J. Lo pezgo mez. and. J. C. Sabina de Lis. Received May 8, 1997; revised January 15, INTRODUCTION


 Irma Lloyd
 1 years ago
 Views:
Transcription
1 journal of differential equations 148, 4764 (1998) article no. DE First Variations of Principal Eigenvalues with Respect to the Domain and PointWise Growth of Positive Solutions for Problems Where Bifurcation from Infinity Occurs J. Lo pezgo mez Departamento de Matema tica Aplicada, Universidad Complutense, 284Madrid, Spain and J. C. Sabina de Lis Departamento de Ana lisis Matema tico, Universidad de La Laguna, La Laguna (Tenerife), Spain Received May 8, 1997; revised January 15, 1998 In this paper the first variation of the principal eigenvalue of &2 in with respect to a general family of holomorphic perturbations of is analyzed. Then, the results from this analysis are used to ascertain the pointwise growth to infinity of the positive solutions of a class of sublinear elliptic boundary value problems with vanishing coefficients at the value of the parameter where bifurcation from infinity occurs Academic Press 1. ITRODUCTIO In this wor we consider the following family of eigenvalue problems &2u=* u in, u= on, (1.1) where is a smooth bounded domain of R, 1, whose boundary possesses a finite number of connected components, 2 is the Laplace operator, and, &, is a family of smooth domains obtained from by the action of an holomorphic family of C 2 diffeomorphisms T :. It is well nown that the lowest eigenvalue _ 1 [ ] of (1.1) is the unique eigenvalue to a positive eigenfunction, denoted by. (x), that _ 1 [ ]is simple and that it varies continuously with, [5]. In fact, the continuous dependence of _ 1 [ ] is valid for rather general elliptic operators, general domains and general perturbations from them (cf. [1, 2, 11]), though it may fail for eumann boundary conditions (cf. [5]). Moreover, in Section Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
2 48 LO PEZGO MEZ AD SABIA DE LIS VII.6.5 of [1] it was shown that the eigenvalues *() of (1.1) are holomorphic in near = when the domain is obtained from by a transformation of the form T (x)=x+r(x) x #, where R(x) is a smooth vector valued function defined in an open set containing the closure. The transformed eigenfunctions (x) :=.(T (x)), x #, are also holomorphic in. Here.( y), y #, is an arbitrary eigenfunction associated to *(). Although the process used in determining the power series in for the eigenvalues and eigenprojections is, in general, rather complicated (cf. Remar 4.18 on p. 46 of [1]), there are several inds of formulas for the first variation of the principal eigenvalue _ 1 [ ], d d _ 1[ ] =. Among them are the following: (6.31) on p. 422 of [1], which is of an abstract nature thus not useful for our purposes herein; and the formula on p. 275 of [13]. amely, 2 d d _ 1[ ] = =& (R, n) \. ds, (1.2) n + where. has been normalized by &. & L 2 ( )=1, and n is the outward unit normal on. Formula (1.2) generalizes to (5.1.1) of [7], found for the special case of twodimensional domains. Besides its intrinsic interest, formula (1.2) has proven to be pivotal in the problem of analyzing the pointwise growth of the positive solutions of a class of sublinear boundary value problems where bifurcation from infinity occurs due to the presence of vanishing coefficients in the model, this being the problem from which our interest in ascertaining the first variation of the principal eigenvalue comes. Thus, in this paper, our attention will be focused on both problems. In Section 2 we extend some of the results of [13] to cover the case of general holomorphic families T of the form T =I+R+O( 2 ) as, (1.3) under minimal regularity requirements on the perturbed domains (C 1, instead of C 3 ), and show in particular that (1.2) remains valid by means of a direct striing proof. In Sections 3 and 4 we analyze the pointwise growth mentioned above. We should point out that (1.2) can not be
3 BOUDARY PERTURBATIO AD POITWISE GROWTH 49 obtained straight ahead from the analysis of [9] and [15], where the corresponding variation of the Green functions was analyzed, nor from the abstract analysis of [16], since the differentiation of the principal eigenvalue entails the differentiation of the minimum of the variational functional with respect to the domain rather than the differentiation of the functional itself. To describe our results, consider the problem &2u=*u&a(x)u r in, u =, (1.4) where is a smooth bounded domain of R, r>1, * # R is regarded as a real parameter, and a # C( ) is a nonnegative weight function, a{, such that D := [x # : a(x)>]//. It will be assumed in addition that D possesses a finite number of connected components, that D consists of a finite number of smooth connected pieces, and that the open set :="D is connected. Under these assumptions it is nown that (1.4) possesses a positive solution if, and only if, _ 1 []<*<_ 1 [ ], and that it is unique if it exists (cf. [3, 6, 12, 14]). Moreover, if we denote it by % *, then lim &% * &, =. * A _ 1 [ ] In Section 4 we use the theory of Sections 2, 3 to analyze the pointwise growth of % * as * A _ 1 [ ]. Our results show that lim % * (x)= for all x #, (1.5) * A _ 1 [ ] uniformly in compact subsets of, and that if a(x) decays sufficiently fast as x approaches D, then lim % * (x)= for all x #D. (1.6) * A _ 1 [ ] For instance, if a(x) is of class C 1 near D, then (1.6) holds (cf. Theorem 4.3 and Remar 4.2).
4 5 LO PEZGO MEZ AD SABIA DE LIS The uniform growth to infinity of % * on any compact subset of is in strong contrast with its behavior in D, where the growth of % * is controlled by the positive solution of the corresponding problem in D subject to infinity boundary conditions. In particular, lim % * (x)< for each x # D. (1.7) * A _ 1 [ ] To prove (1.6) we use some subsolutions built from the principal eigenfunctions of some auxiliary boundary value problems in a family of domains obtained from by an holomorphic family of C 2 diffeomorphisms T satisfying (1.3). The proof of (1.7), being beyond the scope of this wor, will be given in [8]. This paper is organized as follows. Section 2 contains a short selfcontained proof of (1.2). In Section 3 we show that the domain perturbations required for the proofs of (1.5) and (1.6) can be obtained by an holomorphic family of diffeomorphisms. In Section 4 we use the results of Sections 2, 3 to prove (1.5) and (1.6). 2. HOLOMORPHIC DEPEDECE AD FIRST VARIATIOS OF PRICIPAL EIGEVALUES Throughout this section the domain is assumed to be of class C 1. First, we will use the theory of Chapter VII of [1] to show that if, &, is obtained from by a holomorphic family of C 2 diffeomorphisms T, &, then the principal eigenvalue _ 1 [ ] is real holomorphic in. Then we shall find (dd)(_ 1 [ ]) =. Assume that =T ( ), where T : is a family of C 2 diffeomorphisms that can be expressed in the form T (x)=x+ : n=1 with R (n) # C 2 ( ; R ) for each n1, and n R (n) (x) x #, (2.1) lim [&R (n) &, +&D x R (n) &, +&D 2 x R(n) &, ] 1n <+. (2.2) n Here, we have denoted &D x R(n) &, := sup &D x R(n) (x)&, x # 2.
