 with the graduate school lectures, and it's my great pleasure to introduce Zhenya Evgenia Malynnikova from Norwegian University of Science and Technology, who will be talking about the frequency function of solutions to second-order elliptic degrees and zero sets of Laplace-Betteramina items. Thank you. Thank you, Svetlana, for introduction, and thank you, everyone, for coming. It's a great pleasure, and I wanna be here, so thank you for the agonizes for this opportunity. I will use the bell-upboards most of the time. I'll just have some pictures to show you in the middle of the lecture on the screen, but I can start with a very short overview of what you would expect during these four lectures. We will start with a very short note about elliptic degrees and then spend most of the first lecture on agon functions of Laplace-Betteramina operator. If you're working with those, I will not tell you anything new today. Tomorrow, the plan is to look at the frequency function, define it, and see how it helps to understand the properties of growth of solutions of elliptic degrees and also zero sets. And then the last two lectures, I want to prove something, so I will show you how to use frequency function and some iteration schemes to prove a quantitative unique continuation result for elliptic degrees. I will always refer to agon functions and tell you what is the story with agon functions behind. So let me start slowly what kind of PDs we are looking at. For me, elliptic will be always second order, divergence form, no law at the terms, nice operator by uniformly elliptic. I mean that we have a matrix of coefficients with usual condition that this is an elliptic operator, the matrix is always symmetric and uniformly elliptic will mean that I have the same lambda for the whole domain where I'm considering the equation. Well, most of the time, assume that our coefficients are good enough, Lipschitz, so we have an uniform constant age, sorry, uniform constant C such this is true for all points in our domain and this is the regularity that we will assume. It's well known that for such elliptic operators, we have weak unique continuation property. If you have a solution in a connected domain and the solution is zero on an open set, then it's zero in the whole domain. Something that it's very easy to believe when you have the usual or pass operator or you have a elliptic operator with real analytic coefficients, but this is something that is about ellipticity not analyticity of the same say we have the same result for elliptic situation. Moreover, there is strong in incontonation principle tells you that if you have a solution to this equation and one point in this connected domain where solution vanishes of infinite order to formulate it in a right way when we don't have smoothness of solution, we say that the integral over a ball of radius r is bounded by r to the power n for any n or small enough. So we have a vanishing of infinite order in this integral sense, then the function is zero. So this is stronger than the weak in incontonation and strong in incontonation implies that solutions of elliptic equations are not allowed to vanish on sets of positive measure. If you have set of positive measure where solution vanishes, take any density point of this measure and you'll find vanishing of infinite order in this density point. So one of the aims of this for lectures is to prove a quantitative version of this result of non-vanishing on sets of positive measure. It will come back to it in the last two lectures and another aim of the lectures is to connect to what we are going to learn to eigenfunctions so we'll spend most of the time today talking about eigenfunctions. Assume that we have a remaining manifold with nice metric, should be smooth enough to apply our theory and what we need is those two assumptions on the elliptic equation that we get. We'll have a plausible trim operator on this manifold with just on functions usual divergence of the gradient. You can write it down in local coordinates, I won't do it by the way there are lecture notes where most of the things are written down explicitly and also there is an appendix in lecture notes about facts on elliptic PDs that could be useful if you don't remember them. So we are looking at functions that satisfy the eigenvalue equation. Most of the time I will talk about compact manifold M so there is no zero condition here. Sometimes we will assume that it's a manifold with boundary then we're talking about Dirichlet but plus eigenfunctions. For example when M is a domain in RN, you can think about this scheme as description of Dirichlet Laplace eigenfunctions on bounded domains as well. So what we know about this operator, we know that this is self-adjoint, nice operator if you integrate Laplace F times G over manifold as usual, this is minus integral of the product of the gradients and this manifold, sorry this operator has discrete spectrum. So we have eigenvalues for minus