 Hello everyone, welcome to our next TSVP seminar talk. It's my great pleasure to introduce Professor Anders Bjorn from, he's a professor of Linkirk University, Sweden. And he's an expert on potential theory, say. Potential theory is one of the most fascinating fields of mathematics. It's originated from classical electrostatic potential theory, but it's now this field of mathematics in the intersection of partial differential equations from one side, probability theory, functional analysis, measure theory. And Professor Bjorn in particular expertise in abstract potential theory, which is the monograph joined with Jana Bjorn, I think a non-linear potential theory, which is one of the few monographs in this field. And he's talked about this, the clear problem and the boundary regularity. And I would, I have to just one remark is that this is a, I think that Venus, Norbert Venus approve of criteria for the boundary regularity of harmonic functions is a starting point of the potential theory. So that is a great work, which was one of the most, I think prominent results in mathematics proving the criteria for boundary regularity of harmonic functions. And that's a starting point of the potential theory, I think so, and so it's not incidentals as the title of the talk today is related exactly to boundary regularity, please. Thank you for that introduction. I also want to thank OIST and TSVP for the chance of being here. I've been here now for, we have been here now for a month and a half and it's been great time all the time. And I'm still looking forward to almost half a year more. So, so I want to look at the Dirichlet problem and boundary regularity. I would maybe say that the potential theory even started earlier with people like Newton and Gauss and Dirichlet, which were way before Venus, but it really took a different turn of the Venus criteria, that's for sure. And a subtitle there, how to attain the boundary values in a good way. So let me start with looking at boundary value problems. So I'm sure you have all seen this. So if we have a differential operator, for this slide, it doesn't matter what it is. Later I will mainly look at harmonic functions. Then we can solve the operator here. I put simplicity as to an interval under realign. This is something these type of problems are things that we maybe teach even first year undergraduate students to look at. So they solve an equation. They have only first or second order equations with constant coefficients. And then you can put boundary data, boundary values here. I have imposed the condition that the function should have some value at the boundary points, that's for the Dirichlet problem. In down here, I have imposed instead condition on the derivatives, and then it's called the Neumann problem. And then of course we have, in this case, we have two boundary points, so we could have a mixed problem where we have a Dirichlet condition on one at one boundary point and a Neumann condition on the other. I'm not going to be interested in the Neumann problem, but this was just sort of explaining what the boundary value problem is. I'm sure you have all seen this. But I'm interested in the Dirichlet problem. Now, when one studies these kind of questions for PDEs or general question, I mean, this was an ordinary differential equation there. And these on this slide are as well, there are three fundamental questions. There is existence, uniqueness, and then what kind of regularity we can have of the solutions. And regularity could be continuity, C1, CK, then you can have further continuity in between. And so if you don't haven't seen these, you doesn't really matter that much. But C and C1 and CK, I'm sure you have all seen. Just to give an example, here I put up a problem where we have a solution. This is a Dirichlet problem. I have Dirichlet boundary data conditions here. And here, this is the unique solution. So we have uniqueness. Now, if I change the coefficient here to pi squared, then pi squared is an eigenvalue actually of this operator. And we will have more solutions and then they will look like this. So in this case, we have non-unique. So that sort of looks at that. We can have these kind of problems with uniqueness or with non-unique. Okay, those were just appetizers in one dimension. I'm interested in higher dimensions. So two and higher dimensions and later on in metric specs. But let me start a bit slow and start with two dimensions. And then we have a domain. So this is a domain G, some open connected. And I will have a bounded domain for simplicity in the plane or in higher dimension. And I have a continuous function on the boundary. And then what I want to solve, and now I restrict myself to the Dirichlet problem here for the harmonic function, so for the Laplacian. So I want to solve the Laplacian equal to zero in the domain. So that's the same thing as saying that U is harmonic in G. And I want to have U equal to F on the bottom. Here is just the expression for the Laplacian in two variables. And so what do we really mean here? Well, we mean that the solution that we have inside takes these boundary values F of X as limits at all boundary points. This is the boundary DG. And this is called Dirichlet data because we actually look at the value of the function, not that derivatives like we would do in the non-problem or so. So basically we want the solution that is continuous up to the boundary. So in this case, I said we had three sort of key questions. Uniqueness and existence were two of them. Uniqueness is not the problem in this case. If you have two solutions, they have to agree. But existence is not so