Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology pot

vii On the Positive Solutions of the Logistic Weighted Elliptic BVP with Sublinear Mixed Boundary Conditions S.. ON THE POSITIVE SOLUTIONS OF THE LOGISTIC WEIGHTED ELLIPTIC BVP WITH SUBL

Spectral Theory and

Nonlinear Analysis with

Applications to Spatial Ecology

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Spectral Theory and

Nonlinear

Universidad Complutense de Madrid, Spain

Published by

World Scientific Publishing Co Re Ltd

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SPECTRAL THEORY AND NONLINEAR ANALYSIS WITH APPLICATIONS

TO SPATIAL ECOLOGY

Copyright 0 2005 by World Scientific Publishing Co Re Ltd

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Preface

This volume collects the Proceedings of the Complutense International

Seminar Spectral Theory and Nonlinear Analysis that we celebrated in

Madrid in June 14th and 15th, 2004, at the Department of Applied Math- ematics, under the auspices of the Spanish Ministry of Education and Sci- ence, under Grant REN2003-00707, and Complutense University through

an International Seminar budget, which allowed us to invite some of the most renowned experts in these fields Besides the editors, the following experts participated in the International Seminar: F Cobos (Madrid), E

N Dancer (Sydney), I Gohberg (Tel Aviv), D MacGhee (Glasgow), R J Magnus (Reykjavik), J Mawhin (Louvain La Neuve), A G Ramm (Man- hattan, Kansas), B P Rynne (Edinburgh), and A SuArez (Sevilla), among many others that attended some of the talks delivered in that International Seminar, whose kind assistance certainly facilitated its success

Such International Seminar was organized to honor the memory of our friend and colleague J Esquinas, born in March 27th 1960 at Ocaiia (Toledo, Spain), who suddenly died in August 11th 2003 at Covadonga National Park (Asturias, Spain) J Esquinas was a tremendously gifted mathematician who did some seminal contributions to the theory of gen- eralized algebraic multiplicites in the context of bifurcation theory His scientific carrier was as short as intense, since he dedicated many of his efforts to the defense of the rights of the workers in Spanish Universities, becoming a renowned very popular personality in both issues

Besides collecting most of the contributions delivered by the participants

in this Complutense International Seminar , this volume also includes a number of contributions by well recognized experts in Spectral Theory, Differential Equations and Nonlinear Analysis whose mathematical work

is closely related to the one developed by J Esquinas The editors are delighted to thank all of them for their contributions to this so special honoring volume

July 2005 The editors

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Poster of the Seminar

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Contents

Preface v

Poster of the Seminar vii

On the Positive Solutions of the Logistic Weighted Elliptic BVP

with Sublinear Mixed Boundary Conditions

S Cano-Casanova 1 Logarithmic Interpolation Spaces

F Cobos 17 Remarks on Large Solutions

J Garcia-Melicin and J Sabina de Lis 31 Well Posedness and Asymptotic Behaviour of a Closed Loop

Thermosyphon

A Jimknez-Casas 59 Uniqueness of Large Solutions for a Class of Radially Symmetric

Elliptic Equations

J Ldpez-Gdmez 75 Cooperation and Competition, Strategic Alliances, and the

Cambrian Explosion

J Ldpez-Gdmez and M Molina-Meyer 111 Local Smith Form and Equivalence for One-parameter Families

of F'redholm Operators of Index Zero

J Lbpez-Gdmez and C Mora-Corral 127 Multilump Solutions of the Non-linear Schrodinger Equation -

A Scaling Approach

R J Magnus 163

Some Elliptic Problems with Nonlinear Boundary Conditions

C Morales-Rodrigo and A Sua'rez 175

ix

X Contents

Dynamical Systems Method (DSM) and Nonlinear Problems

A G Ramm 201

B P Rynne 229

L Vega 247

J M Vegas 257

Some Recent Results on Periodic, Jumping Nonlinearity Problems

Some Remarks about the Cubic Schrodinger Equation on the Line

Some Remarks on the Invariance of Level Sets in Dynamical Systems

ON THE POSITIVE SOLUTIONS OF THE LOGISTIC WEIGHTED ELLIPTIC BVP WITH SUBLINEAR

MIXED BOUNDARY CONDITIONS*

S CANO-CASANOVA

Departamento de Matemdtica Aplicada y Computacidn

Escuela Te'cnica Superior de Ingenieria Uniuersidad Pontificia Comillas de Madrid,

2801 5-Madrid, S P A I N E-mail: scano@dmc.icai.upco.es

This paper is dedicated, with my greatest admiration, to the great mathematician, colleague and f i e n d J Esquinas Candenas He, joint with Professor J Ldpez Gdmez are the main responsible

of my scientific formation

In this work we prove the uniqueness of positive solution and characterize the existence of positive solutions of a wide class of elliptic BVP of Logistic type with sublinear weighted mixed boundary conditions The results obtained in this work are an extension of the previous one found in S Cano-Casanovag Monotonicity techniques are the main technical tools, used to develop the mathematical analysis

The main goal of this paper is to prove the uniqueness of positive solu- tion and to characterize the existence of positive solutions of the following nonlinear weighted elliptic boundary value problem of Logistic type, with sublinear weighted mixed boundary conditions given by

Problem (1) in the particular case when W = 1 in R and V = 0 on rl

Throughout this work we make the following assumptions:

( a ) The domain R is a bounded domain of R N , N 2 1, of class C2, whose boundary dR = ro U rl, where FO and l?l are two disjoint open and closed subset of d o

( b ) X E R and C stands for a linear second order differential operator of the form

which is uniformly strongly elliptic in 52 with ayij = aji E C ' ( f i ) , ai E C(fi),

zero on any compact subset of [R \ fi:] U [I'l \ dR:]

( d ) The potential W ( x ) is a bounded measurable real weight function

in $2 with arbitrary sign

(e) As far as the nonlinear mixed boundary conditions, b(x) E C(rl) is

a positive function on I'l which is bounded away from zero on I'l n do:,

V ( x ) E C(F1) with arbitrary sign in rl,

C(X) := C - X W ( x ) ,

Positive Solutions of the Logistic Weighted Elliptic BVP 3

which is uniformly strongly elliptic in R with the same ellipticity constant

as the operator L

The following theorem collects the main results of this work

where .?[C(X), B ( - V ( ~ ) ) ] and ' 0 : [,c(x>, 731 stand for the principal eigen-

value in 0 and s2:, respectively, of L(A), subject t o the boundary operators

B ( - V ( z ) ) and D, respectively, being

Moreover, in this case, the positive solution of Problem ( 1 ) ~ is unique and

if we denote it by ux, then ux is strongly positive in R in the sense that

I n particular, ux E C1+a(f=l) f o r all 0 < a < 1 and in addition, ux is a.e

in 52 twice continuously differentiable

In the sequel a function u E Wi(R) is said to be strongly positive in R if

u(z) > 0 for each z E R U r l and apu(z) < 0 for each z E I'o with u(z) = 0 and any outward pointing nowhere tangent vector field /3 E Cl(I'0; RN)

We now introduce a linear boundary operator, which will play a crucial

role throughout this work Under the assumptions of Sec 1, given k ( z ) E

C ( r l ) we will denote by B ( k ( z ) ) the linear boundary operator

B : w;(R) - w,2-+ro) x wi-+rl) defined by

4 S Cano-Casanova

where v := ( ~ 1 , , V N ) E C1(I'l;RN) is any outward pointing nowhere tangent vector field to I'l We want to point out that using the boundary operator B ( k ( z ) ) just defined, Problem ( 1 ) ~ can be written in the form

(L(X) + a(z)uT-l)u = o in R

Now we are going to introduce some basic results concerning the op- erator B ( k ( z ) ) , that we need to develop our work Let us consider the eigenvalue boundary value problem

eigenfunction, unique up to multiplicative constants and called principal eigenfunction of ( L , B ( k ( z ) ) , 0) Thanks to Theorem 12.1 of Amann3, the principal eigenfuntion of (L, B ( k ( z ) ) , R) belongs to Wi(R) for any p > 1 and it is strongly positive in a In fact, cF[L, B ( k ( z ) ) ] is the only eigenvalue

of Problem (7) possessing a positive eigenfunction, and it is dominant in the sense that any other eigenvalue (T of Problem (7) satisfies

Re(c) > dw, B(k(.))l

In the sequel, given any proper subdomain Ro of R of class C2 with dist (rl, dRonR) > 0, we will denote by B ( k ( z ) , no) the boundary operator build up from B ( k ( z ) ) by

and by I$" [C, B ( k ( z ) , Ro)], the principal eigenvalue of (C, B ( k ( z ) , Ro), 00)

To develop the mathematical analysis of the next sections, are essen- tial the different monotonicity properties of op[C, B ( k ( z ) ) ] joint with the continuous dependence of it with respect to perturbations of the domain

and with respect to perturbations of the potential k ( z ) on the boundary,

recently proved by S Cano-Casanova and J L6pez-G6mez6q5

Suppose p > N Then, any function u E Wi(R) is said to be a superso- Zution (subsolution) of Problem ( l ) ~ , if it satisfies

(L(X)u + u(z)uT,B(-V(z) + b(z)u"-1>u) 2 0 (I 0)

Positive Solutions of the Logistic Weighted Elliptic BVP 5

and it will be said that it is a strict supersolution (subsolution), if the respective inequality is strict, where 2 stands for the natural product order

