 Okay, good afternoon, everyone. Welcome to today's TSVP talk. It is my great pleasure to introduce today's speaker, Professor Yana Buillon. Yana is a professor of mathematics in Linköpe University in Sweden, and she also received a PhD in the same university in 1996. And after that, she was a postdoc at the University of Michigan in Ember and then at the Lund University. She has been visiting many institutions, including Charles University in Prague, University of Cincinnati, and the Mittag-Leffler Institute. And Yana is working on analysis on metric spaces, including PDEs, especially PLOPLAS equations. So today, we are very happy. Yana is going to give a general audience lecture related to her work. The title is metric spaces navigating in a world without directions. Thank you, Ching. Can you hear me? And it works. Yes. Thank you, Ching, for the introduction. And I would also like to thank TSVP for letting me spend a time semester here in Okinawa. So it's really exciting to be here. I have prepared this talk for a really general audience, so I'm a little worried that it will be too simple and maybe boring for all of you. Well, let's see. I at least have a lot of pictures. I will not really give any CRMs. So maybe I will just try to give you some ideas where you can see all kinds of different metric spaces and maybe they can be useful for some things. I would also like to apologize to the physicists for my very naive understanding of physics. I will have some examples from, well, pretending to be from physics and the real world and they are probably very naive for the experts. So I will start with a map of the OIST, Institute of Aysa Technology. And when you came here to this room, many of you were coming from the TSVP, so Lab 4. And you can take different ways as you move here. One is if you want to avoid elevators, that's the one I took. Well, then you maybe want to stay on level E and you take the hillside of the building. If you are interested in the shortest way and don't mind elevators, then you go down to level C and take the other side of the building. If it's raining, maybe you go all the way down to skip the rain. If you were a bird or there were skywalks, you could walk directly on the skywalks or you could fly. And all of these give you different routes that they give you different distances as you move from Lab 5 to Lab 4. If you add some more conditions that you want to fulfill on your walk, for example, you want to get a coffee in Cafetancia. Then you walk here first and then you go to Lab 4. And what happens then is that this route through Cafetancia, the TSVP to Cafetancia and then to Lab 4 will necessarily be longer or at least not shorter than the direct route. And this is a triangle inequality that all kinds of distances satisfy. So with that type of distance that whenever you go somewhere and your high priority is coffee, you want to go via Cafetancia. Or it would look like this, like this graph. Then change the middle and then you would have connection to all the labs like that. These types of distances and graph they appear sort of everywhere. So I took another example here from airlines. I guess to this island, all of us have to come by plane. Depending on the airline, there may be different routes. So again, here we have an example from Okinawa to Taipei, there's a direct route with China Air. But if you want to go with the Japanese Airlines, you would have to go probably to Tokyo or at least Osaka. On the other hand, with the Japanese Airlines, Fukuoka and Nagoya have several different flights a day. But with the Sky Team, China Air or other airlines, you might have to go to Taipei or maybe to Seoul and to get to Nagoya. So this is just to give you some ideas of the... In our everyday life, we see all kinds of differences and we don't even think about that. And while we are starting to think about these types of routes, then it does not really matter in which direction we are moving because here with the air flight, first you go to the south and then you go north back, even though you just wanted to go east. So once on the island here in Okinawa, again we can think of different types of different distances and metrics how we measure the way we move around. If the whole island was a homogeneous jungle, then we would just use the standard Euclidean metric where we take the difference in the x-coordinate between two points A and B. So the difference in the x-coordinate squared plus the difference in the y-coordinate squared and then the square root of that. That's the Pythagoras theorem and that gives us the standard shortest distance between two points. Once you move to an inhibited area with say, orthogonal street system, then you have to go along those streets and the distances and the metric changes again. You will be using the L1 metric where you just add the difference in the x-coordinate and the differences in the y-coordinate. And you see also in this picture that what happens here is that there are no longer unique shortest curves. From this point to that point I can go this way or I can go that way and both routes will have the same length. This is just another example. If you forget about everything and just move by car in Okinawa then you get some kind of road system like this that looks a bit like a graph. And again if I want to go from here to here I may have to go first to the north along the E58 and then back along the other E58. So again we are losing the sense of distances so it does not really matter in which direction we are moving. What matters is the length of our path and the distance between various points. So this brings us to the definition of a metric space and I'm sure that everybody knows or most of you know what the metric is but I just want to recall it. So a metric space is any set whatsoever. And on that set we have a so-called metric which is a function of two variables x and y are points in this set and a metric then is the distance between these two points. So it is a non-negative number and it satisfies some properties so if the points are not the same then the distance is always positive if the two points are just one