 So thank you very much, Eliott, for this nice introduction. Yes, I must say I'm really happy to be here. It took a bit of time because actually, I was here visiting professor a few years ago, but then it was during the pandemic, so I couldn't come. Now a bit of delay for the typhoon, but finally I'm here. So I'm really fascinated by this wonderful institution and the whole region here around. So it's really a pleasure. And yes, I will be talking about the team films and a particular review of some results. I was told to give an overview for a general audience. So in particular, I'm talking about the crystal growth of a material, the film material, or on top of another material, the substrate. This can be done in different ways. In our setting, we are considering these two materials, the film and the substrate, and in particular, considering the fact that they can have a different reference lattice at their freestanding equilibrium. And so we have the situation of a vapor that is considered as a reservoir of atoms that are depositing on this substrate. And so we have the substrate vapor interface. And the more atoms are deposited, we can see the film vapor interface. And the substrate vapor interface is covered by the film substrate interface. So this is like the setting. But as I was saying, this will be a review of results from a mathematical point of view. So not numerics, not experimental results, but the objective will be the derivation of variational models that can be treated in the mathematical framework, in the framework of continuum mechanics. And also the justification of those models starting from a microscopic point of view, starting from atomistic models. And so what is called the discrete continuum passage. And then I will have a look of the results about the film geometry, regularity. And at the end, if I have just a reference to the evolution results and some extension to a broader family of problems. So I'm going to say these are some results that, in fact, I include some setting for my PhD thesis as I started working on these exactly at Carnegie Mellon. And with these collaborators, Elisa Davoli in Vienna, Igor Vecic in Zagreb, and then Shokolmatov, Leonhard Kreuz and Francesco Sapia that were through the years hostdocs in my group. And Randy Jurena, that is currently a PhD of mine. So OK, so let's make a step back in the theory, starting from as we are looking for the optimal shape of a crystalline material on top of another, let's look back on the theory about the optimal shape of crystals. So this states back to over a century ago when Gibbs postulated that the surface energy of a crystal would like to have its surface energy minimized. So as the atom is on the surface, means the bonds, you would like those to be in the less amount, the less number. So in a way, this means that surface energy is minimized. And then is a wolf that conjectured that this will be achieved when the distances of the faces of the crystal, this is a picture from his paper, are from a proportional, the distance from one fixed point are proportional to what we would call now surface energy. So this was then proved by Von Lauer, and herring among the sets that were considered by Wolf, so convex polyhedra, and then extended more to the general sets, first by Taylor for boundary sets, and then for the existence, for the uniqueness. So extension to general set, meaning set of, I will refer to this later on, set of finite perimeter that basically we can say set for which the depth boundary that is measurable and with a finite measure. So in the modern terminology, we can rephrase the problem by considering a surface energy defined for these sets that I consider in RD, where this is the dimension, because some results would be, obviously the physical relevant measure is the tree, but some results will be available for two dimensions. And therefore, mathematical point of view, we also consider all dimension in certain case. So I let it be three, d bigger than one, and the energy basically lives on this boundary that is called the reduced boundary is the corresponding notion of the topological boundary in the framework of geometric measure theory. And what does it means? It means that we are considering those points on the boundary for which we can define the uniquely the normal to the set in that point. And then we have the surface tension gamma that depends on the orientation of this normal. And so the integral is of with respect to this, that represent the measure d minus 1, since we are on the boundary of the set. So this has comes from the postulate of Gibbs. We want to minimize the surface energy. So this has a variational nature. This problem has a variational nature. In fact, it is basically the anisotropic version of the isoparametric problem that is considered the beginning of the calculus of variation. So anisotropic version, because we have this density that depends on the orientation. So what is your isoparametric problem is the problem that is known also as Dido's problem. It can be formulated as the Dido's problem. So the problem that was encountered by following the legend told by Virgil by the Queen Dido when she was asking for a land to then create the city that was then founded there, the city of Cartage, when she was asking to the local leader, she just asked for the land covered by the hide of a bull. And so she got the land, but then she was smart enough to cut the hide of the bull in stripes. And then she faced the problem of finding the biggest area that is covered and that is encircled by the stripe. And so this is another formulation of the isoparametric problem. In her case is the circle. In our case, since we have this density that depends on the orientation, is actually the wolf shape. So the theorem by Forsenker and