5 BOUDARY PERTURBATIO AD POITWISE GROWTH 51 In the sequel, for any H # C 2 (, R ), D x H(x) v will stand for the linear action of D x H(x) onv # R, and D 2 H(x)[v x 1, v 2 ] for the bilinear action of D 2 H(x) on (v x 1, v 2 )#R _R. To homogenize the notation we set R () :=I. Thans to (2.2), the series D T x (x)= : n= n D x R(n) (x), (2.3) is absolutely convergent either in C 1 ( ; R 2 ) when =1 or in C( ; R 3 ) if =2. To avoid the difficulty that the underlying Hilbert space depends on, we transform problem (1.1) into a problem in. Let y denote the spatial variable in, and. the principal eigenfunction associated with _ 1 [ ], &. Setting it is easily seen that y=t (x), (x)=. (T (x)), x #, & :, l=1 (D y h, D y h l ) 2 & : x x l l=1 2 y h l x l =_ 1 [ ] in, = on, (2.4) where the function coefficients h i are given by T &1 ( y) :=(h 1 ( y, ),..., h ( y, )), y=t (x), x #. We now analyze the dependence on of the differential operator L(x, D x, ) defined by the lefthand side of (2.4), i.e., It is easily seen that L(x, D x, ) :=& :, l=1 & : l=1 2 (D y h, D y h l ) x x l 2 y h l x l, y=t (x). D y T &1 ( y) y=t (x)=i&d x R (x)+o( 2 ), (2.5) where the corresponding series is absolutely convergent in C 1 ( ; R 2 ). Moreover, D 2 y T &1 ( y) y=t (x)[},}]=&d 2 x R (x)[},}]+o( 2 ), (2.6) where the corresponding series is absolutely convergent in C( ; R 3 ). ow, we are ready to analyze the dependence in of the coefficients of
6 52 LO PEZGO MEZ AD SABIA DE LIS L(x, D x, ). Since the mapping D y h i ( y, ) y=t (x) is the i th component of D y T &1 ( y) y=t (x), (2.5) implies that D y h i ( y, ) y=t (x) can be expressed into a series of powers of which is absolutely convergent in C 1 ( ; R ) for &. Thus, the coefficients (D y h, D y h l ) y=t (x), 1, ln, are real analytic in for &. As for the coefficients 2 y h l, observe that they are the trace of D 2 h y l( y, ) y=t (x), the l th component of D 2 T &1 y ( y) y=t (x), and that, thans to (2.6), D 2 h y l( y, ) y=t (x) can be developped as a power series in that is absolutely convergent in C( ; R 2 ). Thus, the corresponding series for 2 y h l ( y, ) y=t (x) is also absolutely convergent in C( ; R) for &. Therefore, if L(x, D x, ), &, is regarded as a family of closed operators with common domain D(L)=H 1( ) & H 2 ( ) and values in L 2 ( ), then this family is real holomorphic of type (A) in in the sense of Kato (cf. [1, Chapter VII, Section 2]). Indeed, for all u # L 2 ( ) and v # H 1 ( ) & H 2 ( ), the L 2 product ul(},d x, ) v is real holomorphic in for &. Therefore, we find from Theorems 1.7, 1.8 of Chapter VII, Section 1.3, of [1] that _ 1 [ ] and are real holomorphic in for &. In particular, these features mae rigorous the following analysis of the coefficients of (2.4) up to the first order in. Thans to (2.5), we obtain D y h (T (x))=e &D x R (x)+o(2 ), x #, (2.7) where e is the th vector of the canonical basis of R and R (x) is the th component of R (x), 1. Thus, \R (D y h, D y h l ) = l & x l Moreover, it readily follows from (2.7) that 2 y h l (T (x))=&2 x R l (x)+o(2 ), (x)+ R l (x) ). (2.8) x ++O(2 where i =1 if =i and i = if {i. Substituting this relation together with (2.8) into (2.4) gives & :, l=1_ l& \R + R l x l x +& ow, setting 2 x x l + : l=1 2 x R l x l =_ 1 [ ] +O( 2 ). (2.9) = O( 2 ), _ 1 [ ]=_ 1 [ ]+* 1 +O( 2 ), (2.1)
7 BOUDARY PERTURBATIO AD POITWISE GROWTH 53 where. is the principal eigenfunction associated with _ 1 [ ] normalized so that. 2 =1, substituting (2.1) into (2.9), dividing the resulting relation by, and passing to the limit as gives & :, l=1\ R + R l x l x + 2. x x l + : l=1 2 x R l. x l =_ 1 [ ]. 1 +* 1., (2.11) where we have used &2. =_ 1 [].. ow, the Fredholm alternative applied to (2.11) provides us with the following value for * 1 : * 1 = :, l=1. \ R Integrating by parts gives Hence, + R l x l x : x x l. 2 x R l=1 x. (2.12) l l R. * 1 =&2 :. R + :, l=1 x l x x l. 2. =1 x + : R, l=1 x \. 2 x l+. R. * 1 =&2 :. R &_ 1 [ ] :, l=1 x l x x l. 2 =1 x + : R, l=1 x \ A further integration by parts gives R. &2 :., l=1 x l x x l. 2 x l+. (2.13) =&2 (R, {. )({., n) ds&2_ 1 [ ] (R, {. ).. + : R 2.. (2.14), l=1 x l x l x
8 54 LO PEZGO MEZ AD SABIA DE LIS Similarly, and : R, l=1 x \. 2 x l+ = {. 2 (R, n) ds&2 :. R 2., (2.15), l=1 x l x l x &_ 1 [ ] : R. 2 =1 x =2_ 1. (R, {. ), (2.16) since. = on. ow, substituting (2.14)(2.16) into (2.12) yields * 1 =&2 (R, {. )({., n) ds+ {. 2 (R, n) ds. (2.17) On the other hand, since for each x # {. (x)=(. n)(x) n(x), (2.17) reduces to 2 * 1 =& (R, n) \. ds, (2.18) n + which is the wanted value. The previous features can be summarized into the following result. Theorem 2.1. Let /R be a bounded domain of class C 1 and, &, a perturbed family of domains from given by a family of C 2 diffeomorphisms T satisfying (2.1) and (2.2). Let _ 1 [ ] denote the principal eigenvalue of, &. Then, the family of eigenvalue problems (1.1) is real holomorphic in and the first variation of the principal eigenvalue * 1 := (dd)(_ 1 [ ]) = is given by (2.18). Remar 2.1. If R (z)=p(z) n(z), for some nonnegative function p{, then we find from (2.18) that * 1 =& p(. n) 2 ds<. Therefore, in this case _ 1 [ ] decreases linearly to _ 1 [ ]as a. 3. A IMPORTAT EXAMPLE In this section we restrict ourselves to dealing with a particular class of perturbations of which are pivotal for the analysis carried out in Section 4. First, we introduce the family of perturbed domains. Then, we prove that can be obtained from by a holomorphic family of diffeomorphims.
