THE HEAT EQUATION
CRISTIAN E. GUTIERREZ
NOVEMBER 14, 2013
Contents
1. Deduction of the equation
2. Fundamental solution
3. Mean value property
4. The strong maximum principle for the heat equation
5. Uniqueness
6. Tychono counterexample of uniqueness
Refe
PDES I, Math 8141
Prof. Guti rrez
e
Homework 1, First order pdes (Due on 9/10/2013)
F
(t, x) h(t), and x(t) and y(t) are two solutions dened in a
x
neighborhood of t = 0 of x = F(t, x). Prove that
1. Suppose
t
h(s) ds .
x(t) y(t) x(0) y(0) exp
0
2. Le
PDES I, Math 8141
Prof. Guti rrez
e
Homework 3, Divergence theorem (Due on 9/26/2013)
1. Let F(x, y, z) = (x, 1, 0). Verify the divergence theorem in the parallelepiped
[a, b] [c, d] [e, f ] and in the ball centered at (0, 0, 0) with radius R.
2. Calculat
PDES I, Math 8141
Prof. Guti rrez
e
Optional Homework, Physical applications
1. Find the ows of the following vector elds:
(a) F(x, y, z) = (1, 3x2 , 0).
(b) F(x, y, z) = (log( y + z), 1, 1).
(c) F(x, y, z) = (x2 , y2 , z(x + y).
(d) F(x, y, z) = ( y z, z
PDES I, Math 8141
Prof. Guti rrez
e
Homework 4, Gravitational Field (Due on 10/03/2013)
1. Let be a bounded domain in R3 and be bounded and integrable over .
Let
yx
F(x) =
( y)
dy.
 y x3
F(x) is the gravitational force (up to a mutiplicative constant) f
PDES I, Math 8141
Prof. Guti rrez
e
Homework 5, Harmonic Functions (Due on 10/17/13)
1. Consider the polar coordinates in the plane x = r cos , y = r sin . Prove that the
Laplacian in polar coordinates has the form
1 u
1 2 u
u =
r
+ 2 2.
r r r
r
2. Let u
PDES I, Math 8141
Prof. Guti rrez
e
Homework 7, Heat equation (Due on 12/05/13)
1. Let u be a smooth solution to ut u = 0 in Rn (0, ). Prove that for all R
the function u (x, t) = u(x, 2 t) also solves the heat equation. Prove also that v(x, t) =
x Dx u(x
PDES I, Math 8141
Prof. Guti rrez
e
Homework 6, Harmonic Functions (Due on 11/05/13)
1. The function u C() is a viscosity subsolution (supersolution) of the Laplace equation
in if whenever C2 () and x0 are such that u has a local maximum
(minimum) at x0 ,
MATH 8141, PDES
FALL 2010, C. E. GUTIERREZ
PRELIMINARIES
NOTATION
Given 1 , ., n nonnegative integers we set = (1 , ., n ), ! = 1 !.n !,  =

1 + . + n , D =
. Given a complexvalued function f (x), the support of
1
x1 .xn
n
f is the closure of the set
FIRST ORDER PDES
CRISTIAN E. GUTIERREZ
SEPTEMBER 5, 2013
Contents
1. Systems of 1st order ordinary dierential equations
1.1. Existence of solutions
1.2. Uniqueness
1.3. Dierentiability of solutions with respect to a parameter
2. Quasilinear pdes
2.1. Ste
THE GREEN FUNCTION
CRISTIAN E. GUTIERREZ
NOVEMBER 5, 2013
Contents
1. Third Greens formula
2. The Green function
2.1. Estimates of the Green function near the pole
2.2. Symmetry of the Green function
2.3. The Green function for the ball
2.4. Application 1
THE DIVERGENCE THEOREM
CRISTIAN E. GUTIERREZ
SEPTEMBER 24, 2013
Contents
1.
2.
3.
4.
Interpretation of the divergence
The divergence theorem
Coulombs law and Newtons gravitational law
Gauss law
1
3
5
6
1. Interpretation of the divergence
Let F(, t) : Rn R
SOLUTION OF POISSONS EQUATION
CRISTIAN E. GUTIERREZ
OCTOBER 5, 2013
Contents
1. Dierentiation under the integral sign
2. The Newtonian potential is C1
3. The Newtonian potential from the 3rd Green formula
4. Second derivatives
5. Solution in all space
6.
SOLUTION OF THE DIRICHLET PROBLEM
WITH THE METHOD OF BALAYAGE
CRISTIAN E. GUTIERREZ
NOVEMBER 5, 2013
Contents
1.
2.
3.
4.
5.
Maximum principle for subharmonic functions
Properties of subharmonic functions
Harmonic lifting (balayage)
Estimates of derivativ
PDES I, Math 8141
Prof. Guti rrez
e
Homework 2, First order pdes II (Due on 9/19/2013)
1. Solve the following Cauchy problems
2
1. ux = u y , u(0, y) = y2 /2; ANS: u = y2 /(2(1 2x);
2. xux + yu y + ux u y = u, u(s, 0) = s2 ; ANS: u = (4x y)2 /16;
3. x (ux