Forsenker-Muller, in this more general terminology, then says that that surface energy attains a minimum among those set with finite perimeter and the fixed volume V that is a dilation by a certain parameter lambda that is chosen to adjust to the volume constraint of the wolf set. And the wolf set is the one conjectured by wolf that depends on the orientation of the surface tension. And this is actually the unique minimizer. OK, so this is about the optimal shape of crystals in general. But what about in the situation that we are concerned, in which we have also a substrate? So how does this change in the presence of the substrate? So then it was in the bottom that did the following his experiment, did a phenomenological construction in which he was not only considering the film in isotope, but also the substrate with stability. So the capacity of the film to cover to adhere to the substrate. And so in this situation, there is also the possibility for the film to spread on the substrate and to cover it. And this will be then when there is a film layer covering the substrate, the wetting regime, if the substrate is totally wet by the film. So in this setting, let's define the substrate as enough plane. And again, we consider set of finite perimeter that now are contained outside the substrate, the volume constraint. And then we consider the surface tension and isotropic for the film vapor interface gamma F. And then gamma S and gamma Fs for the surface tension of the substrate vapor and the film substrate interfaces, respectively, that we consider here isotropic. So in the energy considered by winter bottom is this one, in which we have the previous energy as before. But then now it's living on the reduced boundary when it's not in that is the portion not intersecting the surface of the substrate. And on the surface of the substrate, we have another weight that is given by this substrate with ability parameter. And so the minimizer constructing is the winter bottom set. So it's a deletion to adjust to the volume constraint of the winter bottom set that is the wolf set with respect now to this gamma F surface tension cut it at a certain level that depends of this parameter. So something that look like in this picture. And in the paper from winter bottom, these are the different regimes that one can have from non-weighting at all in which we have the wolf shape and then cutting more and more to arrive to the extreme situation of total wetting in which we have an infinitesimal layer of atoms. So is this the end of the story? Now we have this optimal shape. Well, there is something we still didn't consider that I said at the very beginning that is the influence of these lattice mismatch between the of the two materials. So it's true that on one side, this surface energy induces this shape would privilege this winter bottom shape. But on the other side, the presence of the lattice mismatch between the optimal equilibrium lattices of the two materials instead creates a large stresses in the film. So the atoms of the film move from their equilibrium to release this elastic energy. And there is a formation of corrugation and isolated islands on the profile of the films. So the possibility are much more. We can have islands on top of the subset that is exposed, islands on top of the wetting layer, or a growth layer by layer. So there is a competition between these two mechanisms, between the surface energy effect and the bulk elastic energy. That is actually the feature of a broad class of problem that we refer sometimes as stress-driven rearrangement in stability, S-T-R-I. OK, but actually it's actually very important, it's very good, because through these multiple possibilities, actually we find that's why team films found so many applications through the years in different fields from optoelectronics, semiconductor devices, photovoltaic devices. And in all, the important thing is to be able to predetermine the shape of the film. So the questions that we would like to answer are what is the determination to have a wetting layer or not, when the subset is, in other words, exposed, covered or not. And this in the literature is related to the angle. So the contact angle formed by the profile of the film when touching the substrate, here denoted by alpha. So another question is to understand how big and what determines really this contact angle. From the literature, this is related to the Young-Dupre low that tells us that the cosine of this angle is given by this ratio, gamma s minus gamma fs, divided by gamma f, this Young-Dupre low. But this was actually introduced in the context of sysile liquid drops, so in the context of fluid mechanics. And it's a question, it's applicability in the context of film crystalline films. So how much the stresses that can be developed in these corners that can be also very, very sharp impacts these angles. From the experimental point of view, as far as I know, this is very difficult to see as these were some methods from a collaborator of mine in Vienna, Rick Debold, in which she had all the tools. But in the end, she had to do some approximation of this angle. So actually, instead, we would like to do this from a mathematical point of view. So using some theoretically, and so from using some variational models. OK, so in the context of this SDRE theory, in the context of these stress-reminar arrangement and stability, we find this model by Spencer Treshev, Spencer, for the tar variation, and there are two types. The sharp interface model, F0, in which there is this continuous transition between the properties of the film and the property of the substrate. And then the transition layer model in which a layer of thickness, small thickness delta is introduced at the interface between the film and the substrate to regularize