Summary: 124A PARTIAL DIFFERENTIAL EQUATIONS
DENIS A. LABUTIN
This is the outline for the course. We emphasize the key ideas and the structure, and often omit the
details. The details will be covered on the lectures. They also can be found in the textbook (W. A. Strauss
PDEs, an Introduction, 2nd ed.) and in the extended UCSB 124-course notes by Viktor Grigoryan. The
two sources contain much more than will be covered during the quarter.
(1) A partial differential equation (in two independent variables) is a relation
F(x, y, u, ux, uy, uxx, uxy, . . .) = 0
for the unknown function u(x, y) . All laws in sciences are formulated in the form of (system of)
PDEs. A function u is a solution of the PDE in a region if
F(x, y, u(x, y), ux(x, y), uy(x, y), uxx(x, y), uxy(x, y), . . .) 0
for (x, y) . Thus it is very easy to check if a given function is a solution of the given PDE.
(2) The general solution to the given PDE is the formula for all solutions. For example, we know
from the calculus that the general solution u(x) of
- u = 0
u = Cex