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00. The expression (12) tends to 0 as e —* 0 in the L2-norm because of the exponential decay of <pi and (¡>2- The expressions (13) and (14) tend to 0 in L2 because they resemble in effect convolutions of functions in L4. (A fuller account giving all details of these limits is in preparation.)

6. Passage to the limit equation

The deduction that the rescaled problem has a solution because the limit problem has a non-degenerate solution is effected by means of the implicit function theorem. A version of this theorem with minimal assumptions, at first sight well adapted for problems of this kind, is stated below.

Let E and F be real Banach spaces, / : x E —» F, fe(x) := f(e, x). Let f€ be differentiate for each e > 0. By a non-degenerate solution of fo(x) = 0 we mean a solution xq for which the Frechet derivative Dfo(xo) is invertible from E to F. Invertibility for a linear map in this context will always mean that the inverse is a bounded linear map from F to E.

Implicit function theorem

Make the assumptions:

(a) The equation fo(x) = 0 has a non-degenerate solution xq.

(c) The limit lim£_o,x—x0 Dfe(x) = Dfo(xo) is attained in the operatornorm topology.

Then for each sufficiently small e > 0 the equation fe(x) = 0 has a unique solution near to xq, which tends to xo as e —> 0.

Now we note a disturbing fact. Condition (c) does not hold in our problem. In fact the limit only occurs in the strong operator topology. This is the case even in the single bump problem, where it arises because in the expression the argument of V, namely e(x — £) = e(x — s) + b converges to b, but not uniformly with respect to a;. In the multi-bump problem similar difficulties arise also in the coupling terms. To counter this unfortunate fact we need to modify the implicit function theorem.

Modified implicit function theorem

Make the assumptions:

(a) The equation fo(x) = 0 has a non-degenerate solution xq.

(c) The inverse of Dfe(xo) exists as an operator from F to E for all sufficiently small e and is uniformly bounded in norm as e —* 0.

(d) The limit lime-to,x-*x0(Dfe(x)-Df£(xo)) = 0 is attained in the operator norm topology.

Then for each sufficiently small e > 0 the equation fe(x) = 0 has a unique solution near to xq, which tends to xo as e —> 0.

The verification of (c) is hard work, (curiously (d) is not so burdensome), and reveals the need for unforeseen hypotheses. It is carried out on the basis of the following strategy. Suppose that Dfe(xo) is a Fredholm operator of index 0. Then invertibility is equivalent to injectivity and establishing the truth of (c) reduces to verifying the following:

Sequence property If ev —* 0 and zv is a sequence of vectors such that \\zu\\e < 1 and Dfiv(x§)zv —> 0 in F then a subsequence of zv tends to 0 in E.

7. Study of the rescaled equations

The rescaled equation for a single lump is:

The derivative of the left-hand side is a linear map IP x W 9 (a, z) i ► -Az{x) + — (V(e(x - £)), </>(*))

r dF dF

We want this to be a Fredholm operator of index 0 for all e. This depends on the observation that the expression in brackets defines a compact operator, and then an additional condition:

Positivity condition There exists h such that ^(a, 0) > h > 0 for all a in the range ofV(x).

In the case of two bumps there are two equations with additional terms such as

with Frechet derivative z2y-> J {■■. )(j>i{x)z2{x + 6 - 6) + • • •

Frechet derivatives of the additional terms are compact operators so the derivative of the rescaled problem is Fredholm with index 0.

Recall that condition (c) reduces to verifying the sequence property. For our problem it reduces to showing that if e„ —> 0 and dF

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