In this section we prove the uniqueness of positive solution of Problem ( l ) ~ ,

if there exists, and we obtain some regularity properties about it

Theorem 3.1 If ux is a positive solution of Problem ( l ) ~ , then

.F[L(X) + a(z)u;-l, B(-V(z) + b(z)u;-')] = 0 , (8)

u x is strongly positive in R, (4) is satisfied and in particular, u~ E C'+"(fi)

f o r all 0 < Q < 1 Moreover, ux is a.e in R twice continuously differen- tiable

for some p > N, satisfies Eq (6) and thanks to the embedding

with the 0 eigenvalue Then, by the uniqueness of the principal eigenpair,

u~ is the principal eigenfunction of Problem ( 6 ) ~ and Eq (8) is satisfied Finally, owing to the structure and regularity of the principal eigenfunction guaranteed by Theorem 12.1 of H Amann3, we have that ux is strongly positive in R and Eq (4) is satisfied The remaining assertions follow again

from the embedding Eq (9) and Theorem VIII.l of E.M Stein" This

completes the proof of the result

Thanks to the previous result, if we set

and we denote by U + the cone of non-negative functions of U and by

F : R X U + - v , 0" : u+ - w ,

6 S Cano-Casanova

the nonlinear operators defined by

3 ( A , u) := L(A)u + a ( x ) u T , (A, u) E IW x ZA+ ,

and

Gv(u) := (u, a,u - V ( x ) u + b(x)uq = B ( - V ( x ) + b(x)u"-1)u), 21 E u+ ,

we have that the positive solutions of Problem ( l ) ~ , are the solutions cou- ples (X,ux) with ux > 0 in R satisfying the system

F ( A , u x ) = 0 in R

On the other hand, D,Gv(ux) can be written in terms of the boundary

opertor B(lc(x)) defined in Eq (5), by

D,Gv(ux)u := B ( - V ( x ) + qb(x)u;-l)u

there exists, is unique and non-degenerate, in the sense that

.T[a3-(A,ux), ~ u G v ( u x ) l > 0 ; i.e., the linearization of Problem ( 1 ) ~ at any positive solution only possesses the null solution

solutions of Problem ( l ) ~ , such that u1 # 212 Then, thanks to Eq (S),

O ~ [ L ( A ) + a ( z ) u ~ - l , ~ ( - v ( x ) + b(x)uf-l)] = 0 , i = 1 , 2 , (10) and the following problem is satisfied

Positive Solutions of the Logistic Weighted Elliptic B V P 7

By construction,

F ( ) 2 TuY-' > u;-', G(-) 2 qu;-l > . (1'4

Now, using the boundary operator B ( k ( z ) ) defined by Eq (5), Problem

( 1 1 ) ~ can be written in the form

{ B(-V(z) + b(z)G(z))(ul - ua) = 0 on dR

Thanks to the monotonicity of the principal eigenvalue with respect to the

potential (cf Proposition 3.3 of S Cano-Casanova and J L6pez-G6mez5) and with respect to the weight on the boundary (cf Proposition 3.5 of S Cano-Casanova and J L6pez-G6mez5), owing to Eq (12), and taking into account Eq ( l o ) , it follows that

To complete the proof of the theorem, it remains to prove the nondegen- eration of each positive solution of Problem ( 1 ) ~ Indeed, if ux is a positive solution of Problem ( l ) ~ , then

cr?[DuF(X, U X ) , DuQv(ux)l =

= CT?[L(X) + Ta(z)u';-l,B(-v(z) + qb(z)u;-l)]

Now, owing to Eq (8), taking into account that T > 1, q > 1, a > 0, b > 0 and that u~ is strongly positive in R, it follows from the monotonicity of the principal eigenvalue with respect to the potential and with respect to the weight on the boundary (cf S Cano-Casanova and J L6pez-G6mez5), that

CJ?[DuF(X, 4, DuGV('ILX)I >

> CT?[C(X) + a(z)ul-l, B(-V(z) + b(z)vq,-l)] = 0 ,

and therefore, the positive solution ux is non degenerate

In this section we characterize the existence of positive solutions of Problem ( 1 ) ~ In the beginning we give the following necessary condition for the existence of positive solution of Problem ( 1 ) ~

8 S Cano-Casanova

Proposition 4.1 If U A is a positive solution of Problem ( l ) ~ , then Eq

(2) is satisfied, where B(-V(z)) and 2) are the boundary operators defined

by (3)

Eq ( 8 ) , and owing to the monotonicity of the principal eigenvalue with

respect to the domain (cf Proposition 3.2 of S Cano-Casanova and J L6pez-G6mez5) and the dominance of the principal eigenvalue under Dirich-

let boundary conditions (cf Proposition 3.1 of S Cano-Casanova and J L6pez-G6mez5), the following is satisfied

0 = cp[L(X) + a ( ~ ) u ; - ~ , B ( - V ( z ) + b(z)u;-')]

< o:'[L(X), B ( - V ( z ) + b(z)u;-', a:)] I c:'[L(X), 27

On the other hand, thanks to the monotonicity of the principal eigenvalue

with respect to the potential (cf Proposition 3.3 of S Cano-Casanova and

J L6pez-G6mez5) and with respect to the weight on the boundary (cf

Proposition 3.5 of S Cano-Casanova and J L6pez-G6mez5), it follows from

Eq (8) that

0 = @[L(X) + a(z>u;-l, B(-V(z) + b(z)uI-')]