these properties between the film and the substrate. So how is the mathematical model? So we have pairs of configuration, first of all. So U and H. So H is the height function that represents the height of the film. So here we are assuming that the profile of the film is parametrizable as the graph of the function. Then omega H represents the region occupied by the substrate and the in light blue, the region occupied by the film that is omega H plus, where that is subject to the volume constraint that we have that I considered before. And then we work in the context of in the theory of small deformation. So we have the U, the variable related to elasticity, so the displacement with respect to the elastic equilibrium of the material. And enough regular even in a weak way so that we can talk about the gradient of U. And so considering the symmetric part of the gradient that in this theory represents the strain to which the film is that is related to the film. And then to represent the fact that we have a mismatch between the lattice mismatch between the two materials, we introduce a mismatch strain is 0, that is 0 in the substrate. So the substrate is at equilibrium. And instead it depends on the parameter is 0 related to the lattice mismatch in the film to represent the fact that the film is strained. So this is 0, in fact, appears in this elastic energy. So in principle, it would be the minimum of the U that it's able to be such that E of U is equal to E0 would be the minimum of this elastic energy. But this is not really possible because being H1 log means that U might need to be also continuous through this interface. So OK. And then so this is the elastic energy that depends on E0. And this W0 is elastic density defined with a spectra tensor that is discontinuous in the film, takes different values in the film and in the substrate. And then we have the surface energy that actually is the same as the one that I introduced at the beginning, the Witter bottom energy up to adding a constant. And we know that finding the minimum with respect to a function with plus or minus a constant is the same. So this is exactly S of omega plus H that I introduced before. OK. So this is for the sharp interface model. So we have a clear discontinuity at the interface between the film and the substrate, both for the surface tension and for the elasticity tensor. Then there is a transition layer model that we don't enter in too much in the definition. It's basically the same energy in which we replace 50C0 and E0 with the other quantity depending on delta. And thanks to an auxiliary function that is interpolating between the quantities of the film and the substrate in this layer depending in the extra variable. But so it has this form. OK. So we have a model. We would like, first of all, to as a mathematician to see if there exists a minimizer. OK. And I will talk here in general to the true model for delta bigger than 0, the transition layer model, and for equal to 0, the sharp interface model. So as I said, this has a variation in nature. So it calls directly for applying calculus variation, and in particular, the direct method of calculus variation. What is this method? Well, it's a method based on four steps. The first is a direct consequence of the definition of infimum and the fact that our space here, x-slip, is non-empty. So we can find a minimizing sequence. What does it mean? A sequence in x-slip whose images approximate the infimum of the energy. And then we have two steps, step two and step three, then somehow are competing one against the other. So solving one can create difficulty in solving the other. Why? Because another way to see the direct method of calculus variation is to the problem of finding a proper topology for which we have both step two and step three that works. What does it mean? For step two, we need a very weak topology. And for step three, we need a strong topology. So we need to find the compromise between these two needs. Why? Because in step two, the step two consists in proving that the space in which we are applying the method is pre-compact. So there exists, we can find a subsequence of our minimizing sequence that is actually converging to some h-bar, u-bar configuration in the space x-slip. So we would like to have a topology that is weak enough so that there are a lot of convergent sequences in the space. And then instead in step three, step three consists in proving that f-delta is lower semi-continuous with respect to the same topology. So here, if we have a lot of convergent sequences, we need to prove this inequality for all of them. So it's easier if we have less of those. So the lower semi-continuous means that f-delta on the limit is less or equal than the limit of the f-delta, the function applied on each element of the sequence. But if we are able to find this topology, then step two and step three are done. And we arrive to the conclusion, basically putting together the other, what we obtained. So we have that the image of the limiting point, h-bar, u-bar, we apply lower semi-continuity by step three. So we have that is less required than the limit. Then this was actually a minimizing sequence. So by uniqueness of the limit is equal to the limit of f-delta. And then since by definition in step one, since it's a minimizing sequence, this limit will be equal to the infimum. It is obviously less than the value on any point that we consider in x-slip. That's why we need that this limit is in x-slip. So given this, then by the squeeze theorem, we have the existence of minimizers. This is basically the proof that it's the same idea of the bias stress theorem. So there is a problem though, in the space that we are considering, actually step two failed, we cannot really prove that it's pre-compact with respect to