: Thus ( u(x,t) = \inf_y \left g(y) + \frac^22t \right ). This is the Moreau envelope of ( g ). For convex ( g ), the infimum is attained at a unique point. For example, if ( g(y) = y^2/2 ), then solving the Euler–Lagrange gives ( y = x/(1+t) ) and ( u(x,t) = \fracx^22(1+t) ).
Maximize over ( x, y ). Using the PDE and convexity, one shows the maximum is nonpositive. Letting ( \varepsilon \to 0 ) yields ( u \le v ). Symmetry gives equality.
Before tackling the exercises, internalize these pillars:
with compact support, integrating by parts to "move" the derivative off the discontinuous solution.
By mastering the concepts and techniques in Evans' PDE solutions Chapter 3, students and researchers can gain a deeper understanding of Sobolev spaces and their applications to partial differential equations.
from the Chapter 3 exercises, or would you like to dive deeper into the Hopf-Lax formula
In Chapter 3 of Evans' PDE textbook, the author discusses various properties of Sobolev spaces. Some of the key properties include: