 Thank you. Thank you. First of all, I want to apologize. I have changed the title of my talk. So for those of you who come here to hear boundary layers, I'm not going to talk about boundary layers. I'm going to talk about more basic stuff, give kind of an introductory course on quantitative homogenization. My student, Jipin Dugger, who actually knows boundary layer better than myself, will give a talk tomorrow morning in the research seminar here. So I'm going to use both the projector and the blackboard. Hopefully that will work well. So let me just spend the first maybe 10 minutes, give you an overview of this course, also the motivation of studying elliptic homogenization. I'm assuming you know nothing about homogenization from the beginning. So we're going to start with elliptic operator. So all of you are taking a course in PDE. The first operator you're looking at is Laplacian. So we're going to write, actually I prefer to write this as a minus Laplacian. So it becomes a positive operator. You can write this in a divergence form with matrix A just being the identity operator. So this operator arises when you try to model the homogeneous and isotropic material or general media here. So this is the simplest elliptic operator, second order you can have. And getting a little more further, we can looking at a general second order elliptic operators in divergence form. So here I'm writing this in the divergence A gradient. And then I write this out using the convention that the summation index, if repeated index, are sum. So the index i and the j are both summed from 1 through d, d being the dimension here. So here you have a variable matrix, which so the operator can be used to describe properties of inhomogeneous material. Inhomogeneous meaning that when you move from point to point, the property, the characteristic of the material changes. So here you deal with a matrix, a d by d matrix, variable matrix here. However, in modern industry and daily life, we deal with a lot of composite materials. So this is actually we're going to talk about today. So what is a composite material? So you have two very different material mixed together, combined in some proper fashion to create a new material which might have the desired property. So the basic components you have here, there are at least two components. One is referred as a matrix or a binder, which holds things together. Another is a reinforcer. Typically it's a fiber, glass fiber or carbon fiber. And these constitutes combine in some organized manner at a very small scale, a very small scale. So if you try to model this using PDE, here we're going to introduce a matrix with a small parameter, epsilon. Epsilon here represents the inhomogeneous scale. So that is the scale that's these two material mixed at here. And we're going to write this matrix in the form of a of x over epsilon. So epsilon appeared, the small parameter, appeared in the denominator of the variable x here. So before the scale here, I'm going to use a variable y. So the ideal case is the periodic case. You mix two material periodically. And beyond that, we can talk about a quasi periodic, almost periodic, or even a realization of a stationary random field here. So here, solving PDE boundary value problem directly is impossible analytically. So in almost all the cases, you're going to have to discretize the equation and change to numerical problem, possibly solving a matrix equation. So if you want to solve this boundary value problem for equation whose coefficient looks like a of x over epsilon, you have to resolve your decomposition in an epsilon scale, which could be very difficult or costly. So in the 60s, in the 70s, in mechanics and in physics, there is how they have developed so-called homogenization theory, which is that using asymptotic analysis to find some effective or averaged homogenized characteristics. So here, we're going to look at how do we describe this theory rigorously in a math manner here. So that is the motivation here. So going back, let's look at the equation we're going to look at in this course. So again, I'm going to look at a second order divergence form. I put a minus here to make sure this is a positive operator. I denote this operator by L sub epsilon. And writing this out in this form. And we're going to make some basic assumptions throughout the course that the matrix will be real bounded and elliptic uniformly. I'll define what that is. And also, in order to carry out homogenization, you will have to assume some structural conditions. Smoothness condition is not enough to do homogenization. So structural conditions, I mean, in this course, we're going to just look at the problem in the periodic setting. But you can also do homogenization in quasi-periodic, almost-periodic statistical homogenous case. That's a course charged smart we're going to talk about next week. So this might be a provider introductory course to his course here. So what's the homogenization theory? So we look at a bounded value problem in a fixed domain omega. And the solution is subject to some boundary conditions. Here we can do a Dirichlet condition or Neumann boundary condition. We'll talk about that in a moment here. And again, this epsilon represents the inhomogeneous scale, the microscopic scale, which is small relative to the linear size of the domain. For instance, the diameter of the domain. And it turns out, as epsilon goes to 0, the solution of this boundary value problem has a limit. Weekly in H1, H1 is the