Optimal pointwise estimate of the gradient of harmonic functions in the unit ball - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:30:48Z http://mathoverflow.net/feeds/question/79642 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79642/optimal-pointwise-estimate-of-the-gradient-of-harmonic-functions-in-the-unit-ball Optimal pointwise estimate of the gradient of harmonic functions in the unit ball Marijan 2011-10-31T16:30:25Z 2011-10-31T22:40:02Z <p>Hello. How to prove that</p> <p>$$\sup_{\gamma\ge 0}\int_0^1 (1+\gamma^2)^{-\frac 12} (1-t^2)^{\frac {n-4}2}( \Phi(\gamma t) +\Phi(-\gamma t))dt$$</p> <p>is achived at $\gamma=0$, where</p> <p>$$\Phi(\zeta)= \int_0^{\frac{\zeta +\sqrt{\zeta^2+1-r^2\left(\frac{n-2}n\right)^2}}{1+r\frac{n-2}n}} \frac{n-r (n-2) + 2 n \zeta \omega -\left(n+r(n-2)\right)\omega^2 } {(1+ \omega^2)^{\frac n2+1} \left( 1+\frac{\left(\frac{1-r}{1+r}\right)^2}{\omega^2}\right)^{\frac n2-1}}d\omega;$$ $n$ is integer $\ge 3,\ \zeta\in \mathbb R,\ 0\le r &lt; 1$.</p> <p>This problem is connected with the very important problem of the optimal poinwise estimate of the gradient of real valued harmonic functions in the unit ball; that is to find the minimal $\mathcal{K}(x)$ for the estimate</p> <p>$$\left|\nabla u(x)\right|\le \mathcal{K}(x),\quad x\in B^n,$$</p> <p>where $u$ is a harmonic function and $\left|u(x)\right|\le 1,\ x\in B^n$. Similar problems were treathed in the papers</p> <p>G. Kresin, V. Maz'ya, Optimal estimates for the gradient of harmonic functions in the multidimensional half-space, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 425--440;</p> <p>G. Kresin, V. Maz'ya, Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball, Journal of Mathematical Sciences 169 (2010).</p> <p>Thank you very much for answers.</p>