point the distance from one point to itself is zero and then we have this triangle inequality that if we go directly from x to y then it's always shorter or the same as going from x to z and then from z to y. So this is the rule that we would take through kafetantia and this would be the direct route from lab 5 to lab 4 and I tried to draw here some examples to give you an idea that it can really be any set that you take it could be some nice open set some domain in Euclidean space in the plane or in the space it could be some graph like this that we saw on the road system it can be fractal set I will come to this one later so don't worry too much about what it looks like but some kind of fractal set and so we have this set with a way of measuring distances called a metric and then one can define new metrics on such a set so the metric is not unique for a given set a given set can have very different types of measuring distances like we saw with the standard homogeneous jungle distance on Okinawa or the orthogonal street system metric in the inhabited areas and so if we have some metric space with a metric we can look at all curves that go between two points and we can define a new metric in this way so imagine that we take some other island which is very highly and then maybe there is a non homogeneous jungle there as well so some places are more difficult to walk in whereas some others are easier and so you take a weight function which somehow describes how difficult it is to walk to move through that area so it could be the slope of the hill that you are climbing or how dense the jungle is at that very point and then given two points in that set so let's say we're going to go from one point to some other point here you look at all curves between these two points like here and you integrate this cost that it gives you but how much effort you have to put into moving from that point to the other one along this curve and then you try to find this infimum that the shortest or the easiest curve that you can take where it takes the least effort or the least time to get from x to y that's a new metric it again still satisfies at least under certain assumptions these properties and so that's a way of changing a metric on a metric space into some other metric and getting a new metric space this is just another example of the same situation so imagine here this island is flat but instead everybody likes beaches so there is a lot of people along the beaches and there is a traffic jam near the coast so if you want to go between these two points here and there you will not travel along the coast because you will have to wait in the traffic jam and fight with all the people who are there but instead you would like to go a bit inside and then back and here the weight function and the cost of your travel will be one over distance to the coast or possibly you could take a power of that as well this type of metric is called the hyperbolic metric and it's used in hyperbolic geometry we heard a talk on this about two weeks ago one example there is the hyperbolic disk which is just a disk but instead of the standard metric standard distances like we have in the plane the weight function is this is roughly the distance to the boundary and the shorter distances the shortest curves the geodesics then look like this these are circles that are perpendicular to the boundary of the hyperbolic disk and this means sort of this weight function means that it takes time it's hard to walk it's hard to move near the boundary of the disk or the island here so you move slow and that's why it costs that and that's why you don't want to travel for too much when you are close to the boundary you go inside like this so these were some examples of how you can make new metrics and I have two more examples that are based on another idea how to make new metrics on your metric spaces and these are based on vector fields they are used in or this falls into the area of submarine and geometry which I'm not going to talk about so these are for me just two examples but the idea is that if you want to go from a point to another point and you want to measure the distance between these two points then you are only allowed to follow curves that have a tangent so your velocity at a given point is prescribed by given vector fields so at every point in the set there are in these examples there are two vectors given one and then other vectors somehow and if you want to move you can only move your velocity can only be a linear combination like here so this is my velocity as I go along this curve and it has to be a linear combination of these two vectors that are prescribed at every point so one example is the so-called Grushin plane so there you have two directions one is called x so at every point you can go in the x direction this is the blue arrow here and then there is a y direction which is moving in the y direction like this the red arrow but the velocity or the speed you can move in the y direction with a certain effort is proportional to the x coordinate so if you are close to the y axis you can only move slowly in this direction because x is small if you are further away from the y axis you can move faster with the same effort and on the y axis x is equal to zero so if I am here on the y axis I'm not allowed to move at all in the y direction so if I want to move somewhere from the y axis I first have to go in the blue direction in one or I can these numbers a and b can be negative so I'm also allowed to go in this direction as well so I have to move away from the y axis first and then I can move along the red arrow and then I can possibly come back to y axis so that would be a way of going from the origin to another point on a y axis and being on Okinawa I saw some analogy between this and the rip current so if you have a rip current here you do not want to swim against it and you do not want to swim with the current so that's the y axis here what you do is you first move away from the current and then you can move in the direction that you want using the red arrow like here another example that is also very useful and very