2002 Mathematics Subject Classification: 35J15, 35J25, 35B05, 35B50The principle eigenvalue and the maximum principle for second-order elliptic equations is studied. New necessary and sufficient conditions for symmetric and nonsymmetric operators are obtained. Applications for the estimation of the first eigenvalue are given

Fabricant, A.

Kutev, N.

Rangelov, T.

Bulgarian Digital Mathematics Library at IMI-BAS

Provided for non-commercial research and educational use. Not for reproduction, distribution or commercial use.PLISKASTUDIA MATHEMATICA BULGARICAПЛИСКАБЪЛГАРСКИ МАТЕМАТИЧЕСКИ СТУДИИThe attached copy is furnished for non-commercial research and education use only. Authors are permitted to post this version of the article to their personal websites or institutional repositories and to share with other researchers in the form of electronic reprints. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to third party websites are prohibited.For further information on Pliska Studia Mathematica Bulgarica visit the website of the journal http://www.math.bas.bg/-pliska/ or contact: Editorial Office Pliska Studia Mathematica Bulgarica Institute of Mathematics and Informatics Bulgarian Academy of Sciences Telephone: (+359-2)9792818, FAX:(+359-2)971-36-49 e-mail: pliska@math.bas.bgPliska Stud. Math. Bulgar. 15 (2003), 161-170 PUSI<&S T U D I A  M A T H E M A T I C A  B U L G A R IC AON PRINCIPLE EIGENVALUE FOR LINEAR SECOND  ORDER ELLIPTIC EQUATIONS IN DIVERGENCE FORMA. Fabricant, N. Kutev, T. RangelovA b s t r a c t . The principle eigenvalue and the maximum principle for second- order elliptic equations is studied. New necessary and sufficient conditions for symmetric and nonsymmetric operators are obtained. Applications for the estimation of the first eigenvalue are given.1. IntroductionThe aim of this paper is to investigate the principal eigenvalue and the related maximum principle for linear second order uniformly elliptic equations in diver­gence form(1) Lu =  — (a,j(x)uXk +  (ij(x)u'j +  V (x)uXj +  b°(x)u in O,(2) a j ( x ) ^ ^ k > /j, |£|2 for every x E Cl, £ E R n, /i =  const > 0.Here О is a bounded domain in R n,(3) oj, a°j € C'i∩), V  € С(й), b° € L ∞ (∩), ∂Sl € C 1, {a j}  = { 4 }and under the repeating indices the summation convention is understood.The case of L°° coefficients or domains with weaker regularity assumptions can be considered in a similar way as in [2] but for simplicity we omit it.2002 Mathematics Subject Classification: 35J15, 35J25, 35B05, 35B50Key words: elliptic equations, maximum and comparison principle, eigenvalue problem162 A. Fabricant, N. Kutev, T. RangelovLet us recall that the maximum principle for the operator L holds if every weak subsolution и E Hq(∩) of (1) is nonpositive, и < 0 in О. The function и E Hq (O) is a weak subsolution of (1) if the integral inequalityJ  ( a j U Xk w Xj  +  CLjUWXj +  VwuXj +  b ° u w ^  d x  <  0 ftis satisfied for every nonnegative function w E Cq(O).The motivation for investigation of this problem is the comparison principle for quasilinear second-order uniformly elliptic equations in divergence form∂(4) Q(u) =  — ——o∙jix, u, Du) +  b(x, u, Du) in O.∂XjIn fact the maximum principle in the linear case is the base for the validity of the comparison principle for weak C 1(0) smooth sub- and supersolutions of (4). More precisely, ifl lwhere St =  v(x) +  t(u(x) — v(x), Pt =  Dv(x) +  t(Du(x) — Dv(x) and u, v E С г(й) are weak sub-and supersolutions of (4), then the validity of the comparison principle for (4) is reduced to the validity of the maximum principle for linear equation (1) with the above coefficients (5). As a consequence of the maximum principle we get immediately the uniqueness and the continuous dependence on the data of the weak solutions of (1) and (4). Moreover, using suitable