This chapter focuses on elliptic partial differential equations of second order, which generalize Laplace's equation. A linear partial differential operator \( L \) is defined as \( Lu := a_{ij}(\mathbf{x}) D_{ij} u + b_i(\mathbf{x}) D_i u + c(\mathbf{x}) u \), and it is elliptic if the symmetric matrix \( [a_{ij}] \) is positive definite for each \( \mathbf{x} \in \Omega \). Nonlinear operators \( Q \) are also considered, which are quasilinear if the coefficient matrix \( [a_{ij}(\mathbf{x}, u, \mathbf{Du})] \) is positive definite. The chapter begins by establishing maximum principles for \( L \) and associated comparison principles for \( Q \). These principles are crucial in the theory of elliptic equations. The text also discusses equations in two variables, such as the minimal surface equation and the equation of prescribed mean curvature, which arise in geometric problems. The latter part of the chapter deals with equations with discontinuous coefficients and nonlinear Dirichlet problems. A key result is the weak maximum principle for \( L_1 \), where \( L_1 u = L_0 u + b_i D_i u \). If \( L_1 u > 0 \) in \( \Omega \), then \( u \) cannot attain a relative maximum at an interior point of \( \Omega \). The chapter also introduces the concept of uniformly elliptic operators and proves a theorem stating that if \( L_1 \) is uniformly elliptic, \( L_1 u \geq 0 \), and \( u \) is continuous at a boundary point \( \mathbf{x}_0 \) with \( \Omega \) satisfying the interior sphere condition, then the normal derivative of \( u \) at \( \mathbf{x}_0 \) is positive.This chapter focuses on elliptic partial differential equations of second order, which generalize Laplace's equation. A linear partial differential operator \( L \) is defined as \( Lu := a_{ij}(\mathbf{x}) D_{ij} u + b_i(\mathbf{x}) D_i u + c(\mathbf{x}) u \), and it is elliptic if the symmetric matrix \( [a_{ij}] \) is positive definite for each \( \mathbf{x} \in \Omega \). Nonlinear operators \( Q \) are also considered, which are quasilinear if the coefficient matrix \( [a_{ij}(\mathbf{x}, u, \mathbf{Du})] \) is positive definite. The chapter begins by establishing maximum principles for \( L \) and associated comparison principles for \( Q \). These principles are crucial in the theory of elliptic equations. The text also discusses equations in two variables, such as the minimal surface equation and the equation of prescribed mean curvature, which arise in geometric problems. The latter part of the chapter deals with equations with discontinuous coefficients and nonlinear Dirichlet problems. A key result is the weak maximum principle for \( L_1 \), where \( L_1 u = L_0 u + b_i D_i u \). If \( L_1 u > 0 \) in \( \Omega \), then \( u \) cannot attain a relative maximum at an interior point of \( \Omega \). The chapter also introduces the concept of uniformly elliptic operators and proves a theorem stating that if \( L_1 \) is uniformly elliptic, \( L_1 u \geq 0 \), and \( u \) is continuous at a boundary point \( \mathbf{x}_0 \) with \( \Omega \) satisfying the interior sphere condition, then the normal derivative of \( u \) at \( \mathbf{x}_0 \) is positive.