NEGATIVE NORM SOBOLEV SPACES AND APPLICATIONS
MARCELO M. DISCONZI
Abstract. We review the denition of negative Sobolev norms. As applications, we derive a necessary and sucient condition for existence of weak solutions of linear PDEs, and give Egorovs
cou
VANDERBILT UNIVERSITY
MATH 294 PARTIAL DIFFERENTIAL EQUATIONS.
HW 2.
Unless stated otherwise, is a bounded domain in Rn .
Question 1. Prove uniqueness of solutions to the Dirichlet problem
u = f
u=g
in ,
on ,
where f and g are given functions (you may hav
VANDERBILT UNIVERSITY
MATH 294 PARTIAL DIFFERENTIAL EQUATIONS.
HW 5.
Question 1. Let be a bounded domain and consider the problem
u u = f,
u = 0,
in ,
on .
(1)
1,2
where f L2 () is given. We say that u W0 () is a weak solution of (1) if
(u, v)1 = (f, v)0
VANDERBILT UNIVERSITY
MATH 294 PARTIAL DIFFERENTIAL EQUATIONS.
HW 6.
Question 1. Let Tn = Rn /Zn be the n-dimensional torus. Consider the problem
Lu + f (x, u) = 0, in Tn ,
(1)
where f C (Tn R), and
Lu = aij ij u + bi i u + cu
is an elliptic operator (not
VANDERBILT UNIVERSITY
MATH 294 PARTIAL DIFFERENTIAL EQUATIONS
HW 1
Let be a domain in Rn , i.e., Rn is an open and connected set contained in Rn . For
concreteness you can imagine that is the ball of radius one centered at the origin.
This assignment invo
VANDERBILT UNIVERSITY
MATH 294 PARTIAL DIFFERENTIAL EQUATIONS.
HW 3.
Question 1. Solve uux + uy = 1 with u = 0 when y = x. Something bad happens if we replace the
condition u = 0 by u = 1.
Question 2. Show that xuy yux = x2 + y 2 has no continuous solutio