Partial Differential Equations -generally, elliptic and parabolic pdes (typified by the Laplace and heat equations). More specifically, overdetermined boundary value problems ( too much data puts constraints on the geometry of the region for existence of a solutiion), maximum principles (where the maximun can be attained and consequences thereof such as bounds for the solution and/or its gradient), temporal and spatial decay results (use differential inequalities or maximum principles to determine the exponential decay of solutions), Liouville theorems (conditions for elliptic systems or fourth order equations to have entire solutions which are constant), radial solutions ( explicit solutions of nonlinear higher ordered elliptic equations). These results are for linear and nonlinear equations.