common is the Heisenberg group which is roughly which is basically the three dimensional space are three but you are only allowed to move in two directions one is this vector field x so again this is moving in the x direction so x now this here x axis is pointing into the board that's how I draw it to get reasonably nice picture so you can move in x direction and plus a little bit in the t direction which is going up but that you can only do how much you can do that depends on the y coordinate so if you are in the origin there are not these parts at all because x and y are zero so you can only move horizontally you can move in the flow direction here in x direction and in the y direction from the other vector field y that's the red arrow here but if you move a little bit on the y axis then here x is still zero so the red arrow will just be moving in the y direction but the x blue arrow suddenly gets a little bit contribution of the t direction going up so this blue arrow points a little bit up so when we are at this point and we want to follow a curve somehow to get from a point to somewhere else this point we can go a little bit up so we can move in this green plane in the origin we can only move in a horizontal plane like that and here is the similar but x and y are interchanged anywhere else on the t axis again x and y are zero so there we only have the horizontal direction so we are in a three dimensional space but we can always only move in two directions we have two dimensional freedom and this again is this one can define a metric and this makes an interesting metric space to study I will not really use it for anything but I wanted to give you the example I want to start talking a little bit about p-harmonic or I will actually only speak about harmonic functions to make life easy but everything I will say would work for p-harmonic functions for those who know what that is so let me just very naively give you an idea why harmonic functions are interesting they appear quite a lot everywhere in a lot of problems so imagine here we have a wall which is insulating our house a room from the outside world now this picture is not really adapted to Okinawa maybe more from Europe somewhere where you would have zero degrees outside and you would have 20 degrees in your room and then the temperature in the wall will be linearly increasing from zero to 20 degrees and you see exactly the same picture if you take a capacitor I think it's called a capacitor not condenser nowadays so you have two plates you put a charge positive negative charge on them then you have electric field between them which is a gradient of some potential you and again if the picture looks like this two parallel plates then the potential will be linear function like this and that's exactly the same picture I have a third picture if you just take a metal rod just a stick like this and again in the wall we keep temperature zero at one end temperature 20 degrees at the other end then if the rod is homogeneous the temperature will increase linearly inside the rod from zero to 20 degrees these are just two other two dimensional examples of the same phenomenon so this time this is not a wall but this is maybe a water pipe like a cut through a water pipe and this is some insulation about it so in the water pipe you have 50 degrees water outside it's zero and then if you look at how the temperature is distributed if you think of this annulus in the insulation of the water pipe it looks like this it's zero here at the annulus so these are you think of the annulus being here so this will be here will be the wall that and then here it is 50 degrees and the same phenomenon would be for the potential if one has a spherical capacitor so this charges positive negative charges onto concentric spheres and the electric field would be in between them like this and then here is just well a little derivation that all these functions here the linear ones and these here are harmonic functions in the set so here in the annulus and here in between the plates or inside in the wall is they satisfy the Laplace equation Laplace operator which is just the sum of second derivatives this is written now in three dimensions here I only have two but you well I guess everybody has seen this before and so harmonic functions satisfy the Laplace equation Laplace operator equal to zero so here I have a P Laplace operator P Laplace equation where which I'm not writing down here and I will not really talk so much about it but basically what I say would apply to it as well in metric space so these are harmonic functions and they are useful so here I have a little bit more on that why would we go into metric spaces so imagine now that we have three rods or we have a network of rods where we are looking at the temperature distribution so the simplicity of three rods like this the end temperatures are 0, 10 and 20 degrees they are all the same length I didn't manage to draw them so but that's what we pretend to and then the temperature on each rod it will be linear and here in this middle point it will be the average of these three temperatures so if I now call the temperature T function U instead what this means that here it's the average so it's a mean value property for that function and one way of writing it is that if I take my point X here in the middle and then I look at all of its neighbors X prime that would be these three neighbors or these four neighbors here in the red picture then the sum of these differences should be zero if you take the sum of these neighbors to the right hand side you easily see that it exactly gives you the mean value property value here is the average of the other three neighbors this was in homogeneous rods if you take rods of different materials and say that the thermal conductivity in one of them in this one would be four times as big as the others and that kind of means that this rod is counted four times and then one gets a weighted average here and the equation that they had here describing the mean value property would be a weighted equation where the thermal conductivity appears so this could be a definition or this