9 BOUDARY PERTURBATIO AD POITWISE GROWTH 55 Let 1 j,1jp, be the components of and n j,1jp, the outward pointing unit normal associated to every 1 j / (cf. [4]). ow, given m1, mp, we pic up m arbitrary components of, say 1 i, 1m, and for each > small enough we consider the domain m :=. =1 [x # R " : d(x, 1 i )<]. (3.1) Observe that is obtained from by enlarging it by an amount just in the direction of the outward unit normals n i of the preselected group 1 i1,..., 1 im of components of, while remains unchanged with respect to the remaining ones. The most simple example is given by the following perturbation := [x # R :1+<&x&<R], <<1, of the annulus :=[x # R :1<&x&<R] where R>1. A less elementary example can be obtained from := >. p i=1 B R i (x i ), where /R is an arbitrary bounded smooth domain, p1, x j #, 1 jp, are p arbitrary points of, and R j >, 1 jp, are p positive real numbers such that B R i (x i ):=[x # R : &x&x i &R i ]/, 1j p. In this case, the perturbation of is defined by choosing 1i 1 <}}}< i m p, # (, min 1m R i ) and taing :=" p B i=1 R i & i (x i ), where i = if i # [i 1,..., i m ] and i = if not. In the proof of Theorem 4.3 of Section 4 it will be seen where our interest into these class of perturbations comes from. In the rest of this section we will prove that they fit into the scenario of Section 2, though a little more regularity on will be required for this. As a consequence from this suplementary regularity it will be seen that, moreover, _ 1 [ ] decays linearly as a. It should be also observed that the linear behavior of the principal eigenvalue is by no means evident even for the simplest case of the annulus when proceeding to the direct analysis of the problem in terms of the underlying Bessel functions. Theorem 3.1. Assume that is a bounded domain of R of class C 3. If is given by (3.1), then for each > sufficiently small, there exists a mapping T : R, such that
10 56 LO PEZGO MEZ AD SABIA DE LIS (i) T # C 2 ( ; R ) and T : is a bijection. (ii) The family T is real holomorphic in for &, in the sense that (2.1) and (2.2) are satisfied. (iii) R 1i =n i if i # [i 1,..., i m ], whereas R 1i = if i [i 1,..., i m ]. Proof. Since the components 1 i of are compact surfaces of class C 3 in R, they are orientable and possess welldefined C 2 outward unit normal fields n i =n i (x) [4]. Let = > be sufficiently small so that the = neighborhoods of 1 i, A i := [x # R : d(x, 1 i )<= ], 1i p, be pairwise disjoint. It is wellnown that for each 1ip, = can be chosen sufficiently small so that A i be a tubular neighborhood of 1 i (cf. [4, II.7]). This means that for every x # A i there exist unique z # 1 i and &= <{<= such that x=z&{n i (z). (3.2) Thus, by reducing once more =, if necessary, the implicit function theorem shows the existence of two unique mappings { i # C 2 (A i ; R),? i # C 2 (A i ; 1 i ), 1ip, such that x=? i (x)&{ i (x) n i (? i (x)), (3.3) for every x # A i and each 1ip. ext, introduce {^ i # C 2 ( ; R) as the extension of { i to by { i (x)== if d(x, 1 i )=. Let n^ i # C 2 ( ; R ) be any regular extension of the vector field n i (? i (x)) to the whole of and consider any function ` # C 3 ([, ); [, )) satisfying `()=1, `({) `({)< for { #[,= 2), and `({)= for {= 2. It is easily seen that for each 1ip the mapping H i (x) :=`({^ i(x)) n^ i(x) x #, is of class C 2, satisfies H i (x)= if d(x, 1 i )= 2, and H i (x)=`({ i (x))_ n i (? i (x)) if d(x, 1 i )<= 2. In particular, H i 1l = li n i, where li =1 if l=i and li = if l=% i. Setting R (x) :=H i1 (x)+}}}+h im (x), it will now be shown how the family T :=I+R satisfies all the requirements of the statement. It is rather clear that they are locally invertible
11 BOUDARY PERTURBATIO AD POITWISE GROWTH 57 mappings of class C 2 satisfying (2.1) and (2.2). So, it suffices to prove that each of them defines a bijection from onto. Since R vanishes in V := {x # : d(x, 1 i ) = 2 for each 1m =, the restriction of T to V equals the identity mapping in V. Moreover, setting U i := {x # : d(x, 1 i ) = 2=, then, either T (x)=x for all x # U i,ifi{i,1m, or T (x)=x+`({ i (x)) n i (? i (x)) for all x # U i, if i=i for some 1m. Thus, from the coordinate representation (3.3), it is easily seen that T also defines a diffeomorphism from U i onto U i _ [x # R " : d(x, 1 i )], when 1m. This completes the proof. ow, from Theorems 2.1, 3.1 the following result is obtained. Theorem 3.2. If is given by (3.1), then the family of eigenvalue problems (1.1) is real holomorphic in and m d d (_ 1[ ]) = =& : =1 1 i\. K 2 n i+ ds<. (3.4) 4. POITWISE DIVERGECE OF POSITIVE SOLUTIOS In this section we analyze the behavior of the positive solutions to the following nonlinear boundary value problem &2u=*u&a(x) u r in, u =, (4.1) where is a bounded domain of R of class C 3, r>1, * # R is regarded as a real parameter, and a # C() is a nonnegative weight function, a=%, satisfying H1. The open set D :=[x # : a(x)>]
12 58 LO PEZGO MEZ AD SABIA DE LIS satisfies D / and it possesses a finite number of connected components D 1,..., D l, such that D i & D j=< if i{ j. Thus, D=D 1 _ }}} _ D l. H2. Every connected component D i is a bounded domain of class C 3. H3. The open set defined by :="D is connected. Remar 4.1. Under these assumptions the subdomain is of class C 3 and = _ D consists of a finite number of components (at least l+1). Since D /, this number is greater than one even in the simplest possible case when D is connected. This explains why we are interested in domains whose boundary exhibits more than one component. ote that the onedimensional version (=1) of (4.1) only requires (H1). In the sequel we will focus our attention on the case 2. The analysis and results can easily be adapted to cover the onedimensional situation as well. The main features concerning the existence, uniqueness, and dependence on * of the positive solutions of (4.1) are summarized in the following result. The existence and uniqueness were found in [3] and [14]. The dependence on * of the positive solutions as well as the validity of the result for general secondorder elliptic operators, not necessarily selfadjoint, was analyzed in [6] and [12]. Theorem 4.1. if, The problem (4.1) possesses a positive solution if, and only _ 1 []<*<_ 1 [ ]. (4.2) Moreover, if such a solution exists, then it is unique, and if we denote it by % *, then % * # W 2, p () & C 1+: ( ) for each p>, where :=1&p, and C 1+: ( ):=[u # C 1+: () :u =]. Furthermore, lim &% * &, =, * a _ 1 [] lim &% * &, =. (4.3) * A _ 1 [ ] In addition, the mapping * % * from _ 1 []<*<_ 1 [ ] into C 1+: ( ) is differentiable, pointwise increasing and v=(d% * d*)#w 2, p () & C 1+: ( ). In this section our goal is to analyze the pointwise growth of % * as * A _ 1 [ ], where, according to Theorem 4.1, bifurcation to positive solutions from infinity occurs. Our first result reads as follows. Theorem 4.2. Let K be a compact subset of. Then lim % * = and lim * A _ 1 [ ] * A _ 1 [ ] d% * = uniformly in K. d*