> o ~ [ L ( X ) , B ( - V ( Z ) + b ( ~ > ~ : - l ) ] > cF[L(X), B ( - V ( z ) ) ]

since b > 0 on rl and U A is strongly positive in fl

Now, in order to give a sufficient condition for the existence of pos- itive solutions of Problem ( l ) ~ , arguing as in Theorem 3 of H Amannl and Theorem 2.1 of H Amann2, taking into account the results found in Theorem 12.1 of H Amann3, and using the Characterization of the Strong Maximum Principle given in the work by H Amann and J L6pez-G6mez4, the following existence result for the solutions of Problem ( 1 ) ~ is satisfied, which we include without proof in order not to enlarge the exposition

l e m ( l ) ~ , such that gA 5 i i ~ Then, Problem ( 1 ) ~ has at least one solution

in the order interval [cA, F A ] More precisely, there exists a minimal solution every solution ux E [cA,iiA] of Problem (1)A satisfies uyin 5 U X I uyax

urnin A ' and a maximal solution uyax in the order interval [zA,ii~] such that

The next two results give sufficient conditions for the existence of positive

strict sub and supersolutions of Problem ( l ) ~ , respectively

Positive Solutions of the Logistic Weighted Elliptic BVP 9

Problem ( 1 ) ~ possesses a positive strict subsolution arbitrarily small and strongly positive in 0

continuous dependence of CTF[L, B(lc(x))] with respect to the weight k ( z )

on the boundary, recently proved in Proposition 3.5 and Theorem 8.2 of S Cano-Casanova and J L6pez-G6mez5, we have that

p+0+ lim ~ F [ c ( x ) , D ( - v ( ~ ) + p)] = @ [ l ( ~ ) , f?(-v(z))I (15)

and

4 [ W > , a(-v(zc))l < F [ W ) , W V ( 4 + PI1 (16)

for all p > 0 Then, taking into account Eqs (14), (15) and (16), it is

possible to take p > 0 such that

4 V ( X ) , f?(-V(z))l < 4 [ W , f ? ( - V ( 4 + PI1 < 0 * (17)

Fix p > 0 satisfying Eq (17) Now, let us consider the positive function

2 = ~ ( p ) c p ( p ) , with ~ ( p ) > 0 small enough, where cp(p) stands for the principal eigenfunction associated to of[L(X), f?( -V(z) + p ) ] , normalized

so that II~(~)IIL,(~) = 1 By construction we have that in R, g satisfies

for ~ ( p ) > 0 small enough Therefore, 3 is a positive strict subsolution of

Problem ( 1 ) ~ for ~ ( p ) > 0 small enough The remaining assertion follows from the fact that the principal eigenfunction p ( p ) is strongly positive in

R (cf H Amann3) This completes the proof

extended to the case when for instance, ặ) belongs t o the general class

of nonnegative measurable potentials d ( R ) of admissible potentials in R introduced in the works due to S Cano-Casanova and J L6pez-G6mez5, S Cano-Casanova7 and S Cano-Casanova and J L6pez-G6mez')

In the following we will suppose that rl = l?: U rq, where r:, i = 1 , 2 are two components of rl, and that R: = 0; U R i E C2, being R6, i = 1 , 2 two components satisfying

o:A[L(x),ăn,R:)] < 0::[L(X),D] (24)

where r = aRA n R Thanks to Eq (21), it follows from Definition 11.2 of

S Cano-Casanova and J L6pez-G6mez5 that

On the other hand, thanks t o the results found in Proposition 3.1 of S Cana-Casanova and J L6pez-G6mez5,

for all n E N and

lim 0 [L(x), ~ ( n , R;)] = 0 [L(x), D I

0 < 07:[L(A),B(n,n;)] < 3 L ( X ) , D ]

(25)

n t m

Thus, owing to Eqs (23), (24) and (25), there exists no E N large enough,

such that for each n 2 no the following is satisfied

(26)

Fix n 2 nọ Now, for each 6 > 0 sufficiently small and i = 1,2, let us consider the 6-neighbourhoods

:= (a; + B ~ ) n R , Njt2 := (r: + B ~ ) n R , @ := (ro + B6), (27) where Bg stands for the ball of radius 6 > 0 centered a t the origin Under the general assumptions, and thanks to Eq (22), it follows the existence of

60 > 0 small enough such that for each 0 < 6 < 60 and i = 1,2,

fiinfiz = 0 , s2Z,nNj>2 = 0 , fiz c R, Enfi; = 0 , fljt2n@ = 0 (28)

Positive Solutions of the Logistic Weighted Elliptic B V P 11

By construction we have that Rk, i = 1 , 2 is a proper subdomain of R;, and

lim a; = at,

in the sense of Definition 6.1 of S Cano-Casanova and J L6pez-G6mez5

Thus, it follows from Theorem 4.2 of J L6pez-G6mez1O, and Theorem 7.1

of S Cano-Casanova and J L6pez-G6mez5, that

12 S Cano-Casanova

where is any regular positive extension of

cp; u cp;,, u d lJ cp;

from (fl n R) U R\ U R? U Nlf2 to d, which is bounded away from zero

in d \ ( f l U R i U 0: U Nil2) Note that $6 exists, since the functions

CP~ĨM: nn, (Pi,nIan\nn, '~2Ian2, and ( ~ 3 I a ~ 1 2 n n are positive and bounded away from zerọ By construction, @(z) > 0 for each z E d and therefore,

it is bounded away from zero in d

To complete the proof of the result it remains to show that there exists

M > 0 sufficiently large, such that Ti = M a provides us with a positive

strict supersolution of Problem ( 1 ) ~

Indeed, by construction we have that in f l n 52 the following holds:

in 52; the following holds:

T

L(X)S+ăz)Tí = M ~ ! , , ( ( T ? ~ [L(X), B(n, Ri)] +ăz)M'-'(cp;,,)'-'), (36)

and in 0: the following holds:

L(X)Ti + ăz)Tí = M ~ ; ( ( T ~ " [ L ( X > , B(-V - S)] + u(z)M'-'(p$)'-'),

and taking into account that p; and ặ) are bounded away from zero in NtS2(cf (c)), there exists M I > 0 large enough such that for each M 2

Positive Solutions of the Logistic Weighted Elliptic BVP 13

and since by construction, the functions u(x), $ J ~ are bounded away from

zero in f i b \ (x U R' U 0 U J V ~ " ~ ) , there exists M2 2 M I > 0 such that for each M 2 $2 - > 0, % 5 5

3,ii - V(z)U + b(z)$ = Mpi(6 + Mq-lb(x)(pi)q-l) 2 M6p; > 0 , (42)

and due to the fact that on r: the following estimate is satisfied,

&Ti - V ( x ) i i + b(x)@ = Mp:,,(-(n + V ( x ) ) + b ( ~ ) M ~ - ~ ( p ; , , ) ~ - ' ) ,

taking into account that pi,, is bounded away from zero on and b ( x )

is bounded away from zero on rl n aR: = r;, we have that there exists

M3 2 M 2 > 0, such that for each M 2 M3 > 0,

a , ~ - V ( Z ) E + ~ ( x ) P > o on r: (43)

Thus, for each X E R satisfying Eq (21), it follows from Eqs (38), (39),

(40), (41), (42) and (43) that for each M 2 M3 > 0, the funcion Ti = M @ ( z )

satisfies

.F(X,ii) > 0 in R

{ B v(u) - > O o n a R and therefore, i := M G ( x ) with M 2 M3 > 0 is a positive strict superso- lution bounded away from zero in fi of Problem ( 1 ) ~

Now, we are ready to prove Theorem 1.1

positive solution of Problem ( 1 ) ~ is Proposition 4.1 We now prove the sufficient condition for the existence of positive solution of Problem ( 1 ) ~

Let X satisfy Eq (2) Then, thanks to Proposition 4.2 and Proposition 4.3, Problem ( 1 ) ~ possesses a positive strict subsolution Z L , arbitrarily small strongly positive in R and a positive strict supersolution E X , arbitrarily large and bounded away from zero in fi Then, taking them such that

particular case when a = 0 in R, ịẹ, 52; = R

the BVP

&u = V ( z ) u - b(z)uQ = 0 o n rl, q > 1,

where b ( x ) E C(rl) is a positive function bounded away f r o m zero o n F1 and

v := ( ~ 1 , , Y N ) E C1(rl, I R N ) is any outward pointing nowhere tangent vector field t o rl Then, Problem (44) possesses a positive solution, if and only i f

~?[c(~),,13(-v(~c>)l < 0 < ?[c(4,q , (45)

ant the remaining assertions of Theorem 1.1 are satisfied

in 0 of Problem (44)x, for each X E IR satisfying Eq (45), it suffices to take the positive function

ii := M v Y ,

for M > 0, n E N large enough and S > 0 small enough, where

@ is the principal eigenfunction associated to the principal eigenvalue

DF[Ỉ(X), B ( n , o)], being

In order to complete the exposition of this work, we include the follow- ing result, which is the counterpart of Theorem 1.1 for the case when the

potential ặ) is bounded away from zero on any compact subset of 52 U rl

Its proof can be easily obtained, arguing as in the previous sections and doing an easy and straight adaptation of the proof of Proposition 4.3

Positive Solutions of the Logistic Weighted Elliptic BVP 15

the B V P

Lu = XW(x)u - a ( x ) u r in R, r > 1,

13,u = V ( x ) u - b(z)uq = 0 o n rl, q > 1,

where the potential a ( x ) is bounded away f r o m zero o n any compact subset

of R U rl, v := (vl, , V N ) E C1(rl,RN) is any outward pointing nowhere tangent vector field to l?l and b ( x ) E C(r1) is a nonnegative function o n

rl Then, Problem (46) possesses a positive solution, if and only i f

References

1 H Amann, Indiana University Mathematics Journal 21, 2, 125-146, (1971)

2 H Amann, Nonlinear elliptic equations with nonlinear boundary conditions,

New Developments in Differential Equations, Ed W Eckhaus, 1976

3 H Amann, Israel Journal of Mathematics, 45, 225-254, (1983)

4 H Amann and J L6pez-G6mez, Journal of Differential Equations 146, 336-

374, (1998)

