MA 3065 Study Guide - Final Guide: Maximum Principle, Maxima And Minima, Viscosity Solution

• School: Minneapolis College of Art and Design
• Department: Filmmaking-Sound
• Course: MA 3065
• Professor: p

Document Summary

What happens if max f < 0: use the maximum principle to show that if. U(x) = f (x) for |x| < 1 and u(x) = 0 for |x| = 1 then u(x) max f. 2n (1 |x|2) for all |x| 1. [hint: note that v = 2cn for v(x) = c(1 |x|2). Choose c so that v u and use the maximum (or comparison) principle. : assume u : r r is a continuous increasing function, but not necessarily di erentiable. Show that u is a viscosity supersolution of u (x) = 0. [hint: you need to show that if u has a local minimum at x0 then (x0) 0, where is a smooth test function. Since u has a local minimum at x0 there exists r > 0 such that u(x) (x) u(x0) (x0) for |x x0| r. Choose x = x0 h for h > 0 and send h 0 to show that (x0) 0.