clear. So Dirichlet from almost 200 years ago solved this problem by minimizing the energy. So this is the energy of a function. You take the gradient of a function and square it and integrate. You can recall maybe I didn't have much physics in high school but I have had this formula with a square and this is sort of where the square comes from. So he minimized this over all functions which are continuous up to the boundary. They take the given boundary date on the boundary and they are C2 inside so we can easily calculate this integral. And if you have a minimizer, then it turns out that it's harmonic. I mean, that was one of the observations he did, I guess. And it then solves the Dirichlet problem because we have the right boundary data. And since the energy here, this is an integral which can never be negative. The energy is always non-negative. It's clear that the infimum of all these energies exist but why is there a minimizer? Infimum here is the greatest lower bound and you can compare it with if we take all the positive real numbers, then the infimum is zero but there is no smallest positive real number. So there's no minimizer in that case. Of course that's a very different problem. That's not our problem but it still emphasizes that the infimum doesn't necessarily have to exist. Dirichlet missed this point. It was later pointed out around 1870 I think by Weierstratt that this was a weak point in Dirichlet's argument. He had another problem where he showed that in that case the minimizer did not exist but it did not show that this one did not work. But at least there was some sort of missing point in the argument. So then if you go to these days to Google, you look at the Dirichlet principle or you can get to Wikipedia. I got this page actually from Wolfram Mass World which says something here. There's something here. There is a U missing. So one shouldn't completely trust everything that's written down. But here at the end I put a square here or the square or a big rectangle rather. Knesser have obtained the valid proof of Dirichlet's principle. So when you read that you think, okay, Knesser filled in the argument, everything is fine. If you read Wikipedia, you get the impression of the same thing but there I think it was Hilbert that was voted. I'm not exactly sure what Knesser and Hilbert did but anyhow, so reading this one thinks, okay, everything is fine but then I wouldn't be giving this talk. So if G is smooth, what is smooth? It's, for instance, locally a graph of a C1 function and this one might be that. Up here it's not even connected but still. But what about the general case? The simplest example of a domain which does not have a C1 boundary is a square. We may, wherever I want to solve the Dirichlet problem in a square and in many other situations. I mean, this is a classical problem that appears all over mathematics and applications. So here, okay, the square is little special. Maybe in one way but it's not covered by the smooth domain. Another situation which is a bit more complicated is if we take a ball and remove a counter set. Here I have a planer counter set and these are just examples. So we could have very complicated geometry and there are many domains. Most domains do not have a smooth boundary. So then there can be a problem. And to see what really can happen, we have these two G examples of what can go wrong. So let's look first at the top here. So here I have a G which is a ball or a disc in two dimensions and I have removed one point here. So it's what I call a punctured ball or punctured disc. This example is due to Zaremba from 1911, just for harmonic functions. So here is the G. I put zero as the boundary values on the circle and one in the middle. That's a perfectly fine continuous function. But when one try to solve this, the only natural solution is sort of missing the value in the middle and the natural solution is constant zero everywhere. And it's greatly wrong at the center of the circle. And we say that zero is an irregular boundary point. I will be a bit more precise with the definition in a little while. So that may look like, okay, you have this isolated point. It's sure did these things go wrong there, but. But Lebesgue, the same guy as with the Lebesgue measure came up with a different example in the next year. It's called the Lebesgue spine. This stopped working. See, okay. So this has to be done in R3. It should be an exponential cast disappearing here. You can think a little bit like an apple and that you have a stalk here. And then it turns out in this situation that the tip of the stalk here, storage in is an irregular boundary point. Here, this one is not isolated and one have to put in the appropriate boundary conditions. But one can put one over this whole cast and then somehow make the function continuous here and go down to zero there and then one see this behavior. So these two examples show that we can have irregular boundary points. And then we, I guess maybe Lebesgue or someone else realized that what one should do is, one should first relax the problem, find somehow a solution, even though it does not quite fulfill the requirements we have and then look at when does that solution fulfill the requirements. And Perron in 1923 came up with a way of solving this. So he looked at, if I have my boundary data here, he looked at functions which are super solutions and somehow lie above these boundary data at all the boundary points here. I have only two and then he took all these super solutions or super harmonic functions and he took the infimum of them. So all of these super solutions should be larger than the