Let us come now to some examples of other variational integrals.
The first generalization of the Dirichlet energy are these quadratic integrals,
which are basically a generalization of the Euclidean norm,
that is the Euclidean norm squared of the gradient of W integrated over omega.
Here instead we're integrating over omega a bilinear form on capital N times small n matrices,
evaluated twice on dW,
which explicitly can be written as this double sum over lm from 1 to capital N,
i.n.j from 1 to small n of aij lm times the integral over omega of dI wL,
dJ wM for W integral over omega with values in Rn.
In the case a is symmetric, and these functions are related to a system of linear PDEs,
which are given by sum over l from 1 to capital N, sum over i.n.j from 1 to small n,
aij lm dI dJ wL equal to 0.
In order omega for every M, here is the indices, which gives the effect that this is a system of equations.
And notice, in the case aij is the Kronecker delta for i and j,
and in L, right, the identity matrix in the other two components, this should give the Laplacian.
Yes, yes, yes, if a is simply the tensor of two identity matrices,
we get here the Euclidean norm, and so of dW, so we get the Laplacian again.
A generalization of these type of equations, and therefore this type of variation integral,
is given only in dimension capital N equal to 1 for the dimension of the values of the function,
just for simplicity. It's given by the integral of omega of 1 half, the sum over i and j from 1 to small n,
aij dI wdJ w minus 1 half C w squared plus f w dx for wd1 with w1 true of omega.
In this case, the integral function involves also x and y, indeed, because it's 1 half the sum over i and j from 1 to n,
aij of x, zi, zj, in this case z is a vector, 9, minus 1 half C of x w squared plus f of x, w, w, that is the number.
And aij and C are in L infinity, and f is in L2 and above constant, coefficients of this function, integral function, not constant.
In the case, aij, the aij's are indeed the same, symmetric.
And f is related to this type of second order linear pd, which is a divergence form second order linear pd with 0th order term and the given data f.
Another generalization of the Dirichlet energy, if you want, in this way we are going to, first we put another type of metric to construct the functional.
So instead of the classical Euclidean scalar product between dW and dW, we get a newer one.
Then, we not only do this, but we subtract some other, we add some other linear terms, sorry, not linear, other terms not involved in the derivatives.
The other possibility is, instead of a power 2, consider another power. So define the p, Dirichlet energy for p between 1 and infinity, through infinity, as Ep of w, which is 1 over p integral over omega of the Euclidean norm of dW to the power p.
So in the dLP norm of dW to the power p divided by p. And of course the integral function is norm of z to the power p divided by p.
In the case p equal 1, the one energy E1 is also called the total variation, and this has a lot of applications in the geometric measure theory, sorry.
It's strongly related to the notion of BV function and to the problems of image reconstruction, that is all to say.
In general, the function of the p energy is related to the p Laplace equation, or system, in the vector case, that is the divergence of the Euclidean norm of du to the power p minus 2 times du is equal to 0.
Notice that if p equal to 2, this is the Laplace equation.
And if p equal to 1, it becomes very funny, because it's the divergence of du over the norm of du, which is a vector of length 1.
So what is varying is only the orientation, so it's very interesting.
If p is between 1 and infinity, we can define also a non-degenerate p energy.
By taking, instead of simply the norm of du to the power p, we take 1 over p, integral over omega, 1 plus the norm of du squared to the power p over 2.
In this sense, it's non-degenerate, because even if w is constant, this is not 0.
And by the way, of course, this is well-defined if omega is a bounded set, of course, because otherwise this is always plus infinity.
And the integral here is 1 over p times 1 plus norm of z squared to the power p over 2.
And the PDE, which arises from it, is very similar to the p Laplace equation, just there is 1 plus norm of du squared to the power p minus 2 over 2.
Just because it should be, it's very similar, you see here, if you take away 1, you simplify it, divide it by 2, you get here.