5 S Cano-Casanova and J L6pez-G6mez, Journal of Differential Equations

6 S Cano-Casanova and J L6pez-G6mez, Nonlinear Analysis T.M.A 47,

7 S CaneCasanova, Nonlinear Analysis T.M.A., 49, 361-430, (2002)

8 S Cano-Casanova and J Lbpez-Gbmez, Electronic Journal of Differential Equations, 74, 1-41, (2004)

9 S Cano-Casanova, O n the existence and uniqueness of positive solutions of

the Logistic elliptic B VP with nonlinear mixed boundary conditions, Nonlin-

ear Analysis, To appear

10 J L6pez-G6mez, Journal of Differential Equations 127, 263-294, 1996

11 E.M Stein, Singular Integrals and Differentiability Properties of Functions,

Princeton University Press, Princeton, NJ.,1970

178, 123-211, (2002)

1797-1808, (2001)

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LOGARITHMIC INTERPOLATION SPACES*

FERNANDO COBOS

Departamento de Ancilisis Matemcitico, Facultad de Matemciticas, Universidad Complutense de Madrid, 28040-Madrid1 SPAIN E-mail: cobosOmat.ucm.es

We review several recent results of Fernhdez-Cabrera, Triebel and the author on logarithmic interpolation spaces based on the real method

The degree of compactness of the embedding from the (fractional) Sobolev

space Hp”’”(S2) into the Orlicz space L,(logL)b(S2) has been studied by

Triebel [29] Here R is a bounded domain in Rn with smooth bound- ary, 1 < p < 00 and b < : - 1 Two years later, Edmunds and Triebel

1131, [14], investigated the behaviour of entropy numbers of the embedding

An important tool to derive these results is a representation theorem

of Zygmund spaces Lp(logL)b(S2) in terms of Lebesgue spaces L,(R) Mo- tivated by this representation, Edmunds and Triebel have studied in [15] abstract spaces based on complex interpolation More recently Fern6ndez- Cabrera, Triebel and the present author [lo] have investigated the corre- sponding spaces using the real interpolation method, giving applications of the results to Lorentz-Zygmund function spaces, Besov spaces of general- ized smoothness and Lorentz-Zygmund operator spaces Next we review these results

(n+w) (0) L, Lp(10gL)b(S22)*

‘Work supported in part by grant MTM2004-01888 of the Spanish Ministerio de Edu- caci6n y Ciencia

17

18 F Cobos

Let R be a domain in Rn with finite Lebesgue measure IRI For 1 < p < 03

and b E R, the Zygmund space Lp(log L)b(R) is defined by all (equivalent classes of) Lebesgue-measurable functions f : R -+ C such that

1 [lf(.)l logb(2 + If(.)I)IPd < O3

n

Clearly, L,(logL)o(R) = Lp(R)

f : R -+ C for which there exists a constant X > 0 such that

For b < 0, the space L,(logL)b(R) is the set of all measurable functions

In the literature this space is also denoted by Lexp,-b(R)

that if 0 < E < p then

Zygmund spaces complement the scale of Lebesgue spaces, in the sense

LP(logL)€(R) c L P W c ~ p ( l o g L ) - € ( R ) , and for -03 < b 2 5 bl < 03

LP+€(W c L,(log L ) b l (0) c -&(log L ) b z ( Q ) c LP-€(R)

Now assume that R has smooth boundary Then it is known that the (fractional) Sobolev space Hp”’”(R) is continuously embedded in

L,(logL)b(R) if, and only if, b 5 i - 1 The embedding being compact if,

and only if, b < i - 1

The study of limiting embeddings of this type goes back to Trudinger

was studied by Triebel[29], who proved that if b < -1 - $ then the entropy numbers of this embedding satisfy

c 1 k - 1 ’ p 5 e k 5 c 2 k - l / p , k E N

Here c 1 , c 2 are positive constants

Let me recall that if T E L ( E , F ) is a bounded linear operator between

the Banach spaces E and F , then the k-th entropy number of T is defined

as

2 k - 1

E > 0 : T ( U E ) & u { b j + EUF} for some b l , , b 2 k - 1 E F

Logarithmic Interpolation Spaces

where U E , UF are the closed unit balls of the spaces E and F , respectively

It is clear that

llTll 2 el(T) 2 e2(T) 2 2 0, and T is compact if and only if limk,, e k ( T ) = 0 Hence, the asymptotic

decay of the sequence (ek(T)) can be considered as a measure of the “degree

of compactness” of the operator T

There is a close relation between entropy numbers and eigenvalues

Namely, if E is a complex Banach space, T E C ( E , E ) is compact and

( X k ( T ) ) is the sequence of all non-zero eigenvalues of T , repeated according

to algebraic multiplicity and ordered so that

IXl(T)I 2 I X 2 ( T ) I L L 0 then it turns out that

I X k ( T ) I I 2 l I 2 e k ( T ) , k E N

This inequality was established by Carl and Triebel [4] and is the basis for many applications In particular, Edmunds and Triebel [12] used this inequality and the entropy estimates mentioned before to study the distri- bution of eigenvalues of some elliptic differential operators

Returning to compact embeddings between function spaces, in 1995

Edmunds and Triebel [13] investigated the behaviour of entropy numbers

of the embedding

(a) Lt Lp(logL)b(fi) where 1 < p < 00, s > 0, b < 0 and l/ps = l / p + s / n In this last paper and

in the paper by Triebel [29] a basic tool for the results was a representation

theorem of Zygmund spaces in terms of L , spaces For 1 < p < 00 the result reads as follows:

Theorem 2.1 Let 1 < p < 00 and b E R Let j o = jo(p) E N such that for all j E W with j L j o ,

20 F Cobos

and (1) defines a n equivalent n o r m in Lp(logL)b(R)

(22)

f : R + C which can be represented as

If b > 0 , then Lp(logL)b(fl) is the set of all measurable functions

The proof requires of the domain R merely that it should have finite

Lebesgue measure Theorem 2.1 can also be found in the book I141 where

they extended statement (a) covering any 0 < p 5 00, as well as p = 1 in

(ii) See also [Ill

As we said, the result is a basic tool for entropy estimates, but it also

has intrinsic interest A consequence of Theorem 2.1 is that assertions

which hold for spaces L p ( R ) , such as properties of integral operators, can

be carried over to Zygmund spaces Edmunds and Triebel have also studied

the spaces that come out by replacing in Theorem 2.1 the space Lp(R) by

the Sobolev space H,S(R) They called these spaces “logarithmic Sobolev spaces”

Constructions of type (1) to (3) have been considered in the framework

of extrapolation theory, especially in the cases p = 1 and p = 00 We refer

to the papers by Jawerth and Milman [19] and Milman [22]

Let me give an application of the case (ii) when p = 1 and b = 1 It

is taken from the book [14] and refers to the Hardy-Littlewood maximal

Logarithmic Interpolation Spaces 21

where the supremum is taken over all cubes Q containing x and with sides parallel to the coordinate axes It is known that there is a positive constant

c such that

C

IIMfllLpcn, 5 - IlfllLp(n, I P 11

P - 1 Hence, using the description of L(1og L)(Cl) and sublinearity of M , it follows that

IIMfllLI(i2) 5 c IlfIIL(logL)(n)l

which is a classical assertion of Hardy and Littlewood from 1930

As it is well known, L, spaces can be obtained by complex interpolation Namely

1

(Lm(R),Li(Q))[e] = L,(Q) if - = 6

So, taking Theorem 2.1 as starting point, Edmunds and Triebel [15] studied the corresponding abstract theory based on complex interpolation In that paper they introduced interpolation spaces which complement the complex interpolation scale and they established the trace theorem for logarithmic Sobolev spaces

P

But L, spaces can also be obtained by real interpolation Indeed,

Therefore, it is also natural to investigate the corresponding abstract theory based on the real interpolation method This was the aim of my joint paper with L.M Fernhndez-Cabrera and H Triebel [lo] Subsequently, we will describe some results of this paper

Let A0 and A1 be Banach spaces with A0 -+ A1 (continuous inclusion) Peetre's K-functional is defined by

K ( t , a) = K ( t , a; Ao, Ai)

=inf{llaollAo + t l l a l l l A l : a = a0 + a l , aj E A j } , t > 0, a E Al For 0 < 6 < 1 and 1 5 q 5 0;), the real interpolation space A O , ~ = (Ao, Al)e,q consists of all those a E A1 having a finite norm

22 F Cobos

Full details on real interpolation can be found in the books [3], [28] or [2]

The theory of real interpolation can be extended by replacing the func- tion te by a more general function parameter e(t) as can be seen in the

papers [24], [17], [18] or [25] For our aim here, the most interesting case is

@e,b(t) = te(l + I logtl)-b, t > 0, where 0 < 8 < 1 and b E R We put

(with the usual modification if q = 00)

the real method we get L,(R) if q = p If q # p we get Lorentz spaces More precisely, for l / p = 8 and 1 5 q 5 00, we have

Here f* is the non-increasing rearrangement o f f

f * ( t ) = inf{b > O : I{ E R : lf(z)I > 6}1 5 t }

and we have put

Lp,p(logL)b(fl) = Lp(logL)b(fl)

Logarithmic Interpolation Spaces 23

Returning to the real interpolation spaces, we always have

= (Ao, Ai)s,qo ~f (Ao, Ai)e,ql = if 1 I qo 5 41 I 00

Moreover, since A0 ~f A1, we have for 1 I p , q 5 00

(Ao, A I ) ~ , ~ ~f (Ao,Ai)e,q if 0 < P < 8 < 1

The next definition is modelled in Theorem 2.1

0 < 8 < 1 and let j o = j o ( 8 ) E N such that, for all j E N with j 2 j o ,

where the infimum is taken over all sequences { a j } satisfying (4) and (5)

(iii) If b = 0, then Ae,q(logA)b = Ae,,

Standard arguments show that Ae,q(logA)b is a Banach space in all

cases of b E R It is independent of j o (equivalence of norms) If we replace the real method by the complex interpolation method, that is to say, if we

replace in (2) A,j,q by (AO,AI)[,~] and in (22) we put (Ao,Al)[xjl instead

Theorem 3.1 Let 1 I q 5 00, 0 < 6' < 1, and b E R Let @ e , b ( t ) =

te (1 + I logtl)-b,t > 0 Then we have, with equivalent norms,

Aee.b;q = A6,q(logA)b

norms are equivalent to each other First note that

Using that A0 -+ Al, one can check that the last expression is equivalent

to (cg=, 2-majqKq(2m, a ) ) : , moreover constants in equivalences can be chosen independent of aj and 0 Similarly