expected solution. So the infimum of all those should also be larger or possibly equal to that. And that creates an upper Perron solution. Then one can do the same thing from below and have a lower Perron solution. And if those two agree, then that should be a good solution. And then the question is, when do they agree? And for harmonic functions, I believe Wiener was the one who proved that a year later, he showed that for continuous functions, they do agree. Then one can also consider this problem now for non-continuous boundaries. Then we can never expect to have continuity at all the boundary points, but this still makes sense. So that makes this a more general method as well. And in that situation, it can happen that we have inequality between these two. And the major question is, when do you have equality? Then you have a good solution, if you don't. And then maybe it's still a reasonable solution one or the other. There are also many uses, but for continuous boundary data, we now have a solution, then we drop the bars and we call this, we denote this by P F. So we have this solution, and then we can look at when is the limit of P F, the given F of X, not just for one solution, F, but for all continuous boundary data. Then if that happens, then we say that X is regular and otherwise it's irregular. And that's how one can really define the irregularity of the Saremba and the Leveg spine examples. And Wiener in 1924, I guess he had a couple of papers in maybe 1924, 1925 on this. Well, one is with this Wiener criterion, which says that a point, or here I have put the point at zero, doesn't have scaling variance in this space. So we can concentrate on looking at when the origin is regular. So zero is regular, if and only if, this condition holds. Now here, he, capacities connected with capacities in physics, but here it's really introduced as a set function, and it was Wiener who introduced it as a set function defined for every set. Exactly how is defined, I don't want to go into that. A bit technical and not so important here. The important thing is maybe that we have an if and only if condition this way, and we can draw some conclusions of it. For instance, we can draw a conclusion that regularity is a local property of the boundary. If I have two domains that look the same, around the point, but different far away, then in here, these terms will be exactly the same from some K onwards, and the first values don't matter because we're only summing finite numbers. So this is a consequence of Wiener's criteria, and there are two domains that look the same as the other ones of Wiener's criteria, and there are other consequences, of course. Another thing that came a few years later is the Kellogg property, which says that the capacity of all the irregular points is zero, and at the same time, one should note that the capacity of the boundary of a domain is positive, so it can never be that all of the boundary points are irregular boundary points. In capacitary sense, most boundary points are regular, but we can still have some irregular boundary points. And capacity is a finer way to measure size than the volume or the big measure. Here I put down some comparison in this case with the half-storey dimension, in case you have heard of that. Also, capacity is not the measure, according to measure theory, but capacities are according to this criterion that it's the right way. We cannot measure this using measures. We need capacity, and capacity is also intimately connected with several of spaces, which are important in various for PDEs and things like that. So they are important for many different other reasons. For all what I've said here about harmonic functions, this was now known, well, not the Kellogg property, but the Wiener criterion is 100 years old this year. These were for harmonic functions. Harmonic functions or the Laplace operator is a linear operator. If we take the sum of two harmonic functions, we get another harmonic function. And that's used extensively in these papers. But I'm interested in the nonlinear theory, which is the later development, starting maybe around the 1960s or so. So if you recall from earlier, I said that the harmonic functions are somehow local minimizers of this energy integral. If we replace the power two here by a power p, we get a nonlinear problem. And we can look at the minimizers for that. These solutions or these minimizers are then called p-harmonic functions. And one can... So they are minimizers of this or equivalently there are solutions of the corresponding Euler Lagrange equation, which looks like this. And I see them as a prototype for a large class of nonlinear elliptic equations and that the main prototype for a nonlinear generalization of harmonic functions. And here one can make it much more general. One can put in coefficients and weights in this equation and so on. But this prototype works well enough for... And I'm not sure the theory holds in that case as well. But let's keep it a bit simpler here. One point here is that regularity says that regularity theory shows that p-harmonic functions are continuous. But in general, they are not C2, which is what you really would want here when you take the gradient and then you take the divergence of the gradient. You would need to have C2 solutions. So one has to understand this equation in a weak or distributional sense. That's also 20th century concepts of how to solve equations that we have. We cannot just understand it in what we now call classical solutions. So for harmonic functions, any weak solution is actually C2 and even C infinity, but that doesn't hold here. So