Laplacian, they are all non-negative and form an increasing sequence. On compact manifold M, the first eigenvalue is zero and the corresponding first eigenfunction is just a constant, you can take function one, it's harmonic there. If we think about domain with a boundary, our first eigenvalue is positive so we have strict inequality here. Let me start with a very elementary example when our manifold is just the union circle. Laplace operator is the usual second derivative so the eigenfunctions are usual cosine and sine functions. Eigenvalues are squares of integers to be more precise, should write it like that. And the corresponding eigenfunctions are linear combinations of cosine and phi and sine. So any eigenfunction that corresponds to eigenvalue n squared is a shifted cosine function, it has n zero points. This is a very trivial simple example, you can change it a little bit, change the metric on the torus. Introduce another metric, not the uniform one and you'll get a standard Sturm-Leuville operator for which the Sturm series say that you have the same phenomenon. You have still n zeros of eigenfunction of number to n and to n plus one. We'll try to down, but it's relatively well-known, I think. In higher dimensions, there is no such good control of the zeros of eigenfunctions but what we know is the current model domain theorem it tells you that if you have Laplace-Biltrami operator on manifold or domain with a boundary and you look at eigenfunction corresponding to the eigenvalue number n, then this eigenfunction, the zero set of this eigenfunction divides the whole manifold into not too many pieces. Let me do it more carefully. So we'll use the notation z of phi all the time, it just, the zero set of any function and the current model domain theorem says that the number of connected components of this set, we call them model domains and this number is bounded by n. So for the first eigenfunction, you have just one little domain and the case of compact manifold, your function was constant, it's zero set as empty and there is only one domain here. In the case of domain with boundary, when you have say domain in Rd with Dirichlet boundary value, current model domain tells you that the first eigenfunction is positive or negative, it doesn't change sign, there is no zero set there, you don't divide this into pieces. It's a textbook theorem and I will just say a couple of words what is nice to know to prove it. We can characterize eigenvalues using minimax property. We know that the first eigenvalue is the minimum of the ratio of this kind just from the formula here. And if you want to construct all other eigenvalues, one way to look for them is to minimize the same ratio but now you're not allowed to use all the functions, you use the fact that you already know the other, the first K minus one functions and look at the minimum over this ratio or the functions that are orthogonal to first K minus one here. From this one, you can prove current model domain theorem using weakening continuation result. This is L2 in the product, this is L2, what is? Yes, orthogonal to. So the useful piece of information in the first eigenfunction is strictly positive up to multiplication by minus one, definitely. And if you have two eigenfunctions with distinct eigenvalues, then they are orthogonal in L2 and if you take all eigenfunctions, they form a basis for L2 of M. Definitely there are some eigenvalues that have non-trivial multiplicity and we have to take the right amount of eigenfunctions for each eigenvalue count the multiplicity. But this formula will do it for you. So two more examples, very standard ones. If instead of the circle would take a sphere in Rd, sorry, I'll take the dimensional sphere in Rd plus one, try to denote the dimension of my manifold by d. So then, what are the eigenfunctions and eigenvalues of the Laplace-Beltrami operator and sphere to find eigenfunctions, you can look at the Laplace in Rd plus one and separate the variables using polar coordinates. You'll see that if P is a polynomial of degree n and homogenous with zero Laplace, so harmonic polynomial in Rd plus one of homogenous degree d, then its restriction to the sphere is an eigenfunction and the eigenvalues n plus d minus one. All such harmonic polynomials give eigenfunctions on the sphere and there are no other ones. These eigenfunctions form a basis in L2 on the sphere. You can write down some simple formulas and look for simple harmonic polynomials and you'll see very nice pictures of zero sets on the sphere. For example, you can see this orange or watermelon zero set on the sphere. Or you can choose your harmonic polynomial to see this nice zero set. You can combine those and see something like that as well. Another very standard example is the torus. Yes, thank you. And you also have simple eigenfunctions. You can divide the variables and look at eigenfunctions of the form that are products of trigonometric functions, say cosines, theta. Then...