This chapter studies elliptic partial differential equations of second order, which generalize Laplace's equation. A linear operator L is elliptic on a domain Ω ⊂ R^n if the matrix [a_ij] is positive definite. A nonlinear operator Q is elliptic if [a_ij(x, u, D u)] is positive definite. These operators are called quasilinear. The function u is typically in C²(Ω) unless stated otherwise. The chapter begins by establishing maximum principles for L and comparison principles for Q, which are crucial for elliptic equations. It also considers equations in two variables arising in geometric problems, such as the minimal surface equation and the prescribed mean curvature equation. The rest of the chapter deals with equations with discontinuous coefficients and nonlinear Dirichlet problems. Section 1 discusses the maximum principle. If L is elliptic, there exists a coordinate transformation that makes the principal part of L the Laplacian. The operator L₁ = L₀ + b_i D_i is considered, and it is shown that if L₁u > 0 in Ω, u cannot attain a relative maximum in the interior of Ω. The minimum eigenvalue λ of the matrix [a_ij] is introduced, and a weak maximum principle is proved for L₁. Theorem 1.1 states that if u is bounded and L₁u > 0 in Ω, then the maximum of u is attained on the boundary of Ω. The analogous result holds for L₁u ≤ 0. The theorem assumes that Ω is bounded, making the boundedness of u redundant in most applications. Theorem 1.2 states that if L₁ is uniformly elliptic on Ω, L₁u ≥ 0, and Ω satisfies the interior sphere condition at a point x₀ on the boundary, then the normal derivative of u at x₀ is positive if u is less than u(x₀) in Ω. The proof involves constructing a function V and analyzing its behavior under the operator L₁.This chapter studies elliptic partial differential equations of second order, which generalize Laplace's equation. A linear operator L is elliptic on a domain Ω ⊂ R^n if the matrix [a_ij] is positive definite. A nonlinear operator Q is elliptic if [a_ij(x, u, D u)] is positive definite. These operators are called quasilinear. The function u is typically in C²(Ω) unless stated otherwise. The chapter begins by establishing maximum principles for L and comparison principles for Q, which are crucial for elliptic equations. It also considers equations in two variables arising in geometric problems, such as the minimal surface equation and the prescribed mean curvature equation. The rest of the chapter deals with equations with discontinuous coefficients and nonlinear Dirichlet problems. Section 1 discusses the maximum principle. If L is elliptic, there exists a coordinate transformation that makes the principal part of L the Laplacian. The operator L₁ = L₀ + b_i D_i is considered, and it is shown that if L₁u > 0 in Ω, u cannot attain a relative maximum in the interior of Ω. The minimum eigenvalue λ of the matrix [a_ij] is introduced, and a weak maximum principle is proved for L₁. Theorem 1.1 states that if u is bounded and L₁u > 0 in Ω, then the maximum of u is attained on the boundary of Ω. The analogous result holds for L₁u ≤ 0. The theorem assumes that Ω is bounded, making the boundedness of u redundant in most applications. Theorem 1.2 states that if L₁ is uniformly elliptic on Ω, L₁u ≥ 0, and Ω satisfies the interior sphere condition at a point x₀ on the boundary, then the normal derivative of u at x₀ is positive if u is less than u(x₀) in Ω. The proof involves constructing a function V and analyzing its behavior under the operator L₁.