Buffon’s needle estimates for rational product Cantor setsAmerican Journal of Mathematics - Tập 136 Số 2 - Trang 357-391 - 2014
Matthew Bond, Izabella Łaba, Alexander Volberg
Let $S_\infty=A_\infty\times B_\infty$ be a self-similar product Cantor set in the complex plane, defined via $S_\infty=\bigcup_{j=1}^L T_j(S_\infty)$, where $T_j:\Bbb{C}\to\Bbb{C}$ have the form $T_j(z)={1\over L}z+z_j$ and $\{z_1,\ldots,z_L\}=A+iB$ for some $A,B\subset\Bbb{R}$ with $|A|,|B|>1$ and $|A||B|=L$. Let $S_N$ be the $L^{-N}$-neighborhood of $S_\infty$, or equi...... hiện toàn bộ
Gradient estimates for variable coefficient parabolic equations and singular perturbation problemsAmerican Journal of Mathematics - Tập 120 Số 2 - Trang 391-439 - 1998
Luis Caffarelli, Carlos E. Kenig
In this article we prove, via monotonicity formulas, interior and boundary gradient estimates for solutions to second order parabolic equations, in divergence form, with Dini top order coefficients. We then prove uniform Lipschitz estimates for solutions of singular perturbation problems, using the previous results, and two phase monotonicity formulas.