Research seminar PDEs

This first post will be dedicated to introduce a math seminar about some classical and new results in PDEs studied in the Universidad Nacional de Colombia in Medellín, this is a joint work with Alexander Muñoz and Cristian Chica.

The aim of this seminar is study and solve some problems related to PDEs. To achieve this goal it is necessary to provide ourselves with many tools from PDE theory, functional analysis, measure theory, stochastic processes and so forth.

The idea is to study a very specific type of operators called elliptic operators, the most important example of these operators is the laplacian. The laplacian of a function {u:{\mathbb R}^n\rightarrow{\mathbb R}} ,twice differentiable, is given by:

\displaystyle \Delta u=\sum_{i=1}^{n} \partial_{ii}u \ \ \ \ \ (1)

This operator appears in many contexts and it has a lot of important (and also beautiful) properties.

To illustrate some of this properties, consider {f} a holomorphic function in a domain {D\subset {\mathbb C}}, it is easy to see that {\Delta \mathfrak{Re}(f)=0} (using the Cauchy-Riemann equations), this conclusion also holds for the imaginary part of {f}. Conversely, if {u} is a real valued function defined in {D} and {D} is simply connected, then there exist {v:{\mathbb C}\rightarrow {\mathbb R}} such that {f=u+iv} is holomorphic and {\Delta v=0}.

A function {g} that satisfies {\Delta g=0} is called harmonic, in this particular example {u} and {v} are both harmonic functions, moreover a simple linear combination of them (over the complex numbers) satisfies an extra condition of analyticity, when these conditions are fullfilled, {u} and {v} are called harmonic conjugates.

Now let {u:{\mathbb C}\rightarrow{\mathbb R}} be a bounded harmonic function, since {{\mathbb C}} is simply connected there exist {v:{\mathbb C}\rightarrow{\mathbb R}} such that {f:=u+iv} is analytic. Given that {u} is bounded, the analytic function {e^f} is also bounded because {|e^f|=e^u}. Applying Liouville’s theorem we conclude that  {e^f}  is a  constant function therefore {u} is constant too.

We have proved that if a harmonic function in {{\mathbb C} \simeq {\mathbb R}^2} is bounded then it must be constant. Actually, this result holds for any harmonic function defined in {{\mathbb R}^n}.

The amazing properties of the solutions of the homogeneus laplace equation suggest us that the harmonic functions are a good candidate to play the role of the holomorphic functions in {{\mathbb R}^n}.

The solutions of the inhomogeneus laplace equation also have a lot of interesting properties, for instance let {D} be a simply connected domain in {{\mathbb C}}, let {f:D \rightarrow {\mathbb R}} be a continuous function and let {g:\partial D\rightarrow {\mathbb R}} be a continuous function, consider:

\displaystyle \begin{cases} \Delta u=f, \, D,\\ u=g, \,\, \partial D. \end{cases} \ \ \ \ \ (2)

Suppose that there exist two continuous solutions of the problem (2), {u_1} and {u_2}. Let us define {u=u_1-u_2}, cleary {\Delta u=0} and {u=0} on {\partial D}. Since {D} is simply connected there exist {v} such that {f=u+iv} is holomorphic in {D}, by our hypotheses is clear that {f} is pure imaginary on {\partial D}, then the holomorphic functions {e^f} and {e^{-f}} have modulus {1} on the boundary so using the maximum modulus principle {e^{u(z)}\leq 1} and {e^{-u(z)}\leq 1} for every {z\in D}. Therefore since {u} is continuous, {u} has to be identically 0 in {D} implying that the solution of our problem is unique.

These proofs can be performed without these amount of hypotheses about the domain (simply connectedness) or about the regularity of the functions (analicity) and in higher dimensions developing the theory of elliptic operators.

I present these results as a motivation for our future work. Hence, if these results are a small part of our future investigation, you can imagine what kind of amazing stuff we can undestand studying more about the PDE theory.

To achieve these goals I propose the following (sketch of) work scheme:

Properties elliptic operators

For this part the plan is study Alex’s thesis to understand the behaviour of the harmonic functions and the solution of elliptic equations in general. A stardand reference for this part is the classic book of Gilbarg and Trudinger.

Spectrum elliptic operators and linear equations

We can study this part from some works from students of our university and also I can teach this part using the results that I had learned from my previous courses. A good reference to study linear PDEs is the PDE course of the professor Fernando Morales.

Functional or modelling spaces

In this part we are going to study the modelling spaces where the solutions of the PDEs naturally appears. For example we are going to study Sobolev spaces, Sobolev embeddings and so forth in this part. Two important references for this part are the books of PDE of Evans  and the book of Functional Analysis of Brezis

Variational and topological methods in linear and nonlinear analysis

In this part we are going to study how can we solve PDEs using energy functionals, to do that is necessary to learn more about nonlinear analysis and also about the topology of the level sets of these functionals (Morse theory), also sometimes is necessary to distinguish solutions of a PDE that come from different methods, in this part we are going to study how can we characterize or identify these solutions using topological and variational methods. Some good references for this part are the books of Chang about Morse theory and nonlinear analysis methods, also the Kesavan’s book of nonlinear analysis and a book about minmax methods of Rabinowitz.

I am attentive to your comments!!.

Note: Many of the references of the seminar like  a student thesis may not be available online so if somebody feels that needs a copy of that thesis to understand this material I will be glad to share them.

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