 So, in our discussion on the second order partial differential equations, so far we have considered partial differential equations second order in two independent variables. In this lecture and the next we are going to discuss second order partial differential equations in more than two independent variables. The first concept that we are going to discuss is what is called characteristic surfaces. This is a generalization of characteristic curves that we have seen in the discussion of second order equations in two independent variables. The outline for today's lecture is we first recall the characteristic surfaces and classification which was done in two dimensions which is same as in two independent variables whatever we have covered for the partial differential equations where we called it was a characteristic curves and we have of course classified based on characteristic curves. So, we will recall that and then we generalize that to higher dimensions and they are now called characteristic surfaces for obvious reasons because they are going to be one dimension less. For example, characteristic curve it is a curve in R2. So, now here we will have to imagine things which are just one dimension less. So, that is why we will come to that then we will understand what this means and then we classify the linear equations in more than two variables. So, linear PDE in D dimensions this how we call it is an equation of this form. This is a most general linear partial differential equation in D independent variables. Often we call that D dimensions PDE in D dimensions means there are D independent variables. Sigma ij equal to 1 to D aij of x dou 2u by dou xi dou xj. This is often called what is called a principal part of the PDE, principal part of the PDE. So, it is that part of the PDE where the highest order derivatives appear. The highest order derivative appears with power 1 coefficients may depend on x, u and first order derivatives does not matter. So, this is what is called principal part. In the context of this D dimensional linear equation the principal part is precisely this because this is where the second order partial derivatives of u appear. If you recall in when the in the case when D equal to 2 the classification and the characterization of characteristic curves is actually based on the principal part. So, similarly it is going to play a role even in higher dimensions. That is why I have introduced the word formally that what is the principal part of a linear partial differential equation. It is that part where the highest order partial derivatives appear and some people call this one as lower order term sometimes because it involves derivatives of u not of second order but of orders less than 2 that is 1 or no derivative. We have to make some place some assumptions on the coefficients without which we cannot do the theory. So, we are going to assume that aij, bi, cd of course they are functions of x defined on a domain that is x belongs to some domain in Rd we also assume that these are continuous functions for simplicity. We most importantly assume that aij equal to aji because aij is a coefficient of dou 2u by dou xi dou xj aji is a coefficient of dou 2u by dou xj dou xi and if you are expecting a twice continuously differentiable function as a solution to this equation then dou 2u by dou xi dou xj is same as dou 2u by dou xj dou xi the order in which you take the second order partial derivative does not matter. Therefore, we can always make this to be symmetric aij equal to aji. So, this is the condition that we must place. So, let us assume this and this x which is in the bold phase actually stands for x1, x2, xd in Rd. So, what is the Cauchy problem for the equation dL that is d dimensional linear equation this is a shortcut form of calling d dimensional linear equation we just call dL. So, what is the Cauchy problem? Take gamma subset of Rd which is a regular hyper surface. Let u and all of its partial derivatives namely ux1, ux2 up to uxd be prescribed on gamma this is often called Cauchy data. Cauchy problem for dL consists of finding a c2 function defined in a neighborhood of gamma because that is where the function u and the partial derivatives are prescribed or even a part of gamma now we are used to this idea that Cauchy problem need not have a solution around entire gamma it could be a part of gamma such that u solves dL it should be a solution of the partial differential equation and the given Cauchy data should be satisfied by u. Now, what is the meaning of gamma inside gamma subset of Rd is a regular hyper surface there are 2 words regular and hyper surface normally hyper surface means somebody who just we just one dimension less in the whole space for example in R3 a plane would be a hyper surface 2 dimensional plane or a sphere that will be hyper surface because a sphere can be described using 2 parameters whereas in R3 we need kind of 3 parameters. So, that is the meaning of the hyper surface and what is regular it is something about smoothness we will see that. So, when d equal to 2 how did we define gamma is a