we have to understand the equation that way. But I have worked a lot with p-harmonic functions on metric spaces. So let me say something about what is a metric space. So a metric space, instead of just the plane or so, we have a set and we have a distance function. And a distance function should satisfy the following these three conditions here. So it's a function taking two variables from the space x. If we have the same point, then the distance is zero. The distance is symmetric and if we have two different points, it's always positive and finite. And then we have the triangle inequality holding. Some of you may have heard Yanas talk or Sylvester's talk, where they also talked about metric spaces here in February. So in metric spaces, we have distances, but we don't have directions like we have in Rn. So it's difficult to talk about the gradient because there we take partial derivatives in the different axial directions and then put them together to a vector. But we can still define something called an upper gradient. So if I have two points x and y and I look at the curve between them, I want u of x minus u of y to be less or equal to the integral of gds over the curve gamma. If this holds and it holds for all curves and with different starting points and I say that g is an upper gradient of u and one can define a minimal upper gradient as well. Here I could make the upper gradient larger. It would still fulfill this. This is basically based on the formula u prime of x dx that we've learned to use already in high school to calculate integrals using antiderivatives. But this then gives us an upper gradient and in metric spaces or in Rn, the upper gradient g u is nothing but the absolute value of the normal gradient. And when we minimized the energy, we were just looking at the absolute value of the gradient and then took the piece power of that and looked at the energy that way. So we can easily define this and look at minimizers this way. But because we don't have the gradient structure, we don't get to an Euler-Arbanche equation. So sometimes I'm saying that I'm working in PDs and I'm looking at other equations. Here I have also put the mu here. So we have two in the metric space have some way to measure volume of sets as well. And this is now measured by mu. And this measure has to satisfy for our theory two conditions called doubling and ponkarenic quality. We don't want to go into what they mean. But with those two conditions, then we can get the quite nice theory. These conditions have to be satisfied at least more quickly. So examples of phases where we satisfy these assumptions. So we can have Rn, we can have closed subsets of Rn, we can have Rn with weights, we can have manifolds, higher-dimensional surfaces or two-dimensional surfaces of course. So all this falls into the theory. Some fractals can have graphs, networks they're called in some applications, can have Heisenberg groups and very regular spaces. So this is the theory of harmonic functions and to some extent pure harmonic functions had been extended to some of these settings before but this is sort of unifying the whole giving one approach to covering all these cases. And there are applications in porous materials and again a motivation. This is a prototype for nonlinear electric equations. More examples with pictures. So here I have the Boncock snowflake curve which should, here it should be that the curve is actually the boundary here. My space is everything inside including the boundary but I don't have the outside. Over here I have what's called the Sierpinski carpet. So first I take a square, I split it into nine smaller squares and I remove the middle one. Then from each of the eight remaining I split that one here into 25 smaller ones and remove the middle one. And then if I do this in a certain way depending on how these factors are then this will also fall within the scope of our theory. Another example is to take this which is the first and the third quadrant and join those into the space. Here they are just connected by one point and that creates interesting counter examples in many situations of what can and cannot happen. And over here I have an example of one-dimensional space glued to two-dimensional space and that can also be covered by the theory. So these are just examples of what can be covered. And then some results that we have. So one result that we obtained about 20 years ago was that if we have a continuous function then we actually, well first of all the PF here is the upper and lower perron solution again and it means that the upper and lower perron solutions agree. So one of the results or parts of that theorem is really that the upper and lower perron solutions agree in this situation. For RN this was known for p-harmonic functions earlier. And then we have an invariance thing here as well. So F is a continuous function and I have a function H which is zero. It's a quasi everywhere here on the boundary and that means that the capacity of the set where it's non-zero the capacity of that set is zero. So I can have a few points on the boundary but the whole boundary where I change the value and I can change it arbitrarily on those. And then with the batteries running out on this one. Now it's working. Then the perron solution does not see the values at those points simply. At least not if F is continuous here. If you change F to non-continuous then we don't have those kind of results in general. This invariance result was new also for unweighted RN. In the linear case where we work with harmonic functions everything is much easier. We can just use linearity like this and then