Harnack inequalities are a set of important results in analysis that provide bounds on the values of positive solutions to certain partial differential equations. They establish that if a function is positive and satisfies a specific type of elliptic or parabolic PDE, then its values cannot vary too wildly over small regions, which implies uniform continuity and boundedness. This property is crucial for understanding the behavior of solutions to differential equations, especially in the context of regularity theory and heat equations.
congrats on reading the definition of Harnack Inequalities. now let's actually learn it.