Ukrainian Mathematical Bulletin

Volume 2, Issue 4, 2005 pp. 507-545.

On a Sharp Condition of Limit Summability of Solutions of Nonlinear Elliptic Equations with $L^1$-Right-Hand Sides

Authors: Alexander A. Kovalevsky
Author institution: Institute of Applied Mathematics and Mechanics of NAS of Ukraine, 74 Rosa Luxemburg Str., 83114, Donetsk Ukraine, alexkvl@iamm.ac.donetsk.ua

Summary: The Dirichlet problem is studied for nonlinear second-order elliptic equations in divergence form with right-hand side $f$ belonging to the class $L^1$. We establish a sharp condition for the function $f$ guaranteeing that the solutions of the problem under consideration belong to a certain limit Sobolev space. This condition is stated in terms of a certain behaviour of integrals of the function $|f|$ over the sets $\{|f|\!\!k\}$ for sufficiently large $k$.

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