regular curve first of all we consider gamma which is a parameterized curve to start with and regularity meant that at every point of gamma there is a well defined tangent. Equivalently it means that a well defined normal exists at every point of gamma even in higher dimensions gamma may be taken as a parameterized surface parameterized by d minus 1 parameters when d equal to 2 the curve prescribed by 1 parameter when you are in Rd it will be prescribed by d minus 1 parameters. So, parameterized by d minus 1 parameters. Regularity means that at each point of gamma in the case of d equal to 2 we said well defined tangent now that it becomes well defined tangent plane exists. Therefore, equivalently a well defined normal exists at every point of gamma. We will come back to this discussion towards the end of this lecture. When d equal to 2 the function u and only its normal derivative were prescribed right recall lecture 3.1, but we found that it is equivalent what is equivalent prescribing function u and only its normal derivative this is equivalent to prescribing u and all its partial derivatives. Why is it so because gamma was a regular we can determine all derivatives if you knew the function and the normal derivative. So, even in higher dimensions one may prescribe the function u and its normal derivative on gamma it can be done. In such a case what do you have all tangential derivatives are completely determined on gamma why is that because the function is given on gamma and they are d minus 1 in number normal derivative is also prescribed. For example, imagine your gamma is let us say in d equal to 3 gamma is x y plane. So, you are prescribing u on x y plane it means automatically u x and u y are determined on the x y plane. So, these are the tangential derivatives which are obviously d minus 1 in number that is 2 in number u x and u y normal derivative is nothing but u z. So, if you give u z you have three derivatives. Now, same thing is true even if gamma is not x y plane gamma is a regular hyper surface. So, if in case of regular hyper surface you are given u on gamma therefore, all tangential derivatives can be determined keep in mind the example of the x y plane as gamma and r d is r 3. Normal derivatives also prescribed thus a total of d directional derivatives are determined along gamma d minus 1 is here and one more here therefore, d minus 1 plus 1 that is d. So, you have d directional derivatives and important thing is the directions are independent directions. Since, gamma is a regular surface prescribing any d directional derivatives in independent directions is equivalent to prescribing the all the d partial derivatives u x 1, u x 2 up to u x d. So, let us review characteristics surfaces and classification as was done in the case of two independent variables. So, let us recall from lecture 3.1. We had a linear system when we are trying to compute the second order derivatives from the PDE and the Cauchy data alone. So, this is the system. F and G was the parameterization of the gamma. Gamma is f s x equal to f s y equal to g s. So, we are trying to determine the second order derivatives then it satisfies a linear system of linear equations non-homogeneous. Determinant of this matrix was what is called delta of s. This is the determinant. If you expand this determinant you get this formula. We did this in lecture 3.1. Now, if delta s equal to 0 then what are the implications? What will happen? So, in other words you have a system let us call this a x equal to b. We have a system a x equal to b and determinant of a equal to 0. What can you say about the solutions of the system a x equal to b? What we can say is there will be b for which there is no solution, there will be for which there will be solution, but the moment you have one solution you have infinitely many solutions. So, you will never have a unique solution either you have no solution or infinitely many solutions. So, if delta is equal to 0 then this system can have no solutions. When will that happen? When the data is incompatible with the PDE. It can have infinite number of solutions. This will happen when you have compatibility between the PDE and the Cauchy data. The next example illustrates these possibilities very clearly. It is a very simple partial differential equation. So, the Cauchy problem is u x y equal to 0 and u of x 0 is given to be f x, u y of x 0 is given to be g x that is u is prescribed and the normal derivative is prescribed on x axis here gamma is x axis. What about this problem? The general solution of the PDE u x y equal to 0 is given by a function of x plus a function of y because you integrate this u x y equal to 0 imagine it is d by dy of u x equal to 0 then u x becomes a function of x and you integrate you get this is a function of x plus function of y. This u satisfies the Cauchy data. What is the Cauchy data? These are the equations. So, when is u x 0 equal to f x substitute here put y equal to 0 you get f of x plus g of 0 must be equal to small f of x. This is the first condition. What about the second condition? You need to differentiate