you just look at this H as it is and know that pH is zero and then that gives the solution. But in the non-linear theory that approach is not possible. More results that we obtain where the Kellogg property I think really the battery is running out on this one. We don't have any replacements I guess. It might take a few minutes. Sorry for that. Let me see if I can point here instead. So the Kellogg property we have at the set of capacity zero points are the regular points are capacity zero. This time the capacity I put the P there. The capacity now depends on P and also on the measure and the metric space. As a consequence of the results we had we also obtained that regularity is the local property of the boundary but here we could not use the linear criterion because we don't have that one in this in full generality. We have one direction but not the other in metric spaces. In RN it's known. And then another result we obtained was this uniqueness result here. So if you have a continuous function then the Perron solution is the unique bounded p-harmonic function which takes the boundary values not everywhere but quasi everywhere on the boundary. And this was also a new result for unweighted aura. And these were 20 year old results but for unbounded domains these turned out to be much more challenging but last year we managed to obtain basically a full generalization of that one and a slight modification of the first CRM now also for unbounded domains. But that took us 20 years to achieve. Okay, regular points now. If we are in RN then the linear criterion imply the smooths and even Lipschitz domains are regular so in particular the square that we had in the beginning a domain is regular if all the boundary points are regular. That's the nicest example. And even here if you have a domain with the snowflake domain then all the boundary points will also be regular in this situation. And if p is greater than n then the capacity of every point is positive and by the Kellogg property it turns out that there cannot be any irregular boundary points. So all domains are regular. Okay, let's see. So that's also nice that in that situation we don't need any condition at all. In metric spaces, sorry, we don't have any way of defining smooths domains in general metric spaces. So of course we can still sort of use the linear criterion when we have it so to deduce but what kind of nice domains do we have in metric spaces? Well, we have walls at least, walls we have in all metric spaces. But it turns out that they can be irregular and it's actually not that difficult to find an example. So if my space, let's say this is my space, it's the first quadrant and I take a ball with exactly this radius. So if this is 1, 1 and I have radius square root 2 then this will be a boundary point. This bit here does not exist in the space, this bit here does not exist so that this ball will go something here and this bit will be boundary as well and this will be an isolated boundary point and that one will be regular if and only if p is greater than 2 but it's irregular if p is less than 2 so I have a ball with an irregular boundary point. With this center it's just this radius that gives an irregular ball but if we work in what's called a Heisenberg group for harmonic functions so for p equal to 2 we have harmonic functions over there that's still in your case and we have it. So for harmonic functions on the Heisenberg group it turns out that the south and north poles are irregular and that's for every ball so no ball is rigged. So balls can be problematic but there are still regular domains. So if we go back a little again and look at the punctured ball on the Lebesgue spine again now in the punctured ball the origin here is regular if and only if the capacity of that point is positive which if it's in Rn happens if p is greater than n but we could have a punctured ball like that in the metric space and then it depends on the metric space and maybe it also depends on where the ball is placed in metric spaces you can have some points with positive capacity and others with zero capacity which can also happen in weighted Rn. Then that's the regularity of this point then we could also have irregular points here on the boundary here on the circle or on the boundary of the ball like if we take the Heisenberg the ball in the Heisenberg group now this one and here we have a different situation and it turns out that I said that these are sort of two examples of irregular boundary points and it turns out actually that the behavior in these two examples are very different now one can show that if you have an irregular boundary point its behavior is either of this type or of this type I don't want to specify exactly what the types are but one can classify the irregular boundary points in two ways and they have drastically different behavior and that's also something one can do in this general theory for p-ammonic functions on metric spaces. I think for harmonic functions that kind of classification fully was done only in about 1980 but then I want to mention also that there so I have so far the Perron solutions is a potential theoretic way of solving the Dirichlet problem. A PDE person would probably take a different approach to the Dirichlet problem. The Dirichlet problem appears we could do this in Rn or in a metric space and we have a boundary domain and then we say that we have a function which lies in the Soblo space whatever that is but the PDE person would work with Soblo spaces quite a bit and for such a function there is a unique p-harmonic function which I denote hg maybe h for harmonic and g for the domain