chosen barrier functions one can estimate the amplitude of the weak solutions of (1) or(4) which is an important step in the proof of the existence of a classical solution by means of the Leray-Schauder fixed point theorem. The maximum principle is important also in the investigations of the asymptotic behaviour of the solutions of linear and quasilinear parabolic equations in divergence form which appear in the population dynamics modeling a population which will persist or will go extinct.There are two type of conditions guaranteeing the validity of the maximum principle. The first of them are necessary and sufficient and are given in [1] for linear equations in divergence form and in [2] for general nondivergence formOn principal eigenvalue for linear second order elliptic equations 163equations. One of the main results in [1] and [2] is that the maximum principle for the operator L holds if and only if the first eigenvalue of L with zero Dirichlet data is positive. It is clear that the positiveness of the first eigenvalue Xl is not easy checkable condition so that this result is more useful for some theoretical investigations. However, there are some qualitative properties of A^  which can be used one to find out lower and upper bounds for the first eigenvalue. For example, A  ^ is an increasing function with respect to the coefficient 6° and a decreasing one with respect to the domain inclusions, i.e. Az,(ft°) > Az,(ft°) if 6° > 6° and Al(^) ≤ if О D Ö . Moreover, A^  is Lipschitz continuous withrespect to the coefficients a*∙, ¥ ,  6° (using the L°° norm) and concave function of b° (see [2]).There are also second type results which are only sufficient but easy checkable conditions for wide class of equations. They are given, for example, in [4], [6],[8] (see also the references there) and guarantee the maximum principle for (1) if one of the following assumptions is satisfied:(i) b° — diva0 > 0  in O, a0 = (a°, ∙ ∙ ∙ , a°);(ii) b° — divb > 0  in O, b=  (b1,--- ,6n);(6) (iii) The matrix A  +  A* is a nonnegative one, where(  ак- Ы \A  — ^ о j  and A* is the conjugate matrix of A.Unfortunately, conditions (6)i,  (6)∏ are not useful for quasilinear equations(4) because the derivatives of the coefficients a!∙, V given by (5) are not under control. That is why (6)i ,  (6)∏ are replaced in the nonlinear case with some additional structure assumptions guaranteeing that aj or Ы are identically equal to zero (see theorem 9.5 in [6]). By the way, (6)i ,  (6)∏ are not sharp even in the linear case because they guarantee that the discrete spectrum of the operator L (or of the formal self-adjoint operator L* of L) is on the right hand side of the origin. However, it is possible the first eigenvalue of L (or L*) to be far from the origin.As for (6)iü,  it seems to be the most general sufficient condition but it is not sharp, too. Following the idea in [7] we obtain that (6)ш is not invariant if equation (1) is rewritten in an equivalent way, for example with(7) Lu  =  — (a,jUXk +  (a°j +  f 3) u)  +  {V +  f J) uXj +  (b° +  d i v / )  и\  ∕  X jfor arbitrary vector f ( x ), / J E C0,1(0). Now (6)ш for equation (1) in the new164 A. Fabricant, N. Kutev, T. Rangelovform (7) isThe matrix A f  +  Aj is a nonnegative one, where(8)Condition (8) can be better than (6)ш for some special choice of / .Starting from the idea of Protter in [7] we consider the whole class of equa­tions (7) instead of (1) and sufficient conditions (8) instead of (6)i∏. Moreover, we prove in section 2 that (8) is also a necessary condition for the validity of the maximum principle for symmetric operators if (8) is taken over the set of all vectors /(ж), / J E C 0,1(0). Unfortunately, the same result is not true fornonsymmetric operators. The reason is that the matrix - ( A f  +  Aj) in (8) corre- ^ zsponds