Journal of Convex Analysis 21 (2014), No. 1, 237--252
Copyright Heldermann Verlag 2014
From the Uniform Approximation of a Solution of the PDE to the L2-Approximation of the Gradient of the Solution
Kakha Shashiashvili
I. Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences, 2 University Street, Tbilisi 0186, Georgia
Malkhaz Shashiashvili
A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 2 University Street, Tbilisi 0186, Georgia
mshashiashvili@yahoo.com
We establish a new energy inequality for the difference of two semiconvex functions in a bounded open convex set D of Rn. This inequality is applied to the L2-approximation problem of the gradient of the unknown solution of the nonlinear elliptic partial differential equation provided that the latter solution is a semiconvex function in D.
Keywords: Semiconvex and semiconcave functions, energy inequality, L-2-approximation, gradient of semiconvex function, convex envelope.
MSC: 26B25, 35J60, 49L25
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