13 BOUDARY PERTURBATIO AD POITWISE GROWTH 59 Proof. Differentiating (4.1) with respect to * gives (&2+ra% * r&1 &*) d% * d* =% * in and (d% * d*)= on. In we have a= and hence, (&2&*) d% * d* =% *. Let. be the principal eigenfunction associated with _ 1 [ ]. Fix and consider c> such that * 1 #(_ 1 [], _ 1 [ ]) % *1 >c. 1 in. Then, thans to Theorem 4.1, for each * #(* 1, _ 1 [ ]) we have % * >% *1 >c. in. Moreover, if * #(* 1, _ 1 [ ]) then the operator &2&* satisfies the strong maximum principle in and hence, d% * c d* >c(&2&*)&1. = _ 1 [ ]&*. in. Since. is bounded away from zero in K, lim * A _ 1 [ ] In addition we also get d% * = uniformly in K. d* # 1[ ]&* 1 % * (x)>% *1 (x)+log \ 1 [ ]&* + for x # K, where #=c inf K.. Thus, both % * and % * diverge to infinity uniformly in K when * A _ 1 [ ]. This completes the proof. K The following result complements Theorem 4.1, providing us with a sufficient condition on the weight function a(x) so that lim % * (x)= for each x #D= ". (4.4) * A _ 1 [ ]
14 6 LO PEZGO MEZ AD SABIA DE LIS Theorem 4.3. Assume that the weight function a=a(x) is of class C 1 in some neighbourhood of the boundary D of its support. Then (4.4) is satisfied uniformly on ". Remar 4.2. Observe that, as a consequence of our assumption, the following holds: {a(x)=, for every x #D. (4.5) evertheless, as it will be seen from the proof of Theorem 4.3, such a restriction can be substantially relaxed by assuming, for instance, that a(x)=o(dist(x, D)) as dist(x, D) +. Proof. by For > small enough, let be the perturbation of defined := _ [x # D : d(x, D)<]. By Theorem 3.2, is a holomorphic perturbation from of the form (3.1) and hence Theorem 3.2 implies where _ 1 [ ]=_ 1 [ ]+* 1 +O( 2 ) as a, (4.6) * 1 <. (4.7) Let. be the principal eigenfunction associated with _ 1 [ ], normalized so that Pic up * satisfying &. &, =1. _ 1 [ ]<_ 1 [ 2 ]<*<_ 1 [ ], (4.8) and consider the function u # C( ) defined by u ( y)= {C. ( y) for y #, for y. where C> is an amplitude constant to be chosen later. It is easily seen that u provides us with a wea subsolution of (4.1) if, and only if, a( y) C r&1. r&1 ( y)*&_ 1 [ ] for all y #. (4.9)
15 BOUDARY PERTURBATIO AD POITWISE GROWTH 61 By (4.8) a sufficient condition for (4.9) is the following a( y) C r&1. r&1 ( y)_ 1 [ 2 ]&_ 1 [ ] for all y #. For y # we have a( y)=. Therefore, u is a subsolution of (4.1) provided a( y) C r&1. r&1 ( y)_ 1 [ 2 ]&_ 1 [ ] for all y # ". (4.1) ow, the choice of the suitable amplitude C=C() requires the analysis of the decay order as of the several quantities involved in (4.1). amely, _ 1 [ 2 ]&_ 1 [ ], sup " a( y) and sup ". ( y). Firstly, thans to (4.6) we have, _ 1 [ 2 ]&_ 1 [ ]=&* 1 2 +O(2 ) as a. (4.11) Moreover, sup " a( y)=o() as. Indeed, for each y # " there exists a unique? i ( y)#d i, for some 1il, such that y&? i ( y) (cf. the proof of Theorem 3.1), and hence, a( y) =a( y)&a(? i( y)) = 1 ({a(ty+(1&t)? i ( y)), y&? i ( y)) dt. Therefore, we find from (4.5) that Finally, lim a a( y) sup =. (4.12) " sup ". ( y)tc as, (4.13) for some positive constant C. To prove (4.13) we proceed separately in each of the regions W i :=[x # D i : dist(x, D i )]. For each > small enough there exists y # W i such that sup Wi. ( y)=. ( y ). We claim that y #D i. Indeed, observe that = in the proof of Theorem 3.1 can be chosen so that. =. (x) in (2.1) satisfy (cf. (3.3)) ({. (? i (x)&{ i (x) n i ), n i ) <, for each x #, dist(x, D i )=, where n i =n i (? i (x)) and 1il. From (2.1) this implies that increases in the direction of &n i, i.e., for {
16 62 LO PEZGO MEZ AD SABIA DE LIS increasing in (3.2). Taing into account. ( y)= (T &1 ( y)), this shows the claim above. Let x be such that y =T (x ). Since y #D i, we have that x =z & n i (z ), where z =? i (x ) (cf. (3.3)). Moreover, since. ( y )= (x ) and (z )=, we find that. ( y )= (x )=& 1 ({ (z &tn i (z )), n i (z ))) dt. (4.14) If we now let we obtain, modulus some subsequence, that z x i for some x i #D i. Therefore, taing limits in (4.14) as we find that sup W i. ( y)=. ( y )=&. n i (x i) +o(), as. Hence, the constant C giving the behavior (4.13) of sup ". ( y) is given by the maximum of the numbers &(. n i )(x i), 1il. Thus, it follows from (4.1), that u is a subsolution of (4.1) if we tae 1 C=C() := sup ". ( y){ _ 1(r&1) 1[ 2 ]&_ 1 [ ] sup " a( y) = 1 = sup ". ( y){ * 1(r&1) 12+O(). (4.15) sup " a( y)= It follows from (4.12), (4.13), and (4.15) that This implies that lim C()=+. lim u (x)=+ uniformly on each compact subset of. To complete the proof of the theorem it suffices to show that lim u (x)=+ on D. This follows from the fact that, inf. ( y)tc 1 y #D
17 BOUDARY PERTURBATIO AD POITWISE GROWTH 63 as for some positive constant C 1. Indeed, the same arguments as in the proof of the estimate (4.13) lead to inf. ( y)=&. (x^ y #D n i) +o(), i as, for some x^ i #D i, and each 1il. So, the constant C 1 is given by Therefore we finally get C 1 = min 1il{ &. (x^ i) = n. i u ( y)c()(c 1 &= 1 ) for each y #D, and = 1 > small. Thus lim u =+ uniformly on D. It should be observed that the problem (4.1) always exhibits arbitrarily large supersolutions in the interior of the positive cone (cf. [6, Lemma 3.4] and [12, Lemma 3.2]). Since (4.1) admits a unique solution % * we obtain u ( y)% * ( y) y # for each _ 1 [ 2 ]<*<_ 1 [ ]. Therefore, the growth to infinity of u leads to the corresponding behavior for % *, and the proof of Theorem 4.3 is completed. K ACKOWLEDGMETS The authors wish to than to DGICYT of Spain for research support under grants DGICYT PB and DGES PB REFERECES 1. J. M. Arrieta, Elliptic equations, principal eigenvalue and dependence on the domain, Comm. Partial Differential Equations 21 (1996), I. Babusa and R. Vyborny, Continuous dependence of eigenvalues on the domain, Czech. Math. J. 15 (1965), H. Brezis and L. Oswald, Remars on sublinear elliptic equations, onlinear Anal. 1 (1986), M. do Carmo, ``Differential Geometry of Curves and Surfaces,'' Prentice Hall Internationals, Englewood Cliffs, J, R. Courant and D. Hilbert, ``Methods of Mathematical Physics,'' Vol. I, Wiley, ew Yor, 1962.