u with respect to y that will give you g prime y and when y equal to 0 that is g prime of 0 that will give you g x. So, these are the two condition that we get. So, if these equations hold then the function g must be constant because g prime 0 is a number that means g is a number. If g is not a constant function then the Cauchy problem does not have a solution that is very clear from this is not satisfied. If g is a constant function then Cauchy problem admits infinitely many solutions. Just convince yourself it is a very simple to show that you can have infinitely many solutions proving this assertion is left as an exercise for you very simple. Now, observe that the Cauchy data is prescribed on the line y equal to 0 that is the x axis which is a characteristic curve for the given PDE. Let us check that delta s this is a formula c f prime square minus 2 b f prime g prime plus a g prime square. What are a b c in our case? a and c are 0 our equation was u x y equal to 0 right. So, a and c are 0 and gamma is x axis therefore f s is s and g s is 0 when we plug in this into this what we get is delta s is 0. So, it is a characteristic curve and clearly u y y cannot be determined along x axis using the equation and the Cauchy data because Cauchy data gives you only u u x and u y anyway is given, but u y y you cannot get because it is not there in the equation. So, now we are ready to define what are characteristic surfaces in higher dimensions. So, the preceding discussion inspires us in generalizing the notion of a characteristic curve to the context of PDEs in more than two independent variables. So, let us define the notion of characteristic surfaces. What is that? A hyper surface in R D is said to be a characteristic surface for a second order linear PDE if at every point of gamma if at every point of gamma there exists at least one second order partial derivative of a function u which satisfies the PDE and that cannot be uniquely determined. That cannot be uniquely determined from the PDE and the Cauchy data that is from the PDE and the values of u and all its partial derivatives along gamma. From this knowledge of u and all its partial derivatives on gamma you cannot determine at least one second order partial derivative uniquely determined cannot be uniquely determined means either you have too many values or no values that is no solutions or infinitely many solutions. If you recall that is what is true right with in two dimensions when delta is equal to 0 we have the two possibilities which were seen in the last example. So, let gamma be a hyper surface now we are going to define a non-characteristic hyper surface. Gamma is said to be a non-characteristic hyper surface. Recall what we have defined in the previous slide what is a characteristic surface. Now we are defining what is called non-characteristic hyper surface. What is that? At every point of gamma all the second order partial derivatives of a solution to the PDE can be determined uniquely from the PDE and the knowledge of u along the gamma and partial derivatives of u along the gamma. Now I would like you to think over the following understand the difference between gamma is not a characteristic hyper surface on the previous slide we defined what is the meaning of gamma is a characteristic hyper surface. Therefore gamma is not a characteristic hyper surface and gamma is a non-characteristic hyper surface there is a difference. I just give you a hint understand the difference between delta S identically equal to 0 on I in fact this is what we called a characteristic curve if this happens and delta S is never 0 that means that curve was non-characteristic curve that is the difference one is not a negation of the other you may call one is a extreme negation of the other that is fine but you must understand the difference between not a characteristic hyper surface and non-characteristic hyper surface both are different. We have a theorem let you be a C1 function on a domain omega in RD let gamma be a regular hyper surface and you be C2 on omega minus gamma that means what we are thinking of is a situation like this you have a omega and you have a gamma then u is C2 here u is C2 here at all these points but u is C1 throughout and jumps across gamma for all secondary derivatives are defined that means that at points here at points on gamma look at any derivative that you think of u x1 x2 this is one partial derivative this jump is defined that means u x1 x2 coming from this domain let us call as omega 1 call this omega 2 at a point here the u x1 x2 this secondary derivative has a meaning that means it is continuous from this side so that means in other words u restricted to omega 1 is actually continuous function of up to closure so that the values of second order partial derivatives make sense on gamma similarly u restricted to omega 2 is C of omega 2 bar not C in fact C2 because C we already assumed u is C1 there is no need of writing C of omega 1 bar or omega 2 bar what we want actually is about second order derivatives have values at points of gamma coming from the right hand side