Okay, concerning the original lecture date, so this is an exception today because for

some other reasons I had to be absent from Erlangen and could not make it this morning,

so we shifted it to this date. So usually we have our lecture on Thursday, let me phrase

it in this way, in the first slot, and my proposal is, and I think you will all agree,

that we start at 8.30. Is this okay?

Yeah, later is not possible because then we will not make it up till 10, but we have this

possibility I think. Okay, so from next week on 8.30 in Übungsraum 3. So, okay, I will

do something in this lecture which I usually not do. I totally deviate from the description

of the module because this description of the module, at least from my point of view,

is far too ambitious and also does not 100% fit with this level you have reached meanwhile,

so there are much more up to the research topics, up to the research line, so maybe

this is something you can do in the third and fourth master semester and we do something

more basic. So I have decided to, as I said already, a little bit to identify a few subjects

we would like to cover. So the first subject, where we know already something about it but

not about the analysis of it, so the first subject will be time dependent PDEs. We have

seen those models already, so we have seen all the time dependent Navier-Stokes equation,

of course also time dependent Stokes equation. If we go down then to the energy conservation,

we have the heat equation and more general versions, we have the diffusion equation.

So this is one class, so the second order equation, second order in space equations

we want to deal with and then of course there is this other class, there is the first order

in space equation, the Euler equations, the wave equations and so on. And, okay, the wave

equations does not come in the form of a second order in space but a second order in time,

first order in space but second order in time, second order in space but can be rewritten

as we have seen as a system of first order equations. So the aim of this first section

is to look at procedures to show existence of solutions for those linear problems. So

first we are still in the realm of linear problems which makes the world much simpler.

Then as a second part, we will see how far we will come there based on that to some extent.

As I said we will look at the Navier-Stokes, not fully of course, and the Euler equations.

So here and then of course the problems become more pronounced. We have systems, we have

nonlinear equations and then this is so to speak the first block dealing with time dependent

equations and the second block going back to stationary equations but now really looking

at the nonlinearity. So the key word would be nonlinear elasticity. So we have touched

this a little bit in the sense that we have derived such models with hyper-elastic material

for example but we haven't said anything. Sorry, sorry, sorry, sorry. This is number

four. This is now the application. So we start with number three. So maybe you have heard

already this key word. So what we have to do a little bit is we have to do a little

bit calculus of variations. So what does that mean? We have seen that a stationary problem

can be written either in the PDE form or in the form minimising energy and of course we

can tackle both forms asking ourselves something about equations, for example the existence

of a solution. So we can either say okay we have, let's go down to the most simple equation,

the Poisson equation. We can either look at the Poisson equation directly at the weak

form, say okay this is a bilinear form, what do we know, can we check the assumptions and

so on. That would be one way to go with that and the other way would be to look at the

energy functional itself. We have a function with a Dirichlet integral to minimise and

what do we know? Of course you know already something about what are conditions for existence

of minima or minimisers in the finite dimensional situation I hope and now but now we are in

infinite dimensional spaces and we have to have similar results and considerations. So

we will go a little bit in the direction you know if we have in finite dimensions or in

one dimension if we have convex functional we can be sure that we have a minimum or a

strict convex function, we have a unique minimum, we try to generalise these notions to not