18 64 LO PEZGO MEZ AD SABIA DE LIS 6. J. M. Fraile, P. KochMedina, J. Lo pezgo mez, and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear equation, J. Differential Equations 127 (1996), P. R. Garabedian and M. Schiffer, Convexity of domain functionals, J. Anal. Math. 2 ( ), J. amelia n, R. Go mezren~ asco, J. Lo pezgo mez, and J. Sabina de Lis, On the uniqueness and pointwise behaviour of positive solutions for a general class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Rat. Mech. Anal., in press. 9. J. Hadamard, Me moires sur le proble me d'analyse relatif a l'equilibre des plaques e lastiques encastre es, Mem. Acad. Sci. Paris 39 (198), T. Kato, ``Perturbation Theory for Linear Operators,'' Classics Math., SpringerVerlag, Berlinew Yor, J. Lo pezgo mez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1996), J. Lo pezgo mez, On the uniqueness of positive solutions for a class of sublinear elliptic problems, Boll. Un. Mat. Ital. B 11 (1997), A. M. Micheletti, Perturbazione dello spettro di un operatore ellittico di tipo variazionale, in relazione ad una variazione del campo, Ann. Mat. Pura Appl. 47 (1973), T. Ouyang, On the positive solutions of semilinear equations 2u+*u&hu p = on the compact manifolds, Trans. Amer. Math. Soc. 331 (1992), J. Peetre, On Hadamard's variational formula, J. Differential Equations 36 (198), J. Simon, Differentiation with respect to the domain in boundary value problems, umer. Funct. Anal. Optim. 2 (198),
and finally, any second order divergence form elliptic operator
Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B
More informationSELFADJOINTNESS OF DIRAC OPERATORS VIA HARDYDIRAC INEQUALITIES
SELFADJOINTNESS OF DIRAC OPERATORS VIA HARDYDIRAC INEQUALITIES MARIA J. ESTEBAN 1 AND MICHAEL LOSS Abstract. Distinguished selfadjoint extension of Dirac operators are constructed for a class of potentials
More informationUPPER AND LOWER SOLUTIONS FOR A HOMOGENEOUS DIRICHLET PROBLEM WITH NONLINEAR DIFFUSION AND THE PRINCIPLE OF LINEARIZED STABILITY
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 30, Number 4, Winter 2000 UPPER AND LOWER SOLUTIONS FOR A HOMOGENEOUS DIRICHLET PROBLEM WITH NONLINEAR DIFFUSION AND THE PRINCIPLE OF LINEARIZED STABILITY ROBERT
More informationVariational eigenvalues of degenerate eigenvalue problems for the weighted plaplacian
Variational eigenvalues of degenerate eigenvalue problems for the weighted plaplacian An Lê Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720 email: anle@msri.org Klaus
More informationMATH 124B Solution Key HW 05
7.1 GREEN S FIRST IENTITY MATH 14B Solution Key HW 05 7.1 GREEN S FIRST IENTITY 1. erive the 3dimensional maximum principle from the mean value property. SOLUTION. We aim to prove that if u is harmonic
More informationThis article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author s institution, sharing
More informationarxiv: v1 [math.ca] 18 Jun 2017
RADIAL BIHARMOIC k HESSIA EQUATIOS: THE CRITICAL DIMESIO CARLOS ESCUDERO, PEDRO J. TORRES arxiv:176.5684v1 [math.ca] 18 Jun 217 ABSTRACT. This work is devoted to the study of radial solutions to the elliptic
More informationRadial Symmetry of Minimizers for Some Translation and Rotation Invariant Functionals
journal of differential equations 124, 378388 (1996) article no. 0015 Radial Symmetry of Minimizers for Some Translation and Rotation Invariant Functionals Orlando Lopes IMECCUNICAMPCaixa Postal 1170 13081970,
More informationPiecewise Smooth Solutions to the BurgersHilbert Equation
Piecewise Smooth Solutions to the BurgersHilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA emails: bressan@mathpsuedu, zhang
More informationarxiv:math.ap/ v2 12 Feb 2006
Uniform asymptotic formulae for Green s kernels in regularly and singularly perturbed domains arxiv:math.ap/0601753 v2 12 Feb 2006 Vladimir Maz ya 1, Alexander B. Movchan 2 1 Department of Mathematical
More informationSELFADJOINTNESS OF SCHRÖDINGERTYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 10726691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELFADJOINTNESS
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationRolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1
Journal of Mathematical Analysis and Applications 265, 322 33 (2002) doi:0.006/jmaa.200.7708, available online at http://www.idealibrary.com on Rolle s Theorem for Polynomials of Degree Four in a Hilbert
More informationLecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationOn semilinear elliptic equations with nonlocal nonlinearity
On semilinear elliptic equations with nonlocal nonlinearity Shinji Kawano Department of Mathematics Hokkaido University Sapporo 0600810, Japan Abstract We consider the problem 8 < A A + A p ka A 2 dx
More informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationDevil s Staircase Rotation Number of Outer Billiard with Polygonal Invariant Curves
Devil s Staircase Rotation Number of Outer Billiard with Polygonal Invariant Curves Zijian Yao February 10, 2014 Abstract In this paper, we discuss rotation number on the invariant curve of a one parameter
More informationAlmost sure limit theorems for Ustatistics
Almost sure limit theorems for Ustatistics Hajo Holzmann, Susanne Koch and Alesey Min 3 Institut für Mathematische Stochasti GeorgAugustUniversität Göttingen Maschmühlenweg 8 0 37073 Göttingen Germany
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationLipschitz shadowing implies structural stability
Lipschitz shadowing implies structural stability Sergei Yu. Pilyugin Sergei B. Tihomirov Abstract We show that the Lipschitz shadowing property of a diffeomorphism is equivalent to structural stability.