the topology. So we cannot conclude in the end that we have directly this minimizer. And this is because by the compactness, by the bounds that we have with our functional, we can have some leap sheets profile. I don't know, there can be some corners, but in particular we can have some situation like this, also very smooth, but with some like ballet like this. And then this ballet in the sequence could actually shrink. And in the limit, what we are going to get is for example, something like this, like a cut. And this is not anymore a leap sheets profile. So that's just without entering in all the discussion here, the actually the basis of the effect that we cannot always get a limit in the x-slip. But actually by applying this theorem and this observation, we can prove that we have convergence of up to extracting a subsequence to a configuration that has the same required regularity for the U variable, but for the H is a bit less regular than leap sheets. It's a function lower than continuous and of bounded variation. So function for which these cuts are possible, okay? And so those function are actually of these types. So the profile now can have portion that are continuous like this gamma graph portion of jumps in yellow and cuts in red. This actually is consistent with the observation which you can have in the experiment, those heels that are shrinking and informed this location at the bottom at the interface with the subset and I believe to, I think to go up and sometimes to the surface. So it's not actually a surprise if we need to extend our space to these big larger class X, but then we have as this is a something happens often in the calculus variation and then you have another problem. Our in both functionals of Delta are actually living on the space X leap. So how that now I find a weak formulation of F that is defined also on these broader class, these larger class X, but it's the same as the previous function on X or as the same, the lowest in the continuity property in this space X. So actually this is F, these functional week formulation has been found. This is the first result that I would like to present and the point is that this is found and it's the same both for starting from the sharp interface model and from the transition layer model. How does it look? So the elastic energy is the same as the sharp interface model. And for the surface energy, we have an energy in which here we don't, we had already gamma F, but here instead of gamma S, we have gamma F, the minimum between gamma F and gamma S minus gamma F S that is the weight ability. And then we have this extra term for the cut part that it's two gamma F and this is not really a surprise because here this portion was counted the gamma F and this portion is also counted gamma F. So when we approximate in this way, the cuts in the end would have the contribution for both sides. And so this is the reason of these two appearing here. Okay, so I was saying that F is found as a weak formulation in two ways. As a relaxation from the sharp interface model, what does it mean? It means to find the largest functional below F zero that satisfy basically step three in the previous procedure. So the largest functional below F zero that is lower semi-continuous so that the diagram method can be then applied. And then as by doing a convergence of the functional F delta where as this layer of thickness delta goes to zero. So in a particular way that is called this gamma convergence. So what is gamma convergence? It's a notion that was introduced by Enio de Giorgi and that commonly cognized as the way of making a convergence between models. So like here in models in which we have these layers to a model in which there is no layer. So the layer disappear. So and why this is considered the conversion to use is because it has a very good property. It assures that the accumulation points of a sequence of minimizers. So let's suppose that U delta bar H delta bar is a minimizer of F delta for each delta. Then if these as an accumulation point or it converges to something then this something will be a minimizer of the other function, the limiting gamma limit F. So it's not only a convergence of energies it's also convergence of minimizers. So when you can find when you have the compactness it's really approximate for delta small you can say that the model F delta approximates the model F and vice versa. Okay, so in both these ways we always get to know the same functional that is here described. And then there was another way we deduce the same functional that is in this other paper that is those with the different settings starting from discrete models. So and here it's again by gamma convergence but not applied to the transition layer but to another quantity. So here we start from atomistic models that depends on the reference lattice that is here taking the triangular lattice with an epsilon parameter between the that is the distance between each side in the reference lattice. And so we perform the gamma convergence by sending epsilon to zero. Okay, so by letting the ender scaling the energy. So by letting the epsilon the atoms the distance between the atoms go to zero and notice here that we don't have these locations. So it's a regular homogeneous lattice between the film and the sub is right. This is the reference because then after we will see. Okay, so let me just give a key feature of these other procedure. So we have these reference lattice and epsilon on these we define an energy. Okay, again, this energy will depends on two on a pair of two variables. One H that is the deposition height but in the discrete these counts basically the pi the number of points in these columns. And then we identify it with the lower semi-continuous piecewise constant interpolation of these columns. And