no problem for the statement of this question by Nate Eldredge.</p> http://mathoverflow.net/questions/60234/characterization-of-hormander-multipliers characterization of Hörmander multipliers Shaoming Guo 2011-03-31T21:05:07Z 2011-03-31T21:10:15Z <p>Denote $S$ as the space of Schwartz functions, for $v\in S'$, the space of tempered distributions, define an operator $T_v:f\in S \to f*v$. Then space of Hormander Multipliers $M^{p,q}$ can be defined as $\{ v\in S': \|T_v\|_{L^p\to L^q}&lt;\infty \}$.</p> <p>Do we know some results on the characterization of $M^{p,q}$, when $p &lt; q$?</p> http://mathoverflow.net/questions/50418/the-upper-semicontinuous-envelope-of-a-lower-semicontinuous-function/50460#50460 Answer by Shaoming Guo for The upper semicontinuous envelope of a lower semicontinuous function Shaoming Guo 2010-12-27T00:20:15Z 2010-12-27T00:20:15Z <p>I want to figure out, in what kind of sense is the upper semi-continuous envelope discontinuous. But that we ask for the function to be lower semi-continuous doesn't play an important role. And the discontinuous points can be dense.</p> http://mathoverflow.net/questions/44332/is-there-an-english-translation-of-the-book-by-guy-barles is there an English translation of the book by Guy Barles? Shaoming Guo 2010-10-31T14:50:02Z 2010-10-31T15:18:18Z <p>is there an English translation of the book by Guy Barles, "Solutions de viscosite des equations de Hamiltion-Jacobi"? Springer-Verlag</p> http://mathoverflow.net/questions/89749/a-slightly-subcritical-elliptic-equation-on-the-ball-blow-up-behavior-near-zero Comment by Shaoming Guo Shaoming Guo 2012-02-28T23:10:13Z 2012-02-28T23:10:13Z Do you want some a priori estimate, and then pass to the limit to prove the existence for the weak solution for the critical power $p=\frac{N+2}{N-2}$? For the variational approach of Yamabe problem or prescribing scalar curvature problem, the strategy is to pass from the subcritical case to the critical case, and to claim that blowing up can not happen in some sense. http://mathoverflow.net/questions/85127/hardy-littlewood-maximal-function Comment by Shaoming Guo Shaoming Guo 2012-01-15T13:19:26Z 2012-01-15T13:19:26Z Is the idea for the proof of interpolation allowed(splitting into two functions)? Or you expect a totally new way of proving? http://mathoverflow.net/questions/74031/what-is-the-idea-of-the-studying-about-interpolation-theory-for-lp-space Comment by Shaoming Guo Shaoming Guo 2011-08-30T17:12:10Z 2011-08-30T17:12:10Z referring to the textbook, I think the one by Lucas Grafakos &quot;Classical Fourier Analysis&quot; is a good choice, and you can see the application in the following chapter, for example the boundedness of Hilbert Operator and other kind of singular integral operator. http://mathoverflow.net/questions/74031/what-is-the-idea-of-the-studying-about-interpolation-theory-for-lp-space Comment by Shaoming Guo Shaoming Guo 2011-08-30T17:05:25Z 2011-08-30T17:05:25Z I think the idea is that, for a function in $L^{p}$, it can be divided into two parts, one is the part larger than 1, which is in $L^{p_1}$, $p_1&lt;p$, and the part less than 1, which is in $L^{p_2}$, $p_2&gt;p$, so we can apply the result of boundedness in both $L^{p_1}$ and $L^{p_2}$, to derive the boundedness in $L^p$. http://mathoverflow.net/questions/73444/is-this-set-of-functions-compact Comment by Shaoming Guo Shaoming Guo 2011-08-22T23:33:57Z 2011-08-22T23:33:57Z is the property that $\varphi$ satisfies here the same with the one in your post, just average of the unit circle? http://mathoverflow.net/questions/73010/sobolev-type-inequalities-involving-affine-metric Comment by Shaoming Guo Shaoming Guo 2011-08-16T19:23:27Z 2011-08-16T19:23:27Z maybe Poincare's inequality can give you some information http://mathoverflow.net/questions/72431/functionals-continuous-with-respect-to-weak-convergence Comment by Shaoming Guo Shaoming Guo 2011-08-09T09:05:35Z 2011-08-09T09:05:35Z I just remembered when I read the book of A. Braides 'Gamma-convergence for beginners', I noticed several remarks like your question, maybe you can check page 55 for some information and the reference there. http://mathoverflow.net/questions/72340/does-this-inequality-hold Comment by Shaoming Guo Shaoming Guo 2011-08-08T17:58:32Z 2011-08-08T17:58:32Z @Willie Wong: fedja is right, and thanks a lot for your comment, I don't know how to close this question. http://mathoverflow.net/questions/72340/does-this-inequality-hold Comment by Shaoming Guo Shaoming Guo 2011-08-08T14:27:47Z 2011-08-08T14:27:47Z @fedja: I tried the case for taking A and B to be harmonic series, it turned out that the above inequality holds. http://mathoverflow.net/questions/72340/does-this-inequality-hold Comment by Shaoming Guo Shaoming Guo 2011-08-08T14:17:29Z 2011-08-08T14:17:29Z @jc I just meet this problem in summing up the dyadic decomposition in PDEs and need this inequality in one step, so I also don't have good motivation. http://mathoverflow.net/questions/72340/does-this-inequality-hold Comment by Shaoming Guo Shaoming Guo 2011-08-08T12:23:01Z 2011-08-08T12:23:01Z sorry, can you be more specific? because I don't quite see that, what I'm thinking now is some extension of Hilbert's double series theorem. http://mathoverflow.net/questions/60427/convergence-of-fourier-series-in-l-infty-norm/60432#60432 Comment by Shaoming Guo Shaoming Guo 2011-04-03T13:07:42Z 2011-04-03T13:07:42Z but as the function and the partial sum of the Fourier series are all continuous functions, if we have $L^{\infty}$ convergence of partial sum, then we can get contradiction? http://mathoverflow.net/questions/60234/characterization-of-hormander-multipliers Comment by Shaoming Guo Shaoming Guo 2011-04-01T11:12:17Z 2011-04-01T11:12:17Z @ Zen Harper: Your comment is very helpful, maybe convolution is always some kind of weird, and also not that &quot;precise&quot;, so it's not easy to get the optimal estimates. http://mathoverflow.net/questions/60234/characterization-of-hormander-multipliers Comment by Shaoming Guo Shaoming Guo 2011-04-01T08:37:19Z 2011-04-01T08:37:19Z @Denis Serre: And also the Lorentz space $L^{r,\infty}$ by the weak type Young's inequality. I'm wondering if Lorentz space is optimal for Young's inequality. http://mathoverflow.net/questions/44332/is-there-an-english-translation-of-the-book-by-guy-barles/44336#44336 Comment by Shaoming Guo Shaoming Guo 2010-10-31T15:32:09Z 2010-10-31T15:32:09Z OK, thank you very much!