More informationNullcontrollability of the heat equation in unbounded domains
Chapter 1 Nullcontrollability of the heat equation in unbounded domains Sorin Micu Facultatea de MatematicăInformatică, Universitatea din Craiova Al. I. Cuza 13, Craiova, 1100 Romania sd micu@yahoo.com
More informationEconomics 204 Fall 2011 Problem Set 2 Suggested Solutions
Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit
More informationHomogenization and error estimates of free boundary velocities in periodic media
Homogenization and error estimates of free boundary velocities in periodic media Inwon C. Kim October 7, 2011 Abstract In this note I describe a recent result ([14][15]) on homogenization and error estimates
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationInternal Stabilizability of Some Diffusive Models
Journal of Mathematical Analysis and Applications 265, 91 12 (22) doi:1.16/jmaa.21.7694, available online at http://www.idealibrary.com on Internal Stabilizability of Some Diffusive Models Bedr Eddine
More informationSimultaneous vs. non simultaneous blowup
Simultaneous vs. non simultaneous blowup Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F.C.E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility
More information1 Introduction and statements of results
CONSTANT MEAN CURVATURE SURFACES WITH BOUNDARY IN EUCLIDEAN THREESPACE Rafael López 1 1 Introduction and statements of results The study of the structure of the space of constant mean curvature compact
More informationA Simple Proof of the Generalized Cauchy s Theorem
A Simple Proof of the Generalized Cauchy s Theorem Mojtaba Mahzoon, Hamed Razavi Abstract The Cauchy s theorem for balance laws is proved in a general context using a simpler and more natural method in
More informationSome Properties of Closed Range Operators
Some Properties of Closed Range Operators J. FarrokhiOstad 1,, M. H. Rezaei gol 2 1 Department of Basic Sciences, Birjand University of Technology, Birjand, Iran. Email: javadfarrokhi90@gmail.com, j.farrokhi@birjandut.ac.ir
More informationVariational Formulations
Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that
More informationThe Skorokhod reflection problem for functions with discontinuities (contractive case)
The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection
More informationRECENT PROGRESSES IN THE CALABIYAU PROBLEM FOR MINIMAL SURFACES. Antonio Alarcón
Matemática Contemporânea, Vol 30, 2940 c 2006, Sociedade Brasileira de Matemática RECENT PROGRESSES IN THE CALABIYAU PROBLEM FOR MINIMAL SURFACES Antonio Alarcón Abstract In the last forty years, interest
More informationHopf Bifurcation in a Scalar Reaction Diffusion Equation
journal of differential equations 140, 209222 (1997) article no. DE973307 Hopf Bifurcation in a Scalar Reaction Diffusion Equation Patrick Guidotti and Sandro Merino Mathematisches Institut, Universita
More informationOn locally Lipschitz functions
On locally Lipschitz functions Gerald Beer California State University, Los Angeles, Joint work with Maribel Garrido  Complutense, Madrid TERRYFEST, Limoges 2015 May 18, 2015 Gerald Beer (CSU) On locally
More informationAn Observation on the Positive Real Lemma
Journal of Mathematical Analysis and Applications 255, 48 49 (21) doi:1.16/jmaa.2.7241, available online at http://www.idealibrary.com on An Observation on the Positive Real Lemma Luciano Pandolfi Dipartimento
More informationON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 167 176 ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Piotr Haj lasz and Jani Onninen Warsaw University, Institute of Mathematics
More information2) Let X be a compact space. Prove that the space C(X) of continuous realvalued functions is a complete metric space.
University of Bergen General Functional Analysis Problems with solutions 6 ) Prove that is unique in any normed space. Solution of ) Let us suppose that there are 2 zeros and 2. Then = + 2 = 2 + = 2. 2)
More informationTHE DISTANCE FROM THE APOSTOL SPECTRUM
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 10, October 1996 THE DISTANCE FROM THE APOSTOL SPECTRUM V. KORDULA AND V. MÜLLER (Communicated by Palle E. T. Jorgensen) Abstract. If
More informationSpace Analyticity for the NavierStokes and Related Equations with Initial Data in L p
journal of functional analysis 152, 447466 (1998) article no. FU973167 Space Analyticity for the NavierStokes Related Equations with Initial Data in L p Zoran Grujic Department of Mathematics, Indiana
More informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationMULTIPLE SOLUTIONS FOR THE plaplace EQUATION WITH NONLINEAR BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 37, pp. 1 7. ISSN: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) MULTIPLE
More informationMcMaster University. Advanced Optimization Laboratory. Title: A Proximal Method for Identifying Active Manifolds. Authors: Warren L.