then we have the elastic variable this discrete orientation preserving the formation. Actually this orientation preserving assumption for this in the discrete is questionable. There are possibility of some papers in which they avoid these in other settings. So this could be a way to try by considering not only nearest interaction but also next to nearest interact atomic interaction. So, and anyhow, this variable is also identified with the affine interpolation in the triangles in the lattice. And through these we have these other energy that it's a discrete energy depending on the lattice that is given by the sum of two big contribution. The contribution of the atomic interaction between substrate atoms, okay, that depends on a potential Vs. And in green is that the interaction between the film atoms here that depends on a potential Vf epsilon. So what are these potentials? We consider like Lena Jones type potential cut at a certain level. So there is a well that in which we have a minimum value gamma S for the substrate interaction and gamma F for the film interaction that is reached at a different distance that bond the optimal bond distance for the substrate here normalized that one and for the substrate for the film at lambda epsilon. So lambda epsilon minus one is the lattice mismatch between the film and represent the lattice mismatch between the film and the substrate. So the reference lattice is the same is homogeneous but the substrate atoms are happy to be there while the film atoms would like to actually be at another different distance lambda epsilon. And this is seen in the energy. Okay, then by proving the gamma convergence we can prove for epsilon that goes to zero that these functional actually converge exactly to this F functional that is so further justification of these functional. And in particular E zero this mismatch strain can be seen as a convergence of here of these, it takes this form. So in which it's zero in the substrate and the Eta for a certain rotation in the film where Eta is the limit with this parameter depending on epsilon on of the discrete lattice mismatch. So to prove this, we need that this limit converts to a real number. So actually as an hypothesis of this conversion from the discrete to the continuum we have that we need to be in a situation of small lattices, lattice mismatch. So under this hypothesis we have this convergence we have this mismatch strain and we get the convergence of the atomistic model to the continuum model. There is the general setting which is for large lattice mismatch in which it is believed that not only the formation happens but there are other ways of stress relief for the atoms. And so the appearance of dislocation between the lattices. But here we didn't have this. Okay, so since we didn't allow that it makes sense that we got as a condition that we have like this is for short for not large mismatch strain. So obviously the goal will be also to address this setting and we started by considering in these two papers a parallel situation, the opposite situation of case the regime A. So in regime A we have the formation but no dislocations. Here we have the opposite. We don't have the formation but we have dislocations. Okay, so in the attempt to reach then to put together everything. And actually this is a bit of the regression because the limiting functional here wouldn't be the energy for team films but the winter bottom energy since we don't have the formation so we get just the surface energy but I would give just an idea of this because actually this is what I did also in that period in which I was formerly even though not present here but I was visiting OISD. So we expect to get in the limit the winter bottom energy. So let me very briefly see what we do in this case with these locations. So first of all here then we have two lattices LF and LS for the film and the substrate and there are then multiple positioning of these lattices and the first observation is that we can reduce to two settings M0 and M1 and we can see that all other positioning are equivalent to these ones. Okay, and I will just refer to M0 for simplicity now but all the results obtained from M0 are with different constants available for M1 and so for all positioning of the lattices if they are not interpenetrating. Okay, so here in particular there are these constant EFS that are presented distance between the two lattices. ES is the optimal distance of the substrate lattice atoms that are it's all filled by atoms, the substrate and then EF is the distance we do in the film lattice and here instead we have a certain configurations of atoms, so yes, that are called like DN, okay. And so also here we consider a discrete energy VN that again is given by two big contributions. So the contribution between film atoms and the contribution between film and substrate atoms. So and here since we are not considering the formation we just consider stickiness potential. So they are, they don't have a well but they are plus infinity then minus, they reach a minimum in minus CF or CS at a certain distance that we saw is EFF or EFS, okay. So this is the situation in this setting with this location and again also here we have a problem with step two. So we have a lack of compactness and in particular we have this problem with respect to two issues. One is the factor of the possibility of having a wetting layer. So a possibility that actually we don't get the winter bottom shape because we get just a line of atoms. So in a set we don't fool the big measure. And the other is the possibility of that we get a cluster of atoms that escape at infinity when the number of atoms tends to infinity. So for the first problem we need to define the wetting threshold under which we get the winter bottom shape. And for the second