is a definition of harmonic function on a graph like this if you have different length and different conductivities and that has to be taken into account but one kind of can write a similar similar description and characterization and interesting thing that will be useful on the soon on the coming slide is that harmonic functions minimize the energy so this is sort of the square root of the gradient something and this is how much it weighs on each edge and one problem that one often studies for harmonic functions or other differential equations is, you want to solve a Dirichlet problem so what you have is you have an open set G which in this case of a graph would be all these red edges that are here the blue dots or purple dots then emo dots would be would be the boundary of this set and so we may have some boundary data prescribed on that set u equal to f on the boundary of the purple boundary and you want to find a function that is harmonic in this red set so it satisfies this equation on every interior point so on these four interior points it would satisfy that mean value property so these were just graphs imagine now that you have a general metric space and you want to look at harmonic functions or more generally p harmonic functions the metric is very general it would look like this so imagine you are looking at temperature distribution in some a mess like that and so how could one define and talk about harmonic functions we don't have any directions we don't have any partial derivative so we cannot really define the Laplace operator in the usual way and this was already something that was visible on the graph because the directions of the edges in the graph and not did not really play any role for the definition of harmonic functions so the main idea how to define harmonic functions on general metric space is the following solutions of the Laplace equation so harmonic functions in our standard world they minimize the energy given by the integral of the square of the length of the gradient modulus of the gradient square so this is the energy of the function and the harmonic functions they try to make this as small as they can among functions that they have the same boundary values so how to make sense of this in a metric space the first step is to replace this integration by something that's reasonable on a metric space and so on the metric space X we put a measure and now well if you are not familiar with measures then a measure is just a way of measuring size of various sets and it does it in an additive way for the joint sets so if a and a prime are two disjoint sets then the total measure the mass or volume or weight of them will be the sum of the individual measures for the these two sets a and a prime so that's a measure a way of measuring size mass sets and then we need to replace this gradient the modulus of the gradient by something to make sense out of this integral and that can be done by so-called minimal week upper gradients and that is defined in the following and now this is long text but I will try to explain it so we have a function U and it's minimal week upper gradient GU will be the smallest non-negative function that controls the function values along curves so if we have two points for any two points X and Y in our space and for almost all curves gamma that connect these two points and almost all means that it's in some sense that can be made precise but I will not do that so for almost all curves the values of the function at this point and at that point they differ about by the curve integral of this gradient GU so it's like the weight we were integrating when getting through the non-homogeneous jungle so this controls these differences for any two points and almost all curves of finite length and then one takes the smallest possible such function GU it can be shown that again under certain assumptions such a function exists that will be our substitute for the modulus of the gradient so just to show you here that this is not coming out of blue this basically generalizes the fundamental theorem of calculus on a real line the difference of function values in two points would be exactly equal to the gradient to the integral of the derivative and here we sort of put absolute values on that and inequality and in Rn in say three or n dimensional Euclidean space this minimal gradient is actually equal to the modulus or at least for nice functions say elliptical functions and on the graphs we basically get what we want as well and this can be defined on any metric space like this one sometimes it's useful and sometimes it's less useful notion but it gives us a possibility to define harmonic functions in the following way so here we have an open set in any metric space X so it will not maybe look this nice be very rough for practice or something but it's an open set in that space and we have a function you blue one here and that function you is called harmonic if it locally minimizes the energy given by the minimal upper gradient so whenever I take some function five that I add to this function which has support inside of this that G so it dies zero before it reaches the boundary of G and then I look at the energy of this modified function it will not be smaller than our function you this means that the function you is harmonic this is the definition and the one can think about what kind of test function do I want to hear use here in RN and the one usually would work with C infinity functions or C zero infinity functions compact support and so on that we cannot do in a metric space because we don't have so smooth functions we don't have derivative but we can anyway talk about lip sheet functions you don't know what the lip sheet function is it's sort of some class of nice functions and those are still possible on a metric space and they give us a large class of this functions to define this notion of harmonic city in a reasonable way so there is I want to give you now some examples first some bad examples if the space is such that there are no curves in it so it would just be a lot of dots everywhere or if there are very few curves in that space then we come