exactly to the symmetric part L q =  - ( L  +  L*) of the operator L and thefirst eigenvalue of Lean be far from the first eigenvalue of Lo, see theorem 3 in section 3.However, over the set of nondegenerate transformations of the operator L pre­serving the first eigenvalue of L, for example, Lu =  e~zL(uez) for £ E C 1 (0), we get as in the previous case a necessary and sufficient condition for the maximum principle for nonsymmetric operators.In this way we prove in section 2 several equivalent formulas for the first eigenvalue Al which are different from the well known results and in many cases are more convenient for lower and upper estimates for A ∙^Using the new expressions for A  ^ we get in section 3 some quantitative prop­erties of the first eigenvalue Al with respect to the coefficients a!∙, V as well as2. M ain results and definitionsIn this section we will recall some definitions for the first eigenvalue Al . If the operator L is a symmetric one, i.e. a® =  fr7', then the variational formulation of Xl is given in the following wayftAs it is well known (see [5]) the above infinum is attained for a function Ф  e Ф  > 0 in O, which solves the equation Ьф = А^</> in О, 0 = 0 on <90with respect to the matrix(9)nwhere the infinum is taken over all functions v E Hq (O), f  v2 dx =  1.On principal eigenvalue for linear second order elliptic equations 165in a weak sense. The function ф is the first eigenfunction of L and every weak solution ф E of the above equation is a multiple of ф.When L is a nonsymmetric operator then the following “max-min” represen­tation formulae for the first eigenvalue Al holds(10) AL =  sup inf (L v /v ), V E W  ,n(0), v > 0 in ft.V xIn order to formulate our results we will rewrite the operator L in the following equivalent way(11) Lu = — (akuXk +  (gj -  c?) u)  + (gj + c?) uXj + b° иXjwhere gJ = + a°), c> = -  a°).The reason is that the influence on A  ^of the coefficients g J from the symmetric part Lq of L is quite different in comparison with the coefficients cJ forming the nonsymmetric part \{L  — L*) of L.For every Lipschitz continuous vector f ( x ), / J E C0,1(0) let us introduce the notationso-Lo(∕) =  ess ( b° +  div /  -  a kA f 3 + gJ) ( f k + gk))(12) яеv <?l 0 = sup a L ( f )fJ£C o-∏∏)where j  = and L q  is the symmetric operatorL q u  =  -  ( c i j U Xk +  g i u j  +  g j u Xj +  b ° u .Now for symmetric operator L q we have the following result.T heorem  1. Let the operator L q satisfy (2) and (3). Then <jl0 =  Al 0 and hence the maximum principle for L q holds if and only if a l 0 > 0.Pr oof .  For arbitrary f ( x ), / J E C0,1(0) we get from (9) the inequalitiesXLo = inf /  (a^vX;ivXk +  2g3vvx . +  ( f 3v2)Xj +  b°v2) dx=  i n f  f  \ a k v Xj +  a f  (gm +  f m ) v  [ v Xk +  a sk (gs +  f s ) v ]+ b° +  d i v / - a ^ ( P  + g J) ( f k + gk) u2|  dx > aLo(f)i.e. ALo > supcrLo(/) =  crLo. ∕166 A. Fabricant, N. Kutev, T. RangelovIn order to prove the opposite inequality we will use a special choice of / .  For every positive constant 5 > 0, there exists from (1.10) in [2] a function u5 £ (7∞(Q) 5 u5 > 0 in O, such that L qu0 > (Al0 — 6)uö. Now for / J =—ак- u*Xk/ и5 — gi we get from (12) the estimateCLo ≥ CLo (/) = ess^inf [Lqu5 j u s) > XLo -  5.After the limit 5 —>► 0 we have the desired inequality <j l 0 > Al0 and hence&L0 =  Al0∙ □As for general nonsymmetric operators L, an equivalent definition of A  ^ by means of cfl is a little bit more complicated. More precisely, let us introduce for every Lipschitz functions £ (ж), f^(x) £ C0,1^ )  the notation(13)0 l( / ,  z ) = essginf ( b °  + di v /  -  +  g3){ fk +  gk) +  c ? z Xj -  { a f j  z X j z X k )aL = sup a L(f ,z )z,p e c 0’1 (si)The following theorem gives the relation between the first eigenvalue A  ^ of the nonsymmetric operator L and the first eigenvalues of the family of suitable chosen symmetric operators.T heorem  2. Let the nonsymmetric operator L satisfies (2) and (3). Then ol — Xl and hence the maximum principle for L holds if and only if cfl > 0. Moreover, the following identity takes place(14) aL = sup \ Mzzec°<l (si)where operator M z is defined asM zu = ^ ( e - zl2L{ezl2u) + ezl2L*{e-zl2u)}= ^(L  + L*) и + '^ c?zXj -  i  akzxjzx^ j  u.Pr oof .  