such that this difference is in the Soblo space whatever that exactly means but it gives a solution to the Dirichlet problem in quite a different way from the Perron solution and one of the results we had and actually this was one of not just the result but it was also one of the tools that we used to obtain some of the results was that if you pick this Soblo function and the representative of it in a reasonable way something called quasi continuous then actually the Perron solution and the Soblo solution agrees for stated like this I have not seen this result for harmonic functions in the literature before our paper and we did it in the nonlinear theory I think it's partly because PDE people work with this type of solution and then you have people working in linear potential theory they don't even know what Soblo spaces are they have heard of them but they stay away from them so it's sort of different communes but when you go to the nonlinear potential theory you need to use Soblo spaces all the time so for us it was natural to think in these terms and we also had this kind of invariance result down here let me end with another example showing some things of what is tricky in this setting there was a paper by Albert Bernstein an American mathematician paper from 1998 where he looked at a very simple situation you have a disk and you have two open arcs on the disk E1 and E2 on the boundary of the disk you could have more arcs as well but keeping it simple we say two arcs and the boundary of these two arcs well there are two boundary points here two here and two here so we have four bounds and now if we take H to be the characteristic or indicator function of those four points then we know from our invariance result that if we have a continuous function on the boundary and we add or change it on those four points the Perron solutions will still be the same well strictly speaking that holds if p is less or equal to 2 because this is now this set has to have capacity of zero that happens if p is less or equal to 2 if p is greater than 2 all points have positive capacity and we cannot use our invariance result okay but what if we take the characteristic function of these arcs so we have these two arcs here we take the boundary function that's one on the union of those two arcs and zero everywhere else this one like makes jumps at the four points and then the question is do we get the same Perron solution for the union of the open arcs and for the closed arcs and this was exactly what Albert Einstein was asking of course for the linear case you use linearity then you just have to calculate the Perron solution for the characteristic function of those four points and that one happens to be zero so then it's sort of trivial and no one would even think of at least not write a paper about it but for p less than 2 or for p greater than 2 this was not known well for p less than 2 it turned out that we could use our results we have to figure out that this characteristic function here is actually a restriction of the Sobler function in the plane and then we could use our invariance result from 2003 so in 2006 we obtained that for p less than 2 for p greater than 2 that approach is not possible for two reasons well one is that these four points now have positive capacity so we don't have the invariance result and the other one is that this characteristic function is no longer a restriction of the Sobler function so both ingredients sort of failed drastically but I managed to find a different way of approaching this and managed to show that yes indeed we have equality here also for p greater than 2 and also that we have this this thing is also holding for p greater than 2 even though now these four points have positive capacity but we still have invariance that way yeah that's what I wanted to say so thank you for your attention thank you very much for the great talk so I'd like to understand this so you have this epiloplasm problem and you want to solve this satiric problem so how does this regularity of the domain for us to solve this PDE depend on the p, the exponent p yes yes but p is bigger than n basically you can put in a boundary data but p is less than n p less than n not sure if this has been written down if I remember correctly you do have the monotonicity of it if you are in a situation where you have the Wiener criterion Janna do you recall? yeah from the Wiener criterion it follows that for larger p it's easier to be regular but I think there is not a proof or I don't know of any proof that actually uses the regularity definition that does not go through the Wiener criterion it's very interesting to have a direct proof yeah and I think that proof I have not seen that anywhere in the literature but we have it written in some notes but in metric spaces where we don't have the Wiener criterion we actually don't know I mean that's what you believe but we don't know and it's kind of hard typically you fix a p and then you do all the theory how what is happening when you change the p is a kind of different matter so I also have a really question it's about the case when p is bigger than n so do you have like more precise continuity like colder continuity or dipshits continuity? yes I mean I just general question I mean even you can in a case like it's a problem of embedding I mean that this is related with the problem of embedding but in other case that's a critical question was p less than n took a long time to complete but the important thing is the capacity because I mean overall this is explained well so regularity is a bound point is decided by pd and geometry of the bound