back here looking at the definition of the minimal upper gradient that was defined through curve if I don't have any curve then this condition is just empty so I may as well take the minimal upper gradient to be zero because I don't have to satisfy anything there are no curves to test it with so in such situation any function has zero as an upper gradient and so this will be zero this will be zero all functions are harmonic and that's not very interesting to study another maybe a little bit more illuminating example is let's take a both so called bow tie so you take the first and the third quadrant in the plane this one and that one X plus and X minus they are only connected through the origin so there are curves between those two parts and in each of these two halves you have plenty of curves that basically behaves like like a plane but the number of curves going through the origin is small it's not enough for these two halves to communicate in an efficient way with each other and that in particular means that if you take this characteristic function you put one on X plus on one half of the bow tie and zero on the other half even though this function is not continuous it will actually have the gradient zero even at this point and so again the energy for that function will be zero and it will be harmonic and that's not really desirable because such a function is not continuous and we would like to have harmonic functions to be continuous so these are bad examples I have some good examples as well and there are two general assumptions that guarantee that the theory of harmonic functions becomes nice and one is that the measure we have put on the metric space is so called doubling measure so if you measure the size of a ball with radius four you can always control it by some constant times the measure of the ball is the concentric ball with radius R that's the doubling condition and there is another requirement so called Poincaré in a quality which I'm not writing down but it roughly controls mean oscillation on function on balls by the gradients so the gradients are not zero all the time or they are not too small they are strong enough to control the function this can be written down but there's no point in doing it here so some good examples of sets or matrices metric spaces that satisfy these two assumptions there's plenty of set spaces and one of them is sufficiently nice set in Euclidean space so in the plane or in Rn and that class is so called uniform domain so here I draw a uniform domain and attempt and these are points where any two points can be connected by a banana curve so what's a banana curve it looks like a banana and the middle of the curve is here that's the black thing going between the two points and the property it has is that as you travel along the curve you can put in a ball inside the domain plays that curve and the size of the ball is proportional to the distance from along the curve from this center from this point to the endpoint so if I start here I need the ball of size zero but as I move along the curve the balls have to grow a little and so this along going along this curve this makes a banana and that has to fit into the set for any pair of points these are uniform domains for example all domains is lip sheet boundary or balls in Rn have this property and so they are nice and the whole theory of harmonic functions on matrix basis works for them another example that looks a little bit more rough is the funk of snowflake which you get by adding triangles like this scaled by one third and so you get some fractal thing that is very very rough here but again since you are always at the freedom of going in some of these triangles you get this banana and this is another example so-called Sherpinski-Karpit I mean these are not my examples so I'm not really giving you any of my theorems I just want to give you the idea of the theory and so this is a Sherpinski-Karpit you take a square cut it in say well you take an odd number sequence of odd numbers I think the first odd number that we have chosen we chose three and then we cut it in three parts in each direction and then we remove the middle square and then we take an X some other odd number that we have chosen for example five and then we each of the remaining squares we cut into five times five squares and again remove the middle one and so one and if these numbers the sequence is chosen so that the sum of one over E n converges then this will produce in the end when you have removed all the squares this will produce a fractal set which actually has this doubling property and the Poincare unit so it is nice for the whole theory of the harmonic function from matrix basis the Heisenberg group and Gaussian plane or other examples so yeah here are some more good examples you can take maybe I'll just give you the examples let me not get to the last slide but it doesn't matter you take you can take finite graphs like this and also some infinite graphs work for example here we have an infinite tree and as long as it has bounded degree like this it can go forever but it never split into more than 3000 new edges then this will satisfy the assumptions one can also play with these a little bit but this will actually satisfy it only locally for on balls up to certain size but if we then change it a little bit and we shrink the edges as we go along and we shrink them geometrically for example exponentially so that in the end generation at the end level they have length of some number between 0 and 1 to power n so they are shrinking then we get a graph that satisfies the doubling condition and Poincare inequality and globally and this is this are some other examples about gluing spaces so we had already this example of a bow tie where when we glued just in the origin that was not okay because there were too few curves going between those two halves so let's make this now infinite infinite chessboard we take a lot of squares that we glue like this at the corners and we just take the black squares for our metric site again the gluing here at these points is too weak there will be too few curves and so the Poincare