or left hand side I have written picture in R2 that is why I am using the word right hand side left hand side so in other words what I am assuming is that this hyper surface gamma cuts omega into two parts one part you call it omega 1 another part you call omega 2 u is C1 on the whole domain omega but on each part u is C2 of omega i bar when you restricted to omega i so that jumps these are well defined quantities so now I am going to talk about that in the next hypothesis assume that at each point of gamma some second order partial derivative has a non-zero jump and let u be a solution to the PDE in both the parts omega 1 and omega 2 as I have written them then conclusion is about gamma gamma must be a high characteristic surface the hyper surface gamma must be a characteristic hyper surface sometimes I am calling characteristic surface sometimes I am calling characteristic hyper surface does not matter because anyway gamma is a hyper surface to start with so it is a characteristic surface suppose it is not a characteristic surface so the proof is going to be like this I assume that the conclusion does not hold and I will show that one of them does not happen in fact I am going to show this does not happen all second order derivatives are continuous across gamma I am going to show that so that will prove this theorem so assume gamma is not a characteristic surface okay then there is a point of gamma at which all second order partial derivatives are determined uniquely call that point P there is at least one point so call that point P all the second order partial derivatives can be uniquely determined from the PDE and the knowledge of u and all its partial derivatives along gamma this is always there okay from this this is what we call Cauchy data in the Cauchy problem from there and the PDE we can determine all second order partial derivatives uniquely at that point now this is where you should compare with what we did in lecture 3.1 expression for them involves expression for what expression for the second order partial derivatives involves u and its partial derivatives along gamma apart from the coefficient functions in the PDE aij, bi, c and d we assumed u is c1 of omega and aij, bi, cd are continuous therefore at P we get the same values for all second order partial derivatives on gamma whether you come from the omega one side or omega two side you have the same value that means no jump across gamma thus all the second order partial derivatives are continuous across gamma and as we discussed at the beginning of the proof this finishes the proof of the theorem how to identify characteristic surfaces this is the next question because what we have defined characteristic surface it is a qualitative definition okay it just says that some partial second order partial derivative is not uniquely determined at each point of gamma such a gamma was called characteristic surface this is a qualitative definition now we would like to have a quantitative version of this so that we can go and find some characteristic surfaces so therefore it is useful to find a quantitative criteria the example which we discussed at the beginning of the lecture suggests the following if dou 2 u by dou x square dou x k square does not feature in the equation then x k equal to 0 would be a characteristic surface in that example what was the example it was u x y equal to 0 right and we have looked at u x 0 is given to be f and u y x 0 this is a normal derivative given to be g and we could not get what is u y y using this PDE and the Cauchy data u y y was missing so if some derivative is missing then that variable equal to 0 will be a characteristic surface this idea so this discussion gives rise to a necessary and sufficient condition for a regular hyper surface now we are going to look at hyper surfaces of this form phi x equal to 0 to be a characteristic surface this means set of all x in r d such that phi x equal to 0 so called level set of the function phi we are going to derive an extreme sufficient condition for regular hyper surface of this form to be a characteristic surface so let gamma be a regular hyper surface defined as the level set of a smooth function phi 1 from r d to r what is the definition set of all x in r d such that phi 1 x equal to 0 and such that gradient of phi 1 is not 0 for each x in gamma so this actually tells you that there is a clearly defined normal at every point of gamma so this is the regularity hypothesis for the hyper surface then the following statements are equivalent what are those gamma is a characteristic surface for d l that is same as saying that this phi 1 gamma is defined through the function phi 1 that phi 1 satisfies this PDE what is the PDE grad phi 1 dot a of x grad phi 1 equal to 0 what is a is the matrix aij which is appearing in the principal part of the equation d l so this is an equation for characteristic surface a level set phi 1 x equal to 0 is a characteristic hyper surface if and only if this equation is satisfied of course this is a first order PDE non-linear okay so consider a non-singular coordinate