On concavity of solutions of the Dirichlet problem for the equation (-Delta)(1/2)phi=1 in convex planar regions

Kulczycki Tadeusz

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On concavity of solutions of the Dirichlet problem for the equation (-Delta)(1/2)phi=1 in convex planar regions

机译：凸平面区等式（-delta）（1/2）PHI = 1的求解求解的求解

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For a sufficiently regular open bounded set D subset of R-2 let us consider the equation. (-Delta)(1/2)phi(x) = 1 for x is an element of D with the Dirichlet exterior condition phi(x) = 0 for x is an element of D-c. Its solution phi(x) is the expected value of the first exit time from D of the Cauchy process in R-2 starting from x. We prove that if D subset of R-2 is a convex bounded domain then phi is concave on D. To do so we study the Hessian matrix of the harmonic extension of phi. The key idea of the proof is based on a deep result of Hans Lewy concerning the determinants of Hessian matrices of harmonic functions.

机译：对于足够常规的开放式界限集D子集R-2，让我们考虑等式。（-Delta）（1/2）Phi（x）= 1对于x是D dirichlet外部条件phi（x）= 0的d的元素是d-c的元素。它的解决方案PHI（x）是从x开始的R-2中的CAUCHY过程的第一个出口时间的预期值。我们证明，如果R-2的D子集是凸有界域，那么PHI是凹陷的D.这样做我们研究了PHI的谐波延伸的Hessian矩阵。证据的关键思想是基于汉斯·乐的深入结果，关于谐波函数的幽灵矩阵的决定因素。

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来源
《Journal of the European Mathematical Society: JEMS》 |2017年第5期|共60页
作者
Kulczycki Tadeusz;
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作者单位

Wroclaw Univ Sci &

Technol Fac Pure &

Appl Math Wybrzeze Wyspianskiego 27 PL-50370 Wroclaw Poland;

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正文语种 eng
中图分类数学;
关键词
Fractional Laplacian; concavity; Hessian matrix; harmonic function; Cauchy process; first exit time;

机译：分数拉普拉斯;凹陷;黑森州矩阵;谐波函数;Cauchy进程;第一次出口时间;

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专利

1. On concavity of solutions of the Dirichlet problem for the equation (-Delta)(1/2)phi=1 in convex planar regions [J] . Kulczycki Tadeusz Journal of the European Mathematical Society: JEMS . 2017,第5期

机译：凸平面区等式（-delta）（1/2）PHI = 1的求解求解的求解
2. Radially symmetric convex solutions for Dirichlet problems of Monge-Ampere equations [J] . Liang Zaitao, Chu Jifeng Mathematical Methods in the Applied Sciences . 2016,第12期

机译：Monge-Ampere方程Dirichlet问题的径向对称凸解
3. Smoothness properties of bounded solutions of Dirichlet's problem for elliptic equations in regions with corners on the boundary [J] . Canadian mathematical bulletin . 1980,第1980期

机译：边界上有角的区域的椭圆方程的Dirichlet问题的有界解的光滑性
4. Multiple periodic solutions of differential delay equations x'(t)=-+ sum _(i=1)~(n-1) delta_if (x(t-r_i)) with delta_i = +-1 [C] . Jibin Li, Xue-zhong He, Weigao Ge US-Chinese Conference on Differential Equations and Applications held in Hangzhou, June 24-29, 1996 . 1996

机译：差分延迟方程x'（t）=-+ sum _（i = 1）〜（n-1）delta_if（x（t-r_i））的多个周期解，其中delta_i = + -1
5. Normalized p-laplacian evolution: boundary behavior of non-negative solutions of fully nonlinear parabolic equations: Gradient bounds for p-harmonic systems with vanishing neumann (dirichlet) data in a convex domain. [D] . Banerjee, Agnid. 2014

机译：归一化的p-拉普拉斯演化：完全非线性抛物方程的非负解的边界行为：具有在凸域中消失的Neumann（dirichlet）数据的p调和系统的梯度界。
6. Some spectrally isolated convex planar regions [O] . Shahla Marvizi, Richard B. Melrose 1982

机译：一些光谱隔离的凸平面区域
7. Numerical Solution of the Simple Monge-Amp\`ere Equation with Non-convex Dirichlet Data on Non-convex Domains [O] . Jensen, Max 2017

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On concavity of solutions of the Dirichlet problem for the equation (-Delta)(1/2)phi=1 in convex planar regions

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