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April  2002, 8(2): 331-359. doi: 10.3934/dcds.2002.8.331

Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations

Xavier Cabré 1,

1. 

Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Av. Diagonal 647. 08028 Barcelona, Spain

Revised  November 2001 Published  January 2002

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.
Keywords: maximum principles, Nonlinear elliptic PDE, fully nonlinear equations., a priori estimates, symmetry properties.
Mathematics Subject Classification: 35J60, 35B5.
Citation: Xavier Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 331-359. doi: 10.3934/dcds.2002.8.331
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