transformation given by x 1, x 2, x d going to eta 1, eta 2, eta d and eta i equal to phi i of x we were given only phi 1 of x therefore eta 1 is phi 1 x now you find phi 2 of x, phi 3 of x, phi d of x so that we have this coordinate transformation defined which is very easy so it can always be done that is an exercise in analysis essentially what we are asking is can you given a function phi 1 with grad phi 1 non-zero can you find phi 2, phi 3 up to phi d such that certain Jacobian is non-zero okay that is the question it is possible. Now under the change of coordinates we have a new function w related to u w of eta 1 eta eta d such that u at the point x is equal to w at phi 1 x, phi 2 x and phi d x now the given PDE will be transformed into the new coordinate system and then we find conditions under which dou 2 w by dou eta 1 square does not appear remember eta 1 equal to 0 is same as set of all x as that phi 1 x equal to 0 is a hyper surface and that is precisely our gamma. So we know that disappearance of this is a sufficient condition for gamma to be a characteristic surface so we ask the question is it also a necessary condition answer is yes when Cauchy data is prescribed on gamma which in the new coordinate system is eta 1 equal to 0 we will not be able to determine this derivative dou 2 w by dou eta 1 square from the PDE and the Cauchy data. As a consequence one of the second order partial derivatives dou 2 u by dou xi dou xj cannot be determined along gamma uniquely okay why because there should be some connection between these derivative this derivative and these derivatives imagine you can find out all of them then can I write this as a combination of that think about this think about chain rule okay. So since dou 2 w by dou eta 1 square will arise only from the principal part of the given PDE namely this part when you change your DL equation into the new coordinate system this one dou 2 w by dou eta 1 square which is a second order derivative it comes only through these terms therefore you will find principal part how this transforms into the new coordinate system this gives rise to this where each aij is this expression. So go back to lecture 3.3 I think where we have done this change of coordinates okay thus the analytic characterization of a characteristic surface gamma is obtained by setting the coefficient of dou 2 w by dou eta 1 square to 0 that means this equal to 0 the above equation is nothing but what we want because this is grad phi 1 dot a grad phi 1 equal to 0 please ignore this number 26 this refers to the equation grad phi 1 dot a grad phi 1 equal to 0 okay. The previous lemma asserts that a hyper surface phi 1 x equal 0 is a characteristic surface if the equation grad phi 1 dot a grad phi 1 is 0 is satisfied. We are interested in knowing whether the equation has any characteristic surfaces a partial answer is provided by the previous lemma it also tells that characteristic surface exists when the equation does not feature the second order derivative with respect to one of the independent variables if the missing variable is xk that is dou 2 u by dou x square is missing then xk equal to 0 is a characteristic surface. Let us look at an example which is wave equation so we will try to determine characteristic surfaces for wave equation. So let gamma be a regular hyper surface defined as the level set of a smooth function phi. Now we have to be careful that wave equation we have x comma t so therefore x in rd and t in r so we write this so gamma is phi of x t equal to 0 and gradient should be non-zero here gradient is not with respect to x and t just to clear any confusion we write this gradient of x comma t of phi x t is non-zero for each x t in gamma. When will this gamma be a characteristic surface for wave equation? Answer the function phi must satisfy this equation grad x t phi dot a grad x t phi equal to 0 and what is a? a is a diagonal matrix minus c square minus c square minus c square in the end 1. So this is like ut t equal to c square u x 1 x 1 plus c square into u x 2 x 2 and so on up to c square u x d x d. So these are d number these 1 so this is a d plus 1 by d plus 1 diagonal matrix. So this equation grad phi dot a grad phi reduces to this equation. This we can write because this is the Euclidean norm is nothing but this whole thing phi x 1 square plus phi x 2 square plus phi x d square is precisely norm grad x phi square Euclidean norm. So this equation exactly this. Now I can get rid of the squares and I get this equation phi t equal to plus or minus c norm grad x phi. Recall from lecture 3.2 that when one of the independent variables is time which is true in the wave equation we will be interested in there the curves of the form t equal to psi of x. Now here we will be interested in surfaces t equal to psi of x. So we are not interested in arbitrary phi of x t equal to 0 but we are interested in t equal to psi of x. We explain the reason why why so because this is going to give the location of the discontinuity that that is what discussed in lecture 3.2. Please go back there and understand again. So in