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Theoretical basis of very general elliptical boundary values of second order.

Now I would like to make a short mention here, that in the case of the existence of the

statements, only required that the right side, we always called that B, that this is a steady

linear form and not that it is as usual, given as linear form integral f, where v then

is equal to the integral of the x, where v is equal to the integral of the y.

Here, in the Hilbert space, the same as the test space, so f here is always taken, the right side, as an L2 function, and that is generally not necessarily necessary to apply leximigram,

it is sufficient that you have a very abstract bilinear form here, as the L2 function, and that is generally not necessarily necessary to apply leximigram,

it is sufficient that you have a very abstract bilinear form here, as the one-dimensional case is possible, if you look at the direct distribution here,

that is, a point evaluation at a point omega within this interval, then it is still a H1 function,

so since v is a H1 function and in the one-dimensional H1, in the space of the steady functions, this expression makes sense,

so that is the point evaluation, that is, the direct distribution in the one-dimensional case, actually a steady linear form on H1,

that can be defined like this, and therefore you can also allow such point sources in the one-dimensional case, i.e. sources or

in the one-dimensional case that this integration is valid, in the two-dimensional case that there is a nice counter example here,

so much just for this additional remark, okay, so I switch back again to English, I want to give a short overview about the result we state, we worked out the last week,

with the solution space here in this sense, our functions have to be in H1, and we have homogeneous Dirichlet boundary conditions on this segment of the boundary gamma 3,

such that our conditions of the theorem of Lex-Mirgram are satisfied, and this theorem ensures us the unique solvability of our boundary value problem,

however these assumptions are that our domain is bounded and is a bounded Lipschitz domain, the coefficients are, with respect to the differential operator, are bounded,

and also the divergence with respect to the advective coefficient is bounded, the right-hand side belongs, is a L2 function,

and well, in the case that we have gamma 1 belongs to the Neumann boundary condition, gamma 2 to the mixed boundary conditions,

if we have one of these conditions, we also assume that the normal direction on the boundary of the coefficient corresponding to the advective term also is a bounded function,

furthermore, this is the quite usual assumption, we assume that the coefficient tensor with respect to the highest order derivative of our elliptic equation, of our elliptic differential operator,

is uniformly elliptic, that means it satisfies this inequality, and on the boundary, our boundary conditions are L2 functions in all three segments, there are some further assumptions,

and these three assumptions are important to assure the ellipticity of our bilinear form, which is a really necessary assumption, condition for the, to apply the L'Ax-Milgram theorem,

which first means that the coefficient with respect to the reaction term, minus one-half over the divergence of the coefficient with respect to the advective term,

is to be greater than zero, in the case of a Neumann boundary condition, we also assume that the normal component of this coefficient is greater than zero, and in the case of a mixed boundary condition, we have this condition.

Well, then we have some further quite technical, complicated, but not that strong conditions, they only ensure that there is a little domain or a little segment on the boundary,

such that all these assumptions are rigorous, that means, for example, there exists a sub-domain belonging to our domain omega, such that this inequality indeed is not only greater than zero, but greater than a scalar, which is indeed greater than zero, and so on.

Quite important is the last assumption, that means, if we have inhomogeneous Dirichlet boundary condition with the boundary condition is G3,

then we assume that there has to be a H1 function on the whole domain omega, such that the restriction of this W function is on this boundary, gamma three is nothing but the boundary condition G3.

So, in these cases, the theory of Lex Milgram ensures us the unique weak solvability of the boundary value problem.

Well, I want to remark briefly the case of fourth order boundary value problems, for example, we're looking the homogeneous Dirichlet problem for bi-harmonic equation,

that means, in the classical sense, we're looking for four times continuously differentiable function U, that satisfies these both equations at the boundary, we have homogeneous Dirichlet boundary and also the normal component of the derivative vanish as well,

and the B Laplacian coincides here with the right hand side. This operator Laplacian square is nothing but we apply the Laplacian operator twice on this function U.

Well, in the one D case, this reduces to only the fourth derivative, that we now have here two different boundary conditions for only one function is really necessary to ensure the uniqueness.

For example, if we skip this condition of the normal component of the derivative, so we only have this condition of the homogeneous Dirichlet boundary condition, then for example, of course, on the right hand side, F equals to zero, then of course, the constant zero function is a solution,

but also maybe on the circle in the two dimensional case, on the unit, in the unit ball, also this function is also a solution, satisfying of course, the first equation,

so the B Laplacian applied on this function equals to zero, and due to the choice of this constant and the radius of this unit ball, it indeed has homogeneous Dirichlet boundary condition.

So, however, for higher ordered boundary value problems, there is indeed, we have indeed to require more boundary conditions.

So, of course, this is the formula we already know from the tutorial that is this holds for arbitrary functions with an H2, belong to H2, and if additionally, function U indeed belongs to H4, then we can replace the function U here by the Laplacian of U,

and we obtain this function, this equation here, that the integral over the product of the Laplacian U times Laplacian V equals to the right hand side here.

The first guy here equals, due to our problem, assuming that the U is the solution of the B Laplacian, of the B harmonic equation, this integral equals to the integral of the right hand side F times V,

and due to our boundary conditions, both surface integrals here vanished.

So, that means that with this solution space, this is the space of all H2 functions, satisfying the boundary conditions, we obtain the following variational formulation.

That means that we are looking for a function belonging to even this function space, such that it fulfills this equation.

Here, our B linear form is, of course, the integral of the product of both the Laplacian of both functions, and the right hand side is, again, the linear form, the integral of the right hand side times the test function.

More general, boundary value conditions of order 2m leads to a variational formulation in Hm, or if we have Dirichlet boundary conditions in Hm indexed by zero.

Okay, so the next quite important and interesting question is about the regularity of the solution.

Often, we need more regularity, not only for our solution of the elliptic boundary value problems, not only that the solution belongs to H1,

but maybe for interpolation of the solution or in context of error estimates or other interesting topics, we are interested in if it is possible to extend somehow the regularity.