subred space. And therefore, strongly in L2, because H1 is compactly embedded in L2. Furthermore, the limit function, U0, is a solution of a boundary value problem for a PDE with constant coefficients, L0. So in this periodic setting, we can actually write down precisely what is the coefficient of L0, which I denote by a hat. So roughly speaking, what this means is that when epsilon is small, very small, although the composite material is highly oscillatory in a very small scale, in a large scale, it simply behaves like a homogeneous material. So in practice, for instance, you can imagine that you can think that you can use the limit, U0, as a approximation for the solution, the true solution, U epsilon. And because U0 solves a boundary value problem with constant coefficient, and therefore it could be relatively easily calculated, computed, compared to the PDE L epsilon. So that is the homogenization theory here. So here, I just simply describe the problem for the second order elliptic equations in linear. I mean, the simplest one you can have. Actually, you can carry this process for any kind of PDEs, elliptic, parabolic, hyperbolic, linear, nonlinear. And so what happens here, you look at a general PDE. Doesn't have to be second order. It can be first order or higher order. You assume that this function capital F has some structural conditions with respect to, say, this variable, x here, the last one here. And you change this variable to x over epsilon, you ask the question that as epsilon goes to 0, does the solution has a limit? And if it does, what is the PDE? What is the effective PDE for the limit function, U0? So these are the qualitative questions, qualitative theory. Once this can be done, and then we can move on to quantitative theory. That is, we're going to consider with, we know it's converged in L2, maybe in LP. And what is the sharp convergence rate of U epsilon to U0? And also, we'll be concerned with the regularity and the geometric properties of the solution that are uniform with respect to the parameter epsilon. So in other words, if you have an estimate, you have a constant C, you ask, can the constant be independent of the parameter? In general, that's not possible because all this regularity theory, except that the Georgian edge theory, carries some smoothness conditions. And once you have the smooth conditions, then your constant always blows up as epsilon goes to 0. It's just simply not going to be uniformly with respect to epsilon. But however, here we have some structural conditions, like periodicity or moving beyond that. So that is the problem here. So here, in this course, we're going to simply deal with the second order elliptic operator in that virgin form. And we're also going to assume that this A is real bounded uniformly elliptic. In other words, the matrix is positively definite. The mu, the constant mu here is positive. And we're also assuming that the coefficients are bounded. You can choose that mu so that the upper bound is mu to the negative 1. So that is the ellipticity condition we assume. The structure condition here is that A is one periodic. That means that it's periodic with respect to the integer lattice. If you have a periodicity with respect to other lattice, you can always change the integer lattice by a linear transformation. Further, if we need some small-scale estimate, we may have to put up a smoothness condition. But these two conditions are totally different here. So this is the setup, the basic assumptions. We're going to deal with. So here is the plan. So today, we're going to talk about the qualitative theory. I already gave an introduction. And then we're going to talk about correctors, effective coefficients, effective operator compactness theorem, and also prove the homogenization of bounded value problems, Dirichlet and Neumann problems. So that's the qualitative theory here. In lecture two, tomorrow, we will looking at the problem of convergent rates. So there is something called the flux correctors, has to be used in order to derive the sharp estimate and if you're smoothing. And then we'll look at the error estimate for two-scale expansions in space H1, and also sharp convergence rate in L2 here. In the last two lectures, three and four, we're going to look at some large-scale regularity. So here, I'm going to present two different approach to the problem. The first one, this was a method of Marco Avalinda and Fanghua back in the 80s, late 80s, by compactness. The problem, the approach originally in the study of minimal surface you will hear this morning. But Avalinda and Fanghua introduced this approach to the study of homogenization problems. In the last lecture, we're going to look at a different approach to the same problem, the larger regularity problem, by the method of Scar Armstrong and Charles Smart using convergence rates. So this is also related to what you hear already in the morning, kind of accesses of approximation there. Initially, I planned to give five lectures. So I also write the last section. By the way, the lecture notes are fairly completed. You should have a copy by now. If not, you should get a copy today here. Each one comes in one section. So in section five, I was going to talk about the uniform Cardinal-Zigmalon estimate. But now you can read this yourself. All right, that's the plan here. OK, so first of all, let's look at the concept of correctors. So on the screen, you saw the definition of correctors. But where is it coming from? How does this arise?