Then there will be a lot of other different types of possible rational integrals, but let us just see a little more carefully what happens in the case p equal 1,
because in the case p equal 1, the geometric case again, also for the p energy, we get the non-parametric area integral.
So S of w is the integral over omega of the square root of 1 plus the Euclidean norm of the gradient of w squared.
The w is scalar, but this is the area of the surface parameterized by w, the surface z equal w of x over omega.
And so, not very surprisingly, looking for the minimum of such operator in a suitable class will give us the minimum surface equation, which you see is exactly case p equal 1 of this equation over here.
Again, similarities. The point is that, why to them to separate them? Because this problem can be solved exploiting the fact that if p is not 1, we have nice properties of the w1p spaces, for instance, the reflexivity.
If p equal 1, you cannot use them.
Let us now talk a little about admissible classes and side conditions, that is, constraints for competitors.
As we saw in the example of the D-Schley energy, the most common thing to do is to put some boundary conditions,
and the most common one is, of course, the D-Schley boundary conditions.
One could also consider the others, the Neumann, the Robin, but, I mean, as I said, Caico's operation is a very vast field, so we need to restrict our study to something, so let's consider this simpler, more classical case.
In which one is basically asking to the function w to have a specific value, say phi, on the boundary of the domain omega.
Of course, as I said, if w is a Sobolev function, what does it mean that it's trace on the boundary of omega? There is a way to make sense of this notion.
But a nice trick to get around this problem is to work on the D-Schley class A, which of course can be defined for every value of p.
So we fix some function u0 in w1p of omega, which we know by the theory of Sobolev functions, it has a trace on the boundary of omega, if omega is nice enough, but if it's not, it does not really matter, because the space A is always well defined,
because we look for functions which are u0 plus some function w, we should have, in some weak sense, zero trace on the boundary of omega, because it's in the closure of the set of Sobolev functions with compact support with respect to w1p norm.
Other type of conditions and constraints are the holonomic side conditions or equality constraints, and this is basically when we ask that the functions w are such that g of x, w of x is equal to zero for x in omega, for some function g defined from omega times Rn in Rk from circuit.
A classical holonomic constraint is asking that your competitors, your minimizers, belong all to some manifolds, the value of your minimizers lives on some n-k dimensional sub-manifold m in Rn, because it's the space of the values of w.
And again, one can ask, what does it mean this condition if w is Sobolev? Sobolev function, priori, is only LP.
Of course, since we ask that Sobolev functions are indeed not only LP functions, the weak derivatives one can prove that they have some sort of regularity, they have fine properties, and in a sense, their point-wise value makes sense up to small sets, much better than what it happens for LP functions.
That's all very nice, but in this case, one can also say w of x in m for almost every x, with respect to the Lebesgue measure.
So, up to a Lebesgue negligible set in omega, the values of w must all live inside this sub-manifold m.
Another example then will be non-holonomic side conditions, or inequality constraints. So basically, before we're asking that some function g of x, w of x is equal to 0, so somehow w solves some sort of implicit function equation problem.
Here instead, we have an implicit function inequality, and this leads us to the so-called obstacle condition, obstacle problem.
We ask that w of x does not belong to some set O for all x in omega, or in the one-dimensional case, this can be rephrased by asking that w of x, the values of w, are always above some given function psi, which is the obstacle.
And again, if the functions are Sobolev, we shall ask that this condition is satisfied almost everywhere.
Finally, but I mean there could be other examples, one can consider the integral constraints, which means we ask that not we have pointwise conditions, or almost everywhere pointwise conditions, but integral conditions, that is, the existence function g, such that the integral of omega of g of x, w of x is equal to 0 for w in our admissible class.
And a very common constraint of this kind is the volume constraint, so the integral of omega of the absolute value of w is equal to a fixed number w between 0 plus infinity.
Why v? Because this integral gives us the volume, the n plus one-dimensional volume of the set of points between the graph of w and omega times 0, if we see this set as a subset of our n plus 1.
In all of this, I'm assuming that w is a scalar function, of course.