Degree of a continuous mapping

Figure degree is a tool of the non-linear analysis to detect the existence of solutions of nonlinear equations. With its help, one can prove, for example, the brouwer between fixed-point theorem, the Borsuk - Ulam or between jordan curve theorem. In the finite-dimensional ( for continuous functions ) it is called Brouwer degree; its extension to Banach spaces ( for compact disturbances of identity ) is called leray - Schauder mapping degree.

• 2.1 Compact disturbances of identity
• 2.2 Compact homotopy
• 2.3 Definition
• 2.4 Example

The Brouwer mapping degree

The Brouwer mapping degree assigns a continuous function for open, restricted and given to an integer. Critical to the applications, the fact that the equation is then solved already when the image is different from zero degrees. Disappears the mapping degree, so no statement can be made for solvability.

Axiomatic definition

The Brouwer mapping degree is a function

Having the following characteristics:

• For everyone.
• Decomposition property:

• Homotopy invariance:

It can be shown that such a function exists and that it is unique.

Important properties of the Brouwer degree

• Is, so the equation is solved on.

• It has so true In particular Figure level is set by the values ​​of unique.

• Beds and in the same connected component of, the following applies We write therefore also short for to indicate that the mapping degree does not depend on the point, but by the component.

• Let and be continuous and the bounded connected components of and, then apply the product formula leraysche which only many summands are different from zero at last.

Representations of the mapping degree

• If in addition, to continuously differentiable and all points are normal, that is, the determinant of the Jacobian matrix is not zero in these points, the following applies Is not continuously differentiable, then you can choose due to the second property of a function that has the same degree as illustration.
• Be back to continuously and continuously differentiable on, not a critical point. Be also a bevy of continuous functions from to to and select all, here denotes the closed ball of radius around zero. Then exists, so that the integral of the formula applies to all.

Number of Turns

The Brouwer mapping degree includes as a special case the important in the theory of functions of turns. If we identify with, so the Brouwer mapping degree is defined for the complex plane. A closed curve can be understood as the continuous image of. With the unit circle ring is referred to the zero point. That is, there exists a continuous and surjective map. If now, as is the term for all continuous continuations of the same number due to the continuity of the mapping degree. It is now

Here denotes a sufficiently small annulus. In particular, in order to justify the last equality sign are a few facts from the topology needed.

The leray - Schauder mapping degree

The leray - Schauder mapping degree is an analogue of the Brouwer degree of ( infinite-dimensional ) Banach spaces. This mapping degree was defined in 1934 by J. Leray and J. Schauder. However, it is not possible to define the mapping degree for arbitrary continuous functions, but we must authorize any compact disturbances of identity.

Compact disturbances of identity

Be Banach spaces and a subset of the Banach space. A function is called a compact operator if

• Is continuous and, if
• Maps bounded sets to relatively compact sets. In other words, is a compact subset of.

An operator can be represented as a compact operator is called compact perturbation of the identity.

Compact homotopy

A compact homotopy is a homotopy between compact operators. It should be open and bounded and for operator-valued function with compact operators. This operator-valued function is called on compact homotopy, if there exists one for each, so that

For all and true.

Definition

Let be a compact perturbation of the identity, open and bounded and. Then the leray - Schauder mapping degree is an integer such that the following properties are valid:

• , Then the equation is solvable.

• Homotopy invariance: Is a compact homotopy on with for all and so the mapping degree is independent of.

Example

The main method for calculating the leray - Schauder mapping degree leads, just like the Brouwer degree, on the homotopy invariance.

If one is interested, for example, of whether the equation has a solution, one first searches a suitable room, so that is a compact operator. To prove the solvability, it now takes to indirectly that on is because otherwise there is nothing to show.

Then you look for a compact homotopy with and for all and. This homotopy should be selected so that you can demonstrate for the leray - Schauder mapping degree. It follows from those for all and thus the existence of a with.

For a concrete example of the initial value problem is

For and given. It can be shown that it has at least one solution, is continuous and if the case may be on a suitable. To see this, you write the system of differential equations in the system

From to integral equations. Since both equations are equivalent, it suffices to show that the integral equation has a continuous solution. This is then also differentiable. Therefore is chosen as the space of continuous function on the interval with the maximum norm. In addition, one sets

Because the set of Arzelà - Ascoli is a compact operator and a compact homotopy. Since the existence of a solution is analyzed, is set. Since it has been assumed, one can show that it is enough to choose a, and receives due to the homotopy invariance

Thus we have shown that the integral equation has at least one continuous solution.

Maps between manifolds

Be

A continuous map between n-dimensional, compact, oriented manifolds. (n is a natural number. )

The orientation of the manifolds induces isomorphisms

The induced homomorphism of f

The multiplication with an integer d, this is the Abbildunsgrad F.