inequality will not hold but we can change the measure a little bit so we put more weight to the points that are close to these connecting points it can be written so it will be basically inverse proportional to the distance for these nodes and by doing that by changing the measure and giving more weight to those points then one gets a space that satisfies the Poincare inequality at least locally but that's usually enough for a lot of the theory and there are some other examples of gluing if you blew two planes and here you take some sort of a cantor set you identify these two points so here you have like secret skywalks in between then that again gives examples of nice metric spaces where a lot of the harmonic and p-harmonic function theory goes through a product of two nice spaces also satisfy those assumptions so what kind of how many minutes how much time do I have or should I already finish maybe I can 10 minutes so what kind of problems can we look at and one of the basic problems is to study the Dirichlet problem I already mentioned in the connection is the graph so you have an open set G so here it will be this although not with the handle something so that can be holding it it can be pretty wild open set as long as it's open and then you have boundary data F given on the boundaries on this boundary and also on this boundary and you are looking for a harmonic function that has these boundary values in some sense it usually cannot have them as a limit at least not at all points if the set is not very nice but in some sense there's a way of attaching a harmonic function to these boundary values that actually will be the last statement that I will give you as a theorem and so that's one problem you want to study sometimes you might want to add Neumann data on a part of the boundary so let's take this part of the boundary and instead of requiring the function to be equal to F you want the normal derivative to be zero so in a word or in a real world situation this might mean that here we know the temperature of this body on the blue part and on the red part we know there is no no flux there is insulation there is perfect insulation so the temperature is not changing here it has zero derivative in the normal direction and so these are two problems this is Dirichlet problem is often called the mixed or sometimes Zaremba boundary value problem the funny thing with metric spaces is that we can actually see in both of these problems is one problem you can see them as Dirichlet problem both of them because by a suitable choice of metric spaces we can get rid of the Neumann data which often are more difficult to handle than the Dirichlet data so this is an example of let's take a unit ball in Rn which is a nice set and if I want to take just the Dirichlet data well then I add the boundary to it consider that as my metric space X so the closed ball will be the space and G is the open set is the open ball and this is the boundary in this metric or this is the boundary in the whole space Rn as well now if we change the outside world change the metric space X and we take it to be the open ball together with this half sphere so we add this to the open ball and together the open ball with this half sphere half of the boundary is a metric space so that's our space that's our universe nothing outside exists that's all we know and it happens to be also quite a nice space so one can do the theory on such a space and in this situation this is not really a boundary because these points are not at all in the space in our universe so these sort of behave like interior points and that automatically forces the harmonic functions to have the zero normal derivative there so in a way this is now just a Dirichlet problem on this set G seen in this metric space with this boundary and this boundary rate so Neumann condition disappears magically so in those previous examples you could say okay you can only do this for very nice sets like balls well in fact the Neumann data should be given or they should at least be contained in a nice set so that one can edit and do sort of get a nice space X so again let's still work on the unit ball like this I take as the metric space X the closed ball and then here I have this blue side here is a closed side but it can be bad it can be rough it can be fractured or whatever and I prescribe the boundary data on this yeah and G will be the rest and actually now these points will be the interior points in this strategy so if you prescribe the Dirichlet data U equal to some boundary data S on this blue contour here that the boundary of G is in the cloud unit ball and then here these points in this world they are not boundary points they are interior points for our set G and the harmonic any harmonic function then here will automatically have to be you'll have zero yeah it'll be constant along the normal derivatives they'll have zero normal derivative so it will automatically satisfy this equation if you see it as a function from just the standard and the reason is that these are interior points in this funny word X and if we then would take and replace this ball by the phone call snowflake which on the previous slide we saw was a nice set so it can serve as a nice metric space then one could make this Neumann data exist on a lot of rough rough boundary of the snowflake as well it doesn't cover all situations but it covers actually quite a lot of rough situations as well and one more situation where changing a metric and using metric sizes can help us to treat more general boundary data is take a disk and cut it here or you could take a ball and cut it take an orange and cut into it with a knife so here you'll get two sides of the cut from above and from below and for some reason many one would like to have boundary data that are zero on the upper part of the cut and one on the lower part of the cut now if this is just seen as a set with this boundary in the plane then the boundary data should be given on this cut as one function not as two different values from above and from below but using a new metric in this cut with this we can actually include these boundary data into