other words we are interested in the level sets of functions of this type t minus psi x what how the equation changes this can equation for a hyper surface to be a characteristic surface will now in terms of psi becomes this norm grad psi grad x psi equal to 1 by c. So when d is equal to 1 this equation is simply this equation. Now we can factorize them like this phi t square plus c square phi x square minus c square phi x square it is there then you have phi t c phi x minus c phi x phi t which gets cancelled. So this is exactly same as this factorization. So two important families of solutions they are the characteristic lines x minus ct equal to constant and x plus ct equal to constant. They come through solutions of these two equations phi t minus c phi x equal to 0 and phi t plus c phi x equal to 0. So now we are going to classify linear equations in more than two variables based on characteristic surfaces. So linear second order PDE is called elliptic if it has no characteristic surfaces. Parabolic if there exists a coordinate system such that one of the independent variables does not appear at all in the principle part when the equation is written in that coordinate system and the principle part is elliptic with respect to all the variables that appear in it hyperbolic if it is neither elliptic nor parabolic. So now we finished this lecture with a comment on the regular hyper surfaces that we started discussing at the beginning of the lecture when we introduced the Cauchy problem. So Cauchy problem is posed on any hyper surface gamma. We did not even define what it is meant right. We went with the understanding of what we know in two dimensions or it is a curve tangent is defined at every point of gamma like that. So naturally we understood by a hyper surface as geometric object in RD which is parameterized by D minus 1 parameters this is correct but we need to add regularity assumptions as was done in two dimensions. In our analysis we assumed that gamma is a level set. So when we looked at the theorem and the lemma we assumed it is a level set of a smooth function with gradient being non-zero. Now is every regular hyper surface defined like this with D minus 1 parametric hyper surface is it level set because we are not if it is not so we are not proved a theorem for a general hyper surface. So are we missing something by analyzing only maybe possibly a special case. So regular surface may be defined in many different but locally equivalent ways. There is no need to worry whether what we did is fine or not do not worry due to the equivalence. For equivalent notions of regular surface there is a wonderful treatment in this book by Wister, Matt and Koch multidimensional real analysis part 2 integration. They have two books on multidimensional real analysis part 1 is differentiation part 2 is integration. Of course if you want to study the part 2 they often refer to part 1 also very beautifully written books. Now let us look at some examples. This equation uxx plus uy plus uzz is of elliptic type. Since there are no associated characteristic surfaces how do I say it is elliptic? I have to ask whether there are characteristic surfaces if there are none it is elliptic. There are none why? Let us look at the equation said to be satisfied by a hyper surface to be a characteristic surface. So assume this gamma given by a level set be a regular surface. Gamma is a characteristic surface if and only if phi 1 satisfies this equation grad phi 1 dot grad phi 1 because the a is identity here. The matrix A that you get out of this equation is identity because only uxx, uy and uzz appear no mixed partial derivatives appear. So half diagonal terms are 0 in the matrix A and diagonal terms are 1. So you have this but what is this? This is mod grad phi 1 square equal to 0. So this will be satisfied if only if grad phi 1 is 0 but we are assuming the set gamma is a regular hyper surface that means grad phi 1 is never 0. So this tells us that there are no characteristic surfaces. Therefore the equation is elliptic type. Look at this equation ut equal to uxx plus uy y. This is a parabolic type. Check the definition. I will not discuss this. Please check that by the definition that is not it is a parabolic equation. Now this equation ut equal to uxx plus uy y it is a hyperbolic type. It is actually a wave equation in two space dimensions. It is a hyperbolic type. Check that it is not elliptic. This is what we have to do. It is not elliptic, it is not parabolic and by definition it is going to be hyperbolic. So let us summarize inspired by discussions of lectures to 3.1, 3.2, 3.3 we defined the notion of a characteristic surface in more independent variables, PDE in more independent variables. Derived a necessary and sufficient condition for a regular hyper surface which are level sets to be characteristic surfaces. And characteristic surfaces for a wave equation are analyzed. That is what we did towards the end. Using the notion of characteristic surfaces we have classified linear partial differential equations in more than two independent variables. Thank you.