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Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations
In these notes we describe the Alexandroff-Bakelman-Pucci estimate
and the Krylov-Safonov Harnack inequality for solutions of
$Lu = f(x)$,
where $L$ is a second order uniformly elliptic operator in nondivergence form
with bounded measurable coefficients. It is the purpose of these notes to
present several applications of these inequalities to the study of nonlinear elliptic
equations.
The first topic is the maximum principle for the operator $L$, and its applications
to the moving planes method and to symmetry properties of positive
solutions of semilinear problems. The second topic is a short introduction to
the regularity theory for solutions of fully nonlinear elliptic equations. We
prove a $C^{1,\alpha}$ estimate for classical solutions, we introduce the notion of viscosity
solution, and we study Jensen’s approximate solutions.