so you work together to create this regularity so capacity depends on p and that basically was how operator influenced this geometry so that's basically what is what p less than n but p bigger than n so embedding just continues but depending on the size of the complement you can get estimates for how fast the solution attains its boundary and models of continuities exactly this is an estimation basically in the case when it is irregular then he has a model of estimation for models of continuity and how fast it attains another question if you ask a question about integration to find the criteria for colder continuity not just continuity, colder continuity operator and geometry is a brought to business any other questions? when you say that in a hexagonal group north pole and south pole like a regular point you mean a cc pole right? it's a pole generated by a kind of ordinary matching not the gauge pole so the pole is looking like an apple shape it's sort of that I'm not an expert on Heisenberg group but we found this result somewhere in the quote he did other questions? in metric spaces how far you go with a validable function is it the wrong solution or a validable function? I mean you can define the Perron solutions for arbitrary boundary data right? I mean do you take just but when they match it's a question basically I would say that this result here that if you have now if this is the Sobler space in the metric setting maybe you don't really have to impose that it's in the global Sobler space but a local Sobler space and you take a quasi-continuous representative I mean from this result Perron and lower Perron solutions agree and that covers basically what more or less all the cases we know when they agree also on RN there are some more results around here where I obtained also some results where we have that they agree but basically on RN and in metric spaces it's the same thing I mean you have this and then you can also use that together with uniform convergence you have a sequence of functions that converge uniformly and all of them are resolute and the limit function will also be resolute so in this result you can get to all the continuous functions because they can be approximated uniformly by Lipschitz functions which lie in the Sobler space and that's actually how we prove the resolutivity for continuous functions but that's basically beyond that we don't really know On the regular domains you can also take arbitrary semi-continuous functions and bounded semi-continuous functions on regular domains are also resoluted so that was, I should have actually said that here when we looked at this Perron solution so this is a lower semi-continuous function we knew already from the book by Heinon and Kippler and Martin from 93 that this or maybe even before that that the upper and lower Perron solutions agree here and also this one is an upper semi-continuous function we knew that the upper and lower Perron solutions agree so we knew that we have one Perron solution there and one there I forgot to mention that but whether they agree that was open for quite some time Bernstein actually asked this question to you Heinon who said that what is in their book was what was known as the state of the art at that time and it did not answer that question so concerning this Perron solution and solid solutions Littmann, Stamperkeva and Berger for elliptic equations I think they also broke the solid solution and Perron Stamperkeva, Stamperkeva, Stamperkeva, Stamperkeva, Berger they looked to uniform to take with bound to measure the coefficients which includes Laplace and Laplace they looked exactly in solid solutions and the double solution I think they were the results of the same Okay that could be I mean I've never definitely seen results in that direction but I think that's a good function which is also continuous but I mean that is for instance in the book by Heinon and Kierkelein and Markiut they have a solid function which is also continuous on the boundary then the two solutions agree and they use this as a tool to prove things This book is actually followed by Littmann, Stamperkeva and Berger and because after that also was a weighted case in the paper by Gelsen and Stamperkeva then the book was kind of built on that because it was exactly the solid solutions and continuous also solid solutions so binocularity but the base of the solid binocularity is the same Okay I'll look back at that paper from 63 and C if I can find it in there Good Other questions? Just a small question So you mentioned that as a gap in the theory this Wiener criterion for metric spaces what's the obstacle there extending into metric spaces? I mean it's constructing the so you have one the sufficiency part that if you have it equal to infinity but in the other one you have to somehow construct your example showing that it's irregular and what one are using to do that are both potentials and they are based on the vector structure so without the vector structure and the equation we're kind of without that tool and that was one of sort of it was a longstanding problem as I showed the sufficiency of the Wiener criterion for PMO like functions in 1970 and then it took until 1994 when Kilpelainen and Mali realized how to construct the examples what Kilpelainen said was that basically you have to find the right test function here so for the vector structure and for the minimizers without the structure that is the necessity as well but there's a wrong power not 1 over P minus 1 but it's 1 over P but that's a necessary condition there is a gap between those thank you any questions so let's have to speak again