June  2010, 28(2): 425-440. doi: 10.3934/dcds.2010.28.425

Optimal estimates for the gradient of harmonic functions in the multidimensional half-space

1. 

Department of Computer Science and Mathematics, Ariel University Center of Samaria, 44837 Ariel, Israel

2. 

Department of Mathematical Sciences, University of Liverpool, M&O Building, Liverpool, L69 3BX, United Kingdom

Received  September 2009 Revised  April 2010 Published  April 2010

A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a multidimensional half-space is obtained under the assumption that function's boundary values belong to $L^p$. This representation is concretized for the cases $p=1, 2,$ and $\infty$.
Citation: Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425
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