To work with the last sum in (6) let j = k+ [logm], where [.] is the greatest

integer function Then j b - m2-j N b [log m] + bk - 2-k We obtain

Logarithmic Interpolation Spaces 25

The proof for the case b > 0 is more involved Details can be found

in [lo] Let me only mention that it is based on the description of real

interpolation spaces in terms of the J-functional

Theorem 3.1 allows us to use the results known on interpolation with

function parameter to study spaces Ae,q(logA)b A first consequence is that

A0 is dense in Ae,q(logA)b if q < 00 Concerning duality, if A0 is dense

in Al, then we have (Al)’ c_t (Ao)’, so we can consider logarithmic spaces generated by the couple {(Al)’, (Ao)’} Call them

AL,q(log A’)b

If q < 00, since A0 is dense in Ae,q(logA)b, we get

(Ail’ L+ (Ae,q(logA)b)’ ~ - t (Ao)’

and we can compare the spaces (Ae,q(log A)b)’ with the spaces AL,,(log A’)b

Using Theorem 3.1 and the duality formula for the real method with a parameter function we derive for 1 I q < 00, l / q + l / q ’ = 1 and 0 < 0 < 1

that

(Ae,q(log A)b)’ = Ai-e,,y (log A‘)-b

In this final section we apply Theorem 3.1 to concrete situations We start

with the couple (L,(R), L1(R)) We get:

Let 1 < p < M, 1 5 q 5 03 and let j o = j o ( p ) E N such that for all j E N

functions f : R + C which can be represented as

(ii) Let b > 0 Then L,,,(logL)t,(R) is the set of all measurable

_ _

b E R and let j o = j o ( 0 ) E N such that, f o r all j 2 j o ,

The proof can be found in [9]

The next application refers to Besov spaces of generalized smoothness Let cp be a C" function in Rn with

supp cp c { E E Rn : IEl 5 2 1 7 cp(t) = 1 if 1c1 I 1

Let j E N and cpj(() = cp(2-jE) - (p(2-j+'() , ( E Rn P u t cpo = cp Then

M

k=O

Logarithmic Interpolation Spaces 27

is a dyadic resolution of unity

(with the usual modification if q = 00) is finite

If b = 0 we get the usual Besov spaces B;,q(Rn) Besov spaces of generalized smoothness Bk;*b)(Rn) are Banach spaces and the norms are equivalent to each other for admissible choices of cp These spaces attracted some attention in connection with fractal analysis and related spectral the- ory A short description can be found in Triebel’s book [30] A detailed

study of these spaces is made in the paper by Moura [23]

Let

with 1 < p < 00,l 5 q 5 00, and -00 < s1 < SO < 00 Then Ao - AI

Moreover (see [8] and [21])

Bk;*b) (W”) = (B;pq(R”) 7 qq (Rn)) e e , b ; q

where s = (1 - @)SO + 8.q Consequently, using Theorem 3.1 we derive:

Corollary 4.2 Let 0 < B < 1 , l 5 q 5 00,b E R, and let 1 < p <

00, oo < s1 < so < 00 and s = (1 - B)so + Bsl

(i) Let b < 0 Then

The last application refers to operator spaces defined by summability

conditions on the singular numbers Let H be a Hilbert space and given any bounded linear operator T E L ( H ) , let { s n ( T ) } be the sequence of the singular numbers of T , defined by

s n ( T ) = inf{llT - RI) : rank R < n } , n E W

For 1 < p < 00,l 5 q 5 00 and b E R, the Lorentz-Zygmund operator space Lp,q,b(H) consists of all T E L ( H ) having a finite norm

(with the usual modification if q = 00)

The space L,,,,b(H) is the component over H of the Lorentz-Zygmund operator ideal that has been studied in [5], [6] and [7] Note that T be- longs to Lp,q,b(H) if and only if { s n ( T ) } belongs to the Lorentz-Zygmund sequence space lp,q(lOgl)b For b = 0 we recover the Lorentz operator

space ( L p , q ( H ) , r p , q ) , and for b = 0 and p = q , we get the Schatten p-class

In order to apply Theorem 3.1 in this context, we put A0 = L l ( H ) and

Logarithmic Interpolation Spaces 29

(ii) Let b > 0 Then C,,,,b(H) consists of all T E C ( H ) which can be

represented as T = C,”=,, Tj with Tj E C P p j ,,(H) such that

Furthermore, the infimum over the expression in (9) is an equivalent norm

an c p , q , b ( H )

Remark 4.1 All results we have shown refer to Banach spaces, but log-

arithmic interpolation spaces can also be considered in the class of quasi- Banach spaces Details can be found in the joint paper by Fernhdez-

Cabrera, Manzano, Martinez and the present author [9] Among other

things, we establish there that statement (ii) in Theorem 2.1 holds for

0 < p < 1 as well, and we give applications to operator spaces on Banach spaces

5 F Cobos, O n the Lorentz-Marcinkiewicz operator ideal, Math Nachr 126

6 F Cobos, Entropy and Lorentz-Marcinkiewicz operator ideals, Arkiv Mat 25