Zeitschrift für Analysis und ihre Anwendungen


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Volume 28, Issue 2, 2009, pp. 129–164
DOI: 10.4171/ZAA/1377

W2,p- and W1,p-Estimates at the Boundary for Solutions of Fully Nonlinear, Uniformly Elliptic Equations

Niki Winter[1]

(1) Institut für Mathematik, RWTH Aachen, Templergraben 55, 52056, AACHEN, GERMANY

In this paper we extend Caffarelli's result on interior $W^{2,p}$-estimates for viscosity solutions of uniformly elliptic equations and prove $W^{2,p}$-estimates at a flat boundary. Moreover we extend a result of A.~\'Swiech and prove $W^{1,p}$-estimates at the boundary. Thereafter we combine these results and prove global $W^{2,p}$-estimates for equations with dependence on $Du$ and $u$. Finally, we show that the previous estimates lead to an existence result for $W^{2,p}$-strong solutions.}

Keywords: Viscosity solutions, uniformly elliptic equations, W2,p-regularity, W1,p-regularity

Winter N. W2,p- and W1,p-Estimates at the Boundary for Solutions of Fully Nonlinear, Uniformly Elliptic Equations. Z. Anal. Anwend. 28 (2009), 129-164. doi: 10.4171/ZAA/1377