the theory that we are dealing with and we can then solve the problem for this type of the boundary of such boundary data and the way to do it is to change the metric a little bit inside in this cut split disk so what we do is that instead of looking at the standard distance between two points x and y we take the shortest distance between these two points which is achieved or can be achieved when we move just inside the set G so we are not allowed to cross this line so this will be the shortest distance between x and y so this kind of opens the split a little bit makes it into Pacman picture like that and this gives us the new metric space is the new metric and this change of the metric is so nice that inside in the set in the domain G it does not change the notion of harmonic functions it's still preserved the harmonic function in the original metric they will still be harmonic in this new one and the other way down and so with this one can solve the problem for more general boundary data like other things that I'm not going to talk about but one can play a little bit more is there are other changes of metric you can take unbounded sets and wrap them and make bounded ones those are often easier to deal with or if for some reason you want to unwrap the bounded sets and make them unbounded and that's possible as well and I guess I should stop and this is my slide and here is just one of our theorems it's not the most deep one but it was easy to formulate it basically gives a unique existence of solutions to the Dirichlet problem where the boundary data for continuous functions are obtained at most boundary points in some sense of capacity and it just illustrates that if you would not have the Poincare inequality then at least in this situation the uniqueness will be lost say in the chessboard with the standard measure where the Poincare inequality does not hold if you want to solve the Dirichlet problem on this set with these blue boundary values then the solution will be determined here on the blue pieces but these two squares are so loosely connected to the rest of the space so the boundary data don't have an influence on what happens here so one can basically choose any two constants for the two red squares and that was it I hope I did not take too much time and thank you all for your attention thank you very much Janna for this great talk are there any questions or comments so in physics we're usually solving the Dirichlet problem on RN looking at a differential equation with boundary conditions and then well the most common thing to do would be to use Green's functions and get an integral solution are there always integral solutions on these more general metric spaces or is it more difficult okay there are two things to it so first I was only talking about harmonic functions which is a linear equation most of the theory we do is actually for so-called p-harmonic functions and those are based on a non-linear operator the p-laplacian and there's no way of using integral representations for the solution if you have a non-linear equation so that's something that we have not really been looking at you can still talk about fundamental solutions that would be solving the equation with Dirichlet at one point but you cannot use them for non-linear equations to represent solutions with certain right hand sides or so by an integral representation and that's even in RN if you have a non-linear equation here there are ways of we don't even have an equation I mean the harmonic functions here even if we talk about harmonic functions they are based on this minimization of the integral and it is not clear not even known and probably not even true in general that a sum of two harmonic functions is harmonic in this generality and they still minimize the energy that I wrote there but the sum will not mess with at least it's not known and so the representation you are asking for is not really existing here and is not studied I did not answer your question at least not satisfactory Any other questions or comments? So you've talked about a lot of harmonic functions or p-harmonic functions so like my point of view is like differential operator of u equals 0 can we have something more or more intricate right hand sides if they have a variational structure? Well yeah because you could be minimizing an integral where you have the energy and then say minus if times function u or something like that that is very little studied in this setting but there should be a lot of things that are doable there as well it's just that there's so many questions already for just minimizing the energy integral or p energy integral so it has not been done but there's definitely something one could study Okay thank you very much Maybe I should add to your question a little bit there's a notion of derivatives that can actually be defined on these metric spaces so called Chieger gradient which is a vector value then one actually gets an equation and for p equal to 2 one would have a Chieger harmonic functions and those would be very much imitating the properties of harmonic functions say from R n so there one knows the linearity and then the integral representations would also at least to some extent be valid Any further questions? I have a quick question so you mentioned this homogeneous Neumann data so can you do non-homogeneous Neumann problems? Well not with this trick that you just forget about it because it's a homogeneous one that's built into that so that there is at least one paper by Nagesan Mungalingam and I guess well at least three more people those three more people one or two papers that they have studied and they have a notion of defining the non-homogeneous Neumann data and again since you don't really have derivatives what's the normal derivative it's not quite clear how to define that and what that would mean because we want to define something that would in a natural way generalize the situation from R n otherwise it's maybe not so useful but there is a paper I can show you Thank you Any other questions or comments? Do we have any questions from online audience? Let's thank Yana again. Thank you very much