Maximum Principles in Differential Equations

apply Theorem assume boundary conditions boundary value problem bounded domain c₁ characteristic coefficients conclude constant D₁ define denote derivative differential inequality disk dx dt eigenvalue elliptic equations elliptic operators example Find upper function w(x given grad Green's function harmonic function Harnack inequality heat equation Hence holds hyperbolic hyperbolic operator hypothesis initial value problem interior point interval K₁ L+ h)[u Laplace operator Lemma lower bounds Math maximum principle nonnegative maximum obtain positive function proof properties R₁ radius result satisfies the inequalities Section Serrin solution subharmonic function subinterval Suppose T₁ T₂ Theorem 11 tion u₁ u₂ unbounded uniformly elliptic uniqueness theorem upper and lower v₁ vanishes x₁ y₁ z₁ z₁(x z₂ z₂(x zero ὃν г₁ ди ду дх მა