Elliptic partial differential equations
The goal of this project is to present a detailed proof of existence and regularity for solutions of linear elliptic partial differential equations.
Regularity forms the essential ingredient of the proof of existence of solutions for both elliptic and parabolic partial differential equations.
Solutions model equilibrium states (elliptic PDE), which form the stationary (time-independent) limit of diffusion-like processes (parabolic PDE).
The project will start with a review of harmonic functions. The Mean Value Theorem will prove their maximum principle, Harnack inequality, Harnack principle, and
interior estimates for derivatives. Then introduce the Green Representation Formula,
the Green function of a domain, the Poisson Integral Formula in the case of the ball, and the Mean Value Property on balls for continuous funcions. The latter characterizes
harmonic functions on the ambient domain.
In the next part the students prove the Weak Maximum Principle, the Hopf Lemma, and the Strong Maximum Principle. These give uniqueness of and bounds for the values
of solutions of the Dirichlet problem for the Poisson equation.
The students prove the Theorem of Perron showing existence of a twice differentiable solution of the Dirichlet problem for the Laplace equation on regular domains.
The Perron construction uses Mean Value Property, Harnack Principle and Maximum Principle.
In the next step, the students move to the Poisson equation and prove interior differentiability estimates, before proving boundary regularity
on the ball with zero boundary data. The boundary regularity requires the Schwarz reflection principle for harmonic functions and inversion of the sphere to a plane.
This leads to a proof of the existence of a unique regular solution of the Dirichlet problem for the
Poisson equation on the ball. The argument uses the Perron method and the continuity method for existence, the above for regularity, and finally the maximum principle for uniqueness.
The project will then, time permitting, discuss Schauder theory, that is, Hoelder estimates for second derivatives of solutions to linear equations. This is the following
question. If we have an equation of the form Lu = f, where L is a linear second order elliptic operator, then when does C^{alpha} regularity of f
imply C^{2,alpha} regularity of u? Next you can move to the L^{p} theory for linear equations;
the analogous question here is when does L^{p} regularity of f imply Sobolev space W^{2,p} regularity of u?
The interior and boundary Schauder estimates then allow to
prove existence of a regular solution for the Dirichlet problem Lu = f in domain A, u = g on the smooth boundary of A. The proof is analogous to the proof for the Poisson
equation, namely by Perron method, regularity, and the continuity method.
Resources for this project
Wikipedia article on elliptic PDE
PDE book by J Jost
Elliptic PDE of second order / David Gilbarg, Neil S. Trudinger.
Notes from a PDE course at MIT
Wikipedia article on Perron method
Wikipedia article on Schauder estimates
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