Okay, so what you would like to do after we have looked a little bit at explicit solution

for the wave equation last time, where we have seen that the wave equation or the solution

of the wave equation behaves differently than the heat equation in the sense that the solution

does only depends on a portion of the data and this information is only transported with

finite speed of propagation, basically given by the constant in front or square root of

the constant in front of the Laplacian.

Everything which is different for the heat equation we would now like to see what about

a general existence result and we prove existence for the wave equation or for a boundary value

problem for the wave equation almost by approximation of course but differently as we did it for

the heat equation namely we use a Gajalkin method.

So we do not discretize in time as we did it for the heat equation getting a sequence

of elliptic boundary value problems forming piecewise constants or piecewise linear functions

in time and correspondingly in space out of that now we keep the time as it is but we

discretize the space.

So the equation again is in some sense a general equation the important thing is only that

we have here the second time derivative and here we allow for a general elliptic operator

which can of course be the minus the Laplacian but also for example also an operator not

being of second order for example it could be Laplacian square the B Laplacian which

will then come up if we look at a vibrating plate not a vibrating membrane and I will

explain a little bit the genera- so again we can think of our example of course our

example of course would be L is minus the Laplacian and our corresponding spaces is

V is being the H1 0 of omega meaning that we are looking at a problem with homogeneous

steric clay conditions and then we have this pivot space L2 of omega but this is only an

example and a general situation which should be allowed and which we can handle in one

step so to speak is a Gelfand triple that we have a Gelfand triple without compactness

so a Gelfand triple of a space V H and the dual V prime without needing compactness

that means the following H is in Hilbert space V is reflexive and separable this separable

is very important in the following of course it would not be separable it would not have

a basis to approximate it with a then finally countable set that's basically what it says

this condition and the norms are correspondently denoted with the indices and then we have

the embeddings V going to H this embedding is supposed to be of course continuous and

dense but we don't need compactness as we would need it for example in the parabolic

case and then correspondingly we have the continuous embedding in the dual space so

these spaces could be those but they could be also in an example of for example of Laplacian

squared we would have here something like the H2 and good question what do we have here

probably also the L2 but I'm not so hundred percent sure okay so and the theorem is now

the following it first deals only with existence but it will also bring us an energy conservation

which is very typical for the heat equation and from this energy conservation it's not

only an estimate we see basically the same way as for the parabolic equations that there

has to be also uniqueness so we have the above okay of course I should still say what would

be the assumptions on the L the general assumptions on the L and the L is so to speak related

to a bilinear form and this bilinear form is supposed to be is supposed to be coercive

and symmetric I think we also need symmetric so we only have really diffusive forces no

advective parts coercive and symmetric in V meaning I think that is very clear to you

meaning that this bilinear form in uses a norm the energy norm which is equivalent to

the norm of V coercive symmetric and continuous so we have the estimates so basically what

we need up to the symmetry what we need in the lux-melgarm theorem so we have the estimate

from below au u is greater or equal than some positive constant norm u square and we have

to estimate from above modulus of au v can be estimated by some constant and norm of

u times norm of v and these both together give the equivalency of the induced energy