Proof of the main theorem

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So there cannot exist any eigenvectorU PL1px dxqforτ1¥β0.

1.3 Proof of the main theorem

The proof of Theorem 1.1.1 is divided as follows. We begin with a result concerning the positivity of the a priori existing eigenvectors (Lemma 1.3.1). We then define, in Section 1.3.2, a regularized and truncated problem for which we know that eigenelements exist (see the Appendix 1.5.2 for a proof using the Krein-Rutman theorem), and we choose it such that the related eigenvalue is positive (Lemma 1.3.3).

In Section 1.3.3, we give a series of estimates that allow us to pass to the limit in the truncated problem and so prove the existence for the original eigenproblem (1.3). The positivity of the eigenvalueλand the uniqueness of the eigenelements are proved in the two last subsections.

1.3.1 A preliminary lemma

Before proving Theorem 1.1.1, we give a preliminary lemma, useful to prove uniqueness of the eigenfunc-tions.

Lemma 1.3.1(Positivity). Consider U andφsolutions to the eigenproblem (1.3).

We definem:inf

x,y x : px, yqPSupp βpyqκpx, yq

(

. Then we have, under assumptions (1.5),(1.6), (1.9) and (1.10)

SuppUrm,8q and τUpxq¡0 x¡m, φpxq¡0 x¡0.

If additionaly 1τ PL10, thenφp0q¡0.

Remark 1.3.2. In case Supp κtpx, yq{x¤yu,then m0 and Lemma 1.3.1 and Theorem 1.1.1 can be proved without the connexity condition (1.10)on the support of β.

Proof. Letx0¡0,we defineF :xÞÑτpxqUpxqe

´x x0

λ βpsq τpsq ds

.We have that F1pxq2e

´x x0

λ βpsq τpsq dsˆ

βpyqκpx, yqUpyqdy¥0. (1.16) So, as soon asτUpxqonce becomes positive, it remains positive for largerx.

We define a:inftx : τpxqUpxq¡0u. We first prove thata¤ 2b. For this we integrate the equation onr0, asto obtain

ˆ a

0

ˆ 8

a

βpyqκpx, yqUpyqdydx0,

ˆ 8

a

βpyqUpyq ˆ a

0

κpx, yqdxdy0.

Thus for almost everyy¥maxpa, bq, ´a

0 κpx, yqdx0.As a consequence we have 1

ˆ

κpx, yqdx ˆ y

a

κpx, yqdx¤ 1 a

ˆ

px, yqdx y 2a thanks to (1.5) and (1.6), and this is possible only ifb¥2a.

Assume by contradiction thatm  a,integrating (1.3) multiplied by ϕ, we have for all ϕPC8c such thatSupp ϕ€r0, as

ˆ ˆ

ϕpxqβpyqκpx, yqUpyqdydx0. (1.17) By definition ofmand using the fact that m a,there exists pp, qqPpm, aqpb,8qsuch that pp, qqP Supp βpyqκpx, yq.But we can chooseϕpositive such thatϕppqUpqq¡0 and this is a contradiction with (1.17). So we havem¥a.

To conclude we notice that onr0, ms, U satisfies

BxpτpxqUpxqq λUpxq0.

So, thanks to the conditionτp0qUp0q0 and the assumption (1.9), we haveU 0 onr0, ms,so ma and the first statement is proved.

Forφ,we define Gpxq:φpxqe

´x

x0 λ βpsq

τpsq ds

.We have that G1pxq2e

´x

x0 λ βpsq

τpsq ds

βpxq ˆ x

0

κpy, xqφpyqdy¤0, (1.18) so, as soon as φ vanishes, it remains null. Therefore φ is positive on an interval p0, x1q with x1 P

R Yt 8u.Assuming thatx1  8and using thatx1¡ambecause´

φpxqUpxqdx1,we can find X¥x1such that

ˆ X

x1

G1pxqdx2 ˆ X

x1

ˆ x1

0

e

´x x0

λ βpsq τpsq ds

φpyqβpxqκpy, xqdy dx 0.

This contradicts thatφpxq0 forx¥x1,and we have proved thatφpxq¡0 for x¡0.

If τ1 PL10,we can takex00 in the definition ofGand soφp0q¡0 orφ0.The fact thatφis positive ends the proof of the lemma.

1.3.2 Truncated problem

The proof of the theorem is based on uniform estimates on the solution to a truncated equation. Let η, δ, Rpositive numbers and define

τηpxq

"

η 0¤x¤η τpxq x¥η.

Then τη is lower bounded onr0, Rs thanks to (1.9) and we denote by µ µpη, Rq : infr0,Rsτη. The existence of eigenelementspλδη,Uηδ, φδηqfor the following truncated problem whenδR µis standard (see Theorem 1.5.2 in the Appendix).

$ enough to satisfy the following lemma.

Lemma 1.3.3. Under assumptions(1.5),(1.8)and(1.13), there exists aR0¡0such that for allR¡R0, and, thanks to Gr¨onwall’s lemma,

τpxqUpxq¥δe

1.3.3 Limit as δ

Ñ

0 for U

ηδ

and λ

δη

Fixη and letδÑ0 (thenRÑ8sinceδRµ2).

First estimate: λδη upper bound. Integrating equation (1.19) between 0 andR,we find λδη ¤δ

ˆ

βpxqUηδpxqdx, then the idea is to prove a uniform estimate on ´

βUηδ. For this we begin with bounding the higher β is nonnegative and locally integrable, and τη is positive. Thanks to assumption (1.12), we know that xη ÝÑ Consequently, if we considerδ¤1 for instance, we obtain

sup

xPp0,xq

τηpxqUηδpxq¤ 1 2Bm{xm

12ρ :C (1.21)

soτηUηδ is uniformly bounded in a neighborhood of zero.