McMaster University Advanced Optimization Laboratory Title: A Proximal Method for Identifying Active Manifolds Authors: Warren L. Hare AdvOlReport No. 2006/07 April 2006, Hamilton, Ontario, Canada A Proximal
More informationTHE POINCAREHOPF THEOREM
THE POINCAREHOPF THEOREM ALEX WRIGHT AND KAEL DIXON Abstract. Mapping degree, intersection number, and the index of a zero of a vector field are defined. The PoincareHopf theorem, which states that under
More informationABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL
ABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL OLEG SAFRONOV 1. Main results We study the absolutely continuous spectrum of a Schrödinger operator 1 H
More informationA Perrontype theorem on the principal eigenvalue of nonsymmetric elliptic operators
A Perrontype theorem on the principal eigenvalue of nonsymmetric elliptic operators Lei Ni And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of
More informationSPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS
SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly
More informationNonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia AlabauBoussouira Collocated feedback stabilization Outline 1 Introduction and main result
More informationOn the FrontTracking Algorithm
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 7, 395404 998 ARTICLE NO. AY97575 On the FrontTracking Algorithm Paolo Baiti S.I.S.S.A., Via Beirut 4, Trieste 3404, Italy and Helge Kristian Jenssen
More informationarxiv:math/ v1 [math.fa] 26 Oct 1993
arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More informationarxiv: v2 [math.ap] 9 Sep 2014
Localized and complete resonance in plasmonic structures HoaiMinh Nguyen and Loc Hoang Nguyen April 23, 2018 arxiv:1310.3633v2 [math.ap] 9 Sep 2014 Abstract This paper studies a possible connection between
More informationLocal mountainpass for a class of elliptic problems in R N involving critical growth
Nonlinear Analysis 46 (2001) 495 510 www.elsevier.com/locate/na Local mountainpass for a class of elliptic problems in involving critical growth C.O. Alves a,joão Marcos do O b; ;1, M.A.S. Souto a;1 a
More informationNotation. 0,1,2,, 1 with addition and multiplication modulo
Notation Q,, The set of all natural numbers 1,2,3, The set of all integers The set of all rational numbers The set of all real numbers The group of permutations of distinct symbols 0,1,2,,1 with addition
More informationCOINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE pharmonic OPERATOR
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 95, Number 3, November 1985 COINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE pharmonic OPERATOR SHIGERU SAKAGUCHI Abstract. We consider the obstacle
More informationExtension and Representation of Divergencefree Vector Fields on Bounded Domains. Tosio Kato, Marius Mitrea, Gustavo Ponce, and Michael Taylor
Extension and Representation of Divergencefree Vector Fields on Bounded Domains Tosio Kato, Marius Mitrea, Gustavo Ponce, and Michael Taylor 1. Introduction Let Ω R n be a bounded, connected domain, with
More informationGeneral Mathematics Vol. 16, No. 1 (2008), A. P. Madrid, C. C. Peña
General Mathematics Vol. 16, No. 1 (2008), 4150 On X  Hadamard and B derivations 1 A. P. Madrid, C. C. Peña Abstract Let F be an infinite dimensional complex Banach space endowed with a bounded shrinking
More informationCongurations of periodic orbits for equations with delayed positive feedback
Congurations of periodic orbits for equations with delayed positive feedback Dedicated to Professor Tibor Krisztin on the occasion of his 60th birthday Gabriella Vas 1 MTASZTE Analysis and Stochastics
More informationarxiv: v1 [math.ap] 16 Jan 2015
Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities Vladimir Lubyshev January 19, 2015 arxiv:1501.03870v1 [math.ap] 16 Jan 2015 Abstract This paper studies a nonlinear
More informationOn supporting hyperplanes to convex bodies
On supporting hyperplanes to convex bodies Alessio Figalli, YoungHeon Kim, and Robert J. McCann Abstract Given a convex set and an interior point close to the boundary, we prove the existence of a supporting
More informationSection 12.6: Nonhomogeneous Problems
Section 12.6: Nonhomogeneous Problems 1 Introduction Up to this point all the problems we have considered are we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationDIFFERENTIAL TOPOLOGY AND THE POINCARÉHOPF THEOREM
DIFFERENTIAL TOPOLOGY AND THE POINCARÉHOPF THEOREM ARIEL HAFFTKA 1. Introduction In this paper we approach the topology of smooth manifolds using differential tools, as opposed to algebraic ones such
More informationNonlinear error dynamics for cycled data assimilation methods
Nonlinear error dynamics for cycled data assimilation methods A J F Moodey 1, A S Lawless 1,2, P J van Leeuwen 2, R W E Potthast 1,3 1 Department of Mathematics and Statistics, University of Reading, UK.
More informationsubgradient trajectories : the convex case
trajectories : Université de Tours www.lmpt.univtours.fr/ ley Joint work with : Jérôme Bolte (Paris vi) Aris Daniilidis (U. Autonoma Barcelona & Tours) and Laurent Mazet (Paris xii) inequality inequality
More informationFinitedimensional spaces. C n is the space of ntuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a prehilbert space, or a unitary space) if there is a mapping (, )
More informationDIMENSIONS OF JULIA SETS OF HYPERBOLIC ENTIRE FUNCTIONS
Bull. London Math. Soc. 36 2004 263 270 C 2004 London Mathematical Society DOI: 10.1112/S0024609303002698 DIMENSIONS OF JULIA SETS OF HYPERBOLIC ENTIRE FUNCTIONS GWYNETH M. STALLARD Abstract It is known
More informationExercise 1. Let f be a nonnegative measurable function. Show that. where ϕ is taken over all simple functions with ϕ f. k 1.
Real Variables, Fall 2014 Problem set 3 Solution suggestions xercise 1. Let f be a nonnegative measurable function. Show that f = sup ϕ, where ϕ is taken over all simple functions with ϕ f. For each n
More informationConvergence rates in l 1 regularization when the basis is not smooth enough
Convergence rates in l 1 regularization when the basis is not smooth enough Jens Flemming, Markus Hegland November 29, 2013 Abstract Sparsity promoting regularization is an important technique for signal
More informationALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE
ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE ALAN CHANG Abstract. We present Aleksandrov s proof that the only connected, closed, n dimensional C 2 hypersurfaces (in R n+1 ) of
More informationSymmetry and monotonicity of least energy solutions
Symmetry and monotonicity of least energy solutions Jaeyoung BYEO, Louis JEAJEA and Mihai MARIŞ Abstract We give a simple proof of the fact that for a large class of quasilinear elliptic equations and
More informationADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS
J. OPERATOR THEORY 44(2000), 243 254 c Copyright by Theta, 2000 ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS DOUGLAS BRIDGES, FRED RICHMAN and PETER SCHUSTER Communicated by William B. Arveson Abstract.