we need to establish a certain mass conservation in the limit. So this is the result with respect to the first problem. So we find this condition that is necessary and sufficient for having as minimizer configuration that are on the first, that are just on this first layer of atoms, okay? So as a confirmation of the, so they tend to the wetting layer. And so we need to put ourselves in this other, the opposite, so the wetting situation that as a note we observe that is less restrictive from what we would get just using the condition that you get from continuum problems. So there is more complexity in the discrete level. Okay, so under this condition, and here I will jump. So the topology that we are considering is actually the weak start convergence with respect to other measures. So we see an equivalence of our problem with respect to these atom position, but instead we state these with respect to measures, but let's keep these and we can prove that we have mass conservation and gamma convergence. So mass conservation meaning in this way. So if we consider a sequence of minimizers and we take for each of them the component, a component with the largest cardinality, then these components we converge to, we take all the mass in the limit. So we have a sequence, the sequence of the components with largest cardinality is converging to it takes all the mass in the limit. And so these replace the lack of compactness that otherwise we have. And then through that topology that I described, we see the convergence to this energy that as you see is exactly as the initial energy that I introduced with the fact that these constant are now characterized with respect to the minimum in the interaction, the atomic interaction. So with respect to CF and CS and Q is the parameter of how the lattice are positioned and here CF. Okay, so this was a digression with respect to what I did in that period I was here at OISD. So, but now I would come back to actually this functional F that has both that energy but also these elastic energy. And since now with the direct method of the calcule variation we have a minimizer, we would like to understand how this minimizer is. So to find some properties of this minimizer. So the first property is this one. So we could prove the internal ball condition. What does it mean that we can follow the profile of the fin with the ball that touches the boundary and as a radius small, but non-zero, so fixed. So this is what implies. It implies that since I need to be able to follow these cuts cannot accumulate. And so the cast points, so this one and the cuts cannot be accumulated. So they are only at most finite number in a interval. And apart from those finite points, we can prove that the profile is locally leapshift. So to sum up, we started from the models from Spencer, Spencer-Tashev, in which the profiles were considered the leapshifts. We couldn't find a minimizer in that setting. So we had to enlarge to function that has also cuts. And then here we recover a bit of regularity by saying, okay, but up to taking away a finite number of points, this is actually locally leapshifts. Okay, so this is a natural processing calculus variation. You enlarge to prove the existence and then you try to recover regularity of your profiles. And then thanks to this and reduce passing to the situation of isotopic material with lamecoeficients that are discontinuous in the film and with respect to the substrate and assuming this condition between the lamecoeficients of the film and the substrate that basically it's classical in transmission problem. So it means that it should mean that the subset is more rigid than the film and this I think could make sense with respect to team films. We can prove that actually the young to pray angle apart from those cast and cut is respected. So the angle with which the profile touches the substrate is the young to pray angle. So it's the cosine of these and it can be zero when gamma f is less than the weight in this when gamma f is less than the weight ability or otherwise is a positive angle. And these are both in the valleys that here are denoted by the family points of valleys VH and in the island border. And in particular, we can also say that there are no valley when sigma is less than one. So this means that valley in valleys the point the angles is always zero and that jumps that touch the substrate are appearing only when the young to pray ideal angle given by the surface tension is 90 degrees. In the cast and cuts, we didn't prove the same but we proved before that these are only in a finite number of them. Okay, this is the proof I will skip and then we can arrive to further regularity by doing a bootstrap argument passing to the Euler Lagrange equation apart from those points where we have cast and cuts the finite number and the contact angle that are different than zero. And then one few minutes just to conclude to make a reference instead to how much time do I have? Yes, I think I should conclude. So about the evolution results. So notice that if formally we consider these reduced energy that depends only on the profile H. So in which you is actually the equilibrium elastic equilibrium depending on omega H then formally we can consider this gradient flow with respect to a certain metric D and with respect to the choice of the metric we can, this actually represent the surface diffusion process so of the automated surface of the film or the evaporation condensation process. More specifically, if we consider the H minus one metric then we have the surface diffusion process and the L2 metric we have the evaporation condensation process. In this regard, it was achieved analogous results in the true setting. So by Fonseca Fusculani-Morini, first in 2D and then in 3D and this was actually part of