Partial Differential Equations: Linear Theory

Prerequisites: Functional Analysis

Sobolev spaces: approximation by smooth functions; weak derivatives; extensions; traces. Hölder spaces. Sobolev inequalities. Kondrachov compactness theorem. Second order elliptic equations: weak solutions; Lax-Milgram theorem; Fredholm alternative; regularity theory; maximum principle. Poincaré inequality. Eigenvalues problems. Linear evolution equations: parabolic equations; hyperbolic equations; semigroup theory. Other topics and applications.

References:
EVANS, L. C. – Partial Differential Equations (Graduate Studies in Mathematics), volume 19. American Mathematical Society, 1998.
MCOWEN, R. – Partial Differential Equations: Methods and Applications, Prentice Hall (2002).

* Standard program. The teacher has the autonomy to make any changes