Maximum Principles in Differential Equations
Maximum Principles are central to the theory and applications of second-order partial differential equations and systems. This self-contained text establishes the fundamental principles and provides a variety of applications.
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admissible domain apply Theorem assume boundary conditions boundary point boundary value problem bounded domain choose coefficients conclude continuously differentiable coordinates D U 3D defined denote differential inequality dimensions disk eigenvalue elliptic equations elliptic operators establish example Find upper formula function w(x functions which satisfy grad Green's Green’s function harmonic function Harnack inequality heat equation Hence holds hyperbolic hyperbolic operator hypothesis initial value problem interior point interval a,b Laplace operator Lemma Let u(x lower bounds maxi maximum principle mean value theorem nonlinear nonnegative maximum nonpositive obtain occur one-dimensional parabolic equations Phragmen-Lindelöf positive constant positive function proof of Theorem properties quantity radius satisfies inequalities satisfies the boundary satisfies the inequalities Section Serrin solution subharmonic function subinterval surface Theorem 11 tion unbounded uniformly elliptic uniformly parabolic uniqueness theorem upper and lower vanishes vector x-axis zero zi(x