Combining (1.20) and (1.22) we obtain

α¥0,DBα: η, δ¡0, To conclude we use the fact that neither the parameters nor Uηδ are negative and we find by the chain rule, forα¥0

¤α and all the terms in the right hand side are uniformly bounded thanks to the previous estimates.

Since we have proved that the familytxατηUηδuδ is bounded inW1,1pR qfor allα¥0,then, because τη is positive and belongs to P, we can extract from tUηδuδ a subsequence which converges in L1pR q

whenδÑ0. Passing to the limit in equation (1.19) we find that

$ more estimate to study the limitηÑ0.

Third estimate: L8bound for xατηUη, α ¥γ. We already know thatxατηUη is bounded for α¥0.

Hence, using Gr¨onwall’s lemma, we find thatVηpxq¤ Ke

2 12ρ

12ρ and consequently xγτηpxqUηpxq¤Ke

2 12ρ

12ρ :C,r xPr0, xs. (1.30) This last estimate allows us to boundUηby xτγ which is inL10by the assumption (1.11). Thanks to the second estimate, we also have that´

xαUηis bounded inL1and so, thanks to the Dunford-Pettis theorem (see [14] for instance),tUηuηbelong to aL1-weak compact set. Thus we can extract a subsequence which convergesL1weak towardU.But for allε¡0, txαUηuηis bounded inW1,1prε,8qqfor allα¥1 thanks to (1.28) and so the convergence is strong onrε,8q.Then we write

ˆ small forη small because of the strong convergence. FinallyUηÝÑ

ηÑ0U strongly inL1pR qandU solution of the eigenproblem (1.3).

1.3.5 Limit as δ, η

Ñ

0 for φ

δη

We prove uniform estimates onφδη which are enough to pass to the limit and prove the result.

Fourth estimate : uniform φδη-bound on r0, As. Let A¡0, our first goal is to prove the existence of a

so, asxηÑx0 and´x0

0 Upxqdx¡0 (thanks to Lemma 1.3.1 and becausex0¡b¥a), we have sup

p0,Aq

φδη ¤C0pAq. (1.31)

Fifth estimate : uniformφδη-bound onrA,8q. Following an idea introduced in [111] we notice that the equation in (1.19) satisfied byφδη is a transport equation and therefore satisfies the maximum principle (see Lemma 1.5.3 in the Appendix). Therefore it remains to build a supersolutionφthat is positive at xR,to concludeφδηpxq¤φpxqonr0, Rs.

This we cannot do onr0, Rs,but on a subintervalrA0, Rsonly. So we begin with an auxiliary function ϕpxqxk θ withkandθ positive numbers to be determined. We have to check that onrA0, Rs

τpxq B

Bpxq pλδη βpxqqϕpxq¥pxq ˆ

κpy, xqϕpyqdy δφδηp0q, i.e.

pxqxk1 pλδη βpxqqϕpxq¥

2θ 2 ˆ

κpy, xqykdy βpxq δφδηp0q. Fork¥2,we know that´

κpy, xqxykkdy¤c 1{2 so it is sufficient to prove that there existsA0¡0 such that we have

pxqxk1 pλδη βpxqqpxk θq¥p2θ 2cxkqβpxq δC0p1q (1.32) for allx¡A0,whereC0is defined in (1.31). For this, dividing (1.32) byxk1τpxq,we say that if we have

p12cqpxq

τpxq ¥k 2θβpxq δC0p1q

xk1τpxq , (1.33)

then (1.32) holds true. Thanks to assumptions (1.8) and (1.13) we know that there existsk ¡ 0 such that for anyθ¡0,there existsA0¡0 for which (1.33) is true onrA0, 8q.

Then we conclude by choosing the supersolutionφpxq C0pθA0qϕpxqso that φpxq¥φδηpxq onr0, A0s,

and onrA0, Rs,we have

$

&

%

τpxq B

Bxφpxq pλδη βpxqqφpxq¥pxq´x

0 κpy, xqφpyqdy δφδηp0q, φpRq¡0,

(1.34)

which is a supersolution to the equation satisfied byφδη.Thereforeφδη ¤φuniformly in η andδ and we get

Dk, θ, C s.t.η, δ, φδηpxq¤pCxk θq. (1.35) Equation (1.19) and the fact thatφδη is uniformly bounded inL8locpR qgive immediately that Bxφδη is uniformly bounded inL8locpR , τpxqdxq, so inL8locp0,8qthanks to (1.9).

Then we can extract a subsequence oftφδηuwhich convergesC0p0,8qtowardφ. Now we check thatφ satisfied the adjoint equation of (1.3). We consider the terms of (1.19) one after another.

Firstpλδη βpxqqφδηpxqconverges topλ βpxqqφpxq inL8loc.

ForBxφδη, we have anL8 bound on each compact ofp0,8q.So it convergesL8weaktowardBxφ.

It remains the last term which we write, for allx¡0, ˆ x

0

κpy, xqpφδηpyqφpyqqdy¤}φδηφ}L8p0,xq ÝÑ η,δÑ00.

The fact that´ in Theorem 1.1.1 also follow from those uniform estimates. It remains to prove that λ ¡ 0 and the uniqueness.

1.3.6 Proof of λ

¡

0

We prove a little bit more, namely that λ¥ 1

2sup

x¥0

tτpxqUpxqu. (1.36)

We integrate the first equation of (1.3) between 0 andxand find 0¤λ

For the eigenvectors we use the General Relative Entropy method introduced in [98, 99]. ForC¡0,we test the equation onU1 against sgn UU12 C

and then

Dans le document The DART-Europe E-theses Portal (Page 35-45)