More informationDIFFERENTIABLE PERTURBATION OF UNBOUNDED OPERATORS. Andreas Kriegl, Peter W. Michor
Math. Ann. 327, 1 (2003), 191201 DIFFERENTIABLE PERTURBATION OF UNBOUNDED OPERATORS Andreas Kriegl, Peter W. Michor Abstract. If A(t) is a C 1,α curve of unbounded selfadjoint operators with compact
More informationCONTROLLABILITY OF NONLINEAR DISCRETE SYSTEMS
Int. J. Appl. Math. Comput. Sci., 2002, Vol.2, No.2, 73 80 CONTROLLABILITY OF NONLINEAR DISCRETE SYSTEMS JERZY KLAMKA Institute of Automatic Control, Silesian University of Technology ul. Akademicka 6,
More informationThe double layer potential
The double layer potential In this project, our goal is to explain how the Dirichlet problem for a linear elliptic partial differential equation can be converted into an integral equation by representing
More informationGAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n
GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. A. ZVAVITCH Abstract. In this paper we give a solution for the Gaussian version of the BusemannPetty problem with additional
More informationA NOTE ON PROJECTION OF FUZZY SETS ON HYPERPLANES
Proyecciones Vol. 20, N o 3, pp. 339349, December 2001. Universidad Católica del Norte Antofagasta  Chile A NOTE ON PROJECTION OF FUZZY SETS ON HYPERPLANES HERIBERTO ROMAN F. and ARTURO FLORES F. Universidad
More informationREACTIONDIFFUSION EQUATIONS FOR POPULATION DYNAMICS WITH FORCED SPEED II  CYLINDRICALTYPE DOMAINS. Henri Berestycki and Luca Rossi
Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 REACTIONDIFFUSION EQUATIONS FOR POPULATION DYNAMICS WITH FORCED SPEED II  CYLINDRICALTYPE
More informationImplicit Functions, Curves and Surfaces
Chapter 11 Implicit Functions, Curves and Surfaces 11.1 Implicit Function Theorem Motivation. In many problems, objects or quantities of interest can only be described indirectly or implicitly. It is then
More information1 1 u(0) = 0 = u(tt).
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 306, Number 2, April 1988 A CLASS OF NONLINEAR STURMLIOUVILLE PROBLEMS WITH INFINITELY MANY SOLUTIONS RENATE SCHAAF AND KLAUS SCHMITT ABSTRACT.
More informationBellman function approach to the sharp constants in uniform convexity
Adv. Calc. Var. 08; (): 89 9 Research Article Paata Ivanisvili* Bellman function approach to the sharp constants in uniform convexity DOI: 0.55/acv060008 Received February 9 06; revised May 5 06; accepted
More informationON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.
More informationGroup construction in geometric Cminimal nontrivial structures.
Group construction in geometric Cminimal nontrivial structures. Françoise Delon, Fares Maalouf January 14, 2013 Abstract We show for some geometric Cminimal structures that they define infinite Cminimal
More informationMath 225B: Differential Geometry, Final
Math 225B: Differential Geometry, Final Ian Coley March 5, 204 Problem Spring 20,. Show that if X is a smooth vector field on a (smooth) manifold of dimension n and if X p is nonzero for some point of
More informationSymmetry of entire solutions for a class of semilinear elliptic equations
Symmetry of entire solutions for a class of semilinear elliptic equations Ovidiu Savin Abstract. We discuss a conjecture of De Giorgi concerning the one dimensional symmetry of bounded, monotone in one
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationCOMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
Dynamic Systems and Applications 22 (203) 37384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute
More informationTrace Class Operators and Lidskii s Theorem
Trace Class Operators and Lidskii s Theorem Tom Phelan Semester 2 2009 1 Introduction The purpose of this paper is to provide the reader with a selfcontained derivation of the celebrated Lidskii Trace
More informationProduct metrics and boundedness
@ Applied General Topology c Universidad Politécnica de Valencia Volume 9, No. 1, 2008 pp. 133142 Product metrics and boundedness Gerald Beer Abstract. This paper looks at some possible ways of equipping
More informationSome aspects of vanishing properties of solutions to nonlinear elliptic equations
RIMS Kôkyûroku, 2014, pp. 1 9 Some aspects of vanishing properties of solutions to nonlinear elliptic equations By Seppo Granlund and Niko Marola Abstract We discuss some aspects of vanishing properties
More informationLERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION
2005Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationNONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction
Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp. 1 1. ISSN: 1076691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER
More informationarxiv: v3 [math.cv] 26 Nov 2009
FIBRATIONS AND STEIN NEIGHBORHOODS arxiv:0906.2424v3 [math.cv] 26 Nov 2009 FRANC FORSTNERIČ & ERLEND FORNÆSS WOLD Abstract. Let Z be a complex space and let S be a compact set in C n Z which is fibered
More informationNOTES ON EXISTENCE AND UNIQUENESS THEOREMS FOR ODES
NOTES ON EXISTENCE AND UNIQUENESS THEOREMS FOR ODES JONATHAN LUK These notes discuss theorems on the existence, uniqueness and extension of solutions for ODEs. None of these results are original. The proofs
More informationDEGREE AND SOBOLEV SPACES. Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 13, 1999, 181 190 DEGREE AND SOBOLEV SPACES Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg Dedicated to Jürgen
More informationON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE. Sangho Kum and Gue Myung Lee. 1. Introduction
J. Korean Math. Soc. 38 (2001), No. 3, pp. 683 695 ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE Sangho Kum and Gue Myung Lee Abstract. In this paper we are concerned with theoretical properties
More informationSimultaneous vs. non simultaneous blowup
Simultaneous vs. non simultaneous blowup Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F..E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility
More informationAn introduction to some aspects of functional analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied athematics http://jipam.vu.edu.au/ Volume 4, Issue 5, Article 98, 2003 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES D. ESKINE AND A. ELAHI DÉPARTEENT
More informationELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS EXERCISES I (HARMONIC FUNCTIONS)
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS EXERCISES I (HARMONIC FUNCTIONS) MATANIA BENARTZI. BOOKS [CH] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Interscience Publ. 962. II, [E] L.
More informationUniquely Universal Sets
Uniquely Universal Sets 1 Uniquely Universal Sets Abstract 1 Arnold W. Miller We say that X Y satisfies the Uniquely Universal property (UU) iff there exists an open set U X Y such that for every open
More informationDisconjugate operators and related differential equations
Disconjugate operators and related differential equations Mariella Cecchi, Zuzana Došlá and Mauro Marini Dedicated to J. Vosmanský on occasion of his 65 th birthday Abstract: There are studied asymptotic
More information