my thesis and now we in two dimension and now we did the three dimension. In both setting, we prove short time existence of a regular evolution and the stability of Lyapunov or asymptotic type of the flat configuration. So if we start with the initial configuration close to the flat configuration then in this model we could prove that we remain close to that flat configuration. Though these both were done under a curvature higher perturbation. So there is an extra term added here in the energy that basically penalized corners on the profile. So I don't have time to go more in details for this. And then there is a direction of work on extending these results to the general family of SDRE. So to take out, so we took out actually the graph this assumption on the boundary that is a graph or other boundary constraint first in 2D and now recently in 3D. So I mean in any dimension actually it is in any dimension in which so we don't have graph like a boundary constraint. We have an isotropic surface tension for both material and we have the delamination. We have the possibility of delamination between the film and the substance. Actually this is an extension, but those need a need because otherwise we don't have lower semi-continuous as we could have a component of the film that is falling on the substrate and there this could create problem in the lower semi-continuity. So this then becomes a very general model that can include a different application not only team films, but crystal cavity, calpillarity drop, delamination and cracker setting. In the situation with down the mismatch strain but with a Dirichlet condition and only in the wetting regime there is also this result by Chris Malle and Friedrich. And very recently we're about to complete these work on which we actually let free the substrate profile. So it's not anymore like a fixed set but it's a free graph. So the film is a set of five perimeter and the subset is a free boundary that is parameterized as the graph. And so these are the works that in a way I try to summarize and to put together for these presentations and with these I thank you for the attention. Thank you very much for a highly informative very a lot of information. Clearly done, great deal of work. Are there any questions? Ah, yes. No, actually there is no restriction about the scene so it's for film. So there is not a restriction with respect to the film it's free, yeah. Are there any other questions? Do you have anything? Is in this L, in this setting here for the subset is meant to be longer, bigger than like this other dimension but in this way one could consider that it's thin if you. You mentioned orientation preserving deformations. Yeah. I was wondering, I did not understand fully like the importance of it and if you let go of that. No, no, this is a technical requirement that now, yes, this one that I mean it can be considered a non-interpretation condition but so for discrete material for atoms it's not so good. So actually I was saying that this is a technical needed hypothesis that could be avoided by using some of these techniques here and the difference with respect to this work and these other work that are different setting though so is that here we are considered only first neighbor interaction of atoms but by allowing also for not only the next atom interaction with the next atom but the next two nearest neighbors. So not just the one in the reference lattice that are bonded but also the next one, this one for example for this one, then one could think to avoid this hypothesis but it's something we should try to do. It's not, it's just an idea of what we could do to avoid this hypothesis. Other questions? Yeah, there is a reference lattice in this way but so the fact that here we have a well and the energy is depends on the difference with respect to the displacement with respect to the site. So the reference lattice is I and J, so it's fixed and then we have a displacement Y from those reference and the interaction that we are paying is like with respect to this well, with respect to the formed. So with the displacement from that, but yeah. Maybe I would add to that, I think the underlying idea is that there is a separation of time scales so that the time scales associated with the vibrations of the atoms about their lattice sites is much faster than the time scale associated with, let's say the deposition process or the growth of the film. And also you have no inertia. Yeah, no, it's a simplified setting for sure. And yeah, it goes, it's a... Questions from the, yes, go ahead, please. Could you explain a little bit more about the curvature, higher order resolution? Yes. My understanding is that curvature is second order. So what do you mean by higher order? Yes, it's basically we add to the functional, the integral of the square of the curvature. So this gives a contribution and the four, so because for the L2, this becomes of the fourth order, the equation. And this then with the curvature regularization is of the sixth order, okay? And yeah. Wouldn't that go back to the idea of next nearest neighbor interaction? Oh, in this, for evolution, yeah, we are not considering from the discrete to the continuous, so. But in essence, when you have the gradient to energy, what you are accounting for, let's say from a discrete perspective is the presence of next nearest neighbor interaction. I see, okay, okay, interesting. This goes back to Fermi-Pasto-Ruzan and many others. Okay, okay. That's it. No Zoom questions. Thank you very much. Paolo, thank you for an excellent and as I said, very informative talk. And please, how many more days will you be here? Yes, I will next week also. Next week. So I encourage people to meet in person with Paolo. Thank you. I would be very glad to have an interaction. Thank you.