From (9) and the chain of inequalitiesAmz — inf f v M zv d x =  inf f  e~z/2v L(ez/2v) dx уеЩ(п)п уеЩ(п)п< f  e~z/2wL(ez/2w) dx =  ( f  e~z(j)L(j)dx j ( f  е~гф2 dx j =  Aln \ n  J \ n  JOn principal eigenvalue for linear second order elliptic equations 167where w =  е~г!2ф ^ f  е~гф2 dx^j and ф is the first eigenfunction of L, we get the estimate(15) sup Xmz ≤ A ∙^zIn order to prove the opposite inequality let us consider a sequence ∩j ofexpanding C°° smooth subdomains of O, UfL∙ = O, Al(^) = ∏m ^(fL ∙), wherej —>■ ooAl(^j) is the first eigenvalue of L in Oj. If ф > 0, ф > 0 are the first eigenfunc­tions of L and L* respectively, we consider the truncated functions! kj for X E О and ln(0/,0) > k j , ln(0/^) for ж E О and raj < ln(0^) < k j ,raj for ж E О and ln(0/^) <  raj,where kj =  supln(</>/?/>), raj =  inf ln(0/'0). fij∙Simple computations give us the identity М^-г; = A ^  in Öj, г; =  (фф)1/2. Since V > 0 in Oj it follows from corollary 2.1 in [2] that Ам j ≥ Using the monotonicity of Am j with respect to the domain inclusions, after the limit j  ^  oo we get the inequality sup Amz (&k) ≥ for every к =  1, 2, ∙ ∙ ∙ .zFrom theorem 1 we havesup Amz (&k) — SUP ess i∏f2 ∕b° + d z Xj -  ^a jzxjzxk +  div/ -  a j ( f J +  ^ ) ( / fe +  / )= °ь{^к)  i.e. aL(£lk) = sup A Mz(^k) > A l -After the limit к —>► oo, from (15) we obtain the final result (14)ol =  Al  = sup XMz ∙ □Using theorem 2 we will give here different variants of cfl or equivalently for AL which are useful for the investigations of the qualitative properties of A^  in section 3.P roposition  1. Let the operator L satisfy (2), (3) and gi, cJ E (70,1(O). Then the identity(17) AL — ctl — sup ess inf b° +  divf  +  а кс^ск — a k( f i  +  g^)(fk +  gk) zec^(∩) xeQ *-holds, where / J = =LcJ — gi +  akzXk.R em ark  1. For the special choice of ∕  and £ in (17), / J = cJ — gi =  — a® z =  0 we get immediately from (17) the condition (6)i and for / J = —cJ — gi =  —fc7, z — 0 respectively, the condition (6)∏.168 A. Fabricant, N. Kutev, T. Rangelov3. P rop erties  o f th e  principal eigenvalueIn this section we will give some applications of theorems 1, 2 and propositions1 for qualitative properties of A ∙^ For this purpose let us recall the well known monotonicity and concavity properties of with respect to b°: Xl is an increas­ing and concave function with respect to 6°.For the time being it is not known whether a similar monotonicity result forXl is true with respect to the matrix or coefficients a!∙, fr7, respectively g∙7,cJ. To give some particular answer of these questions we will need the followingproperties of Xl .T heorem  3. Let the operator L satisfy (2), (3). Then the inequalities(18) Xl0 < XL < Xl xhold, where L q is the symmetric part of L and L\ =  L q +  a kc^ck.Moreover, if additionally ah, gi, c∙7 E C 1(0) then(19)(i) Xl =  Xlq фь =  Фь0 in О йт(сф2ь^) =  0 in O,where фL^  фLo are the first eigenfunctions of L and L q respectively;(ii) Xl =  A !^ c∙7 = - a kzXk for some z E C 1(0) and more precisely,∆iz = In {фь/фь*)Pr oof .  By integration by parts we get immediately the estimateAl = f  фь^фь dx = f  а^(фь )Х] (фь)хк + 2gj Фь(Фь)х, + Ь°ф2ь dx n n> inf f  ( t f v XjvXk +  2gJvvXj +  b°v2) dx = XLo.veH0(∩)∩ V ∕Since c?zXj — \a k zx .zxk < a kc^ck we get from (14) and theorem 2 the in­equalities AL =  &L — sup Amz ≤ ^Li j where M z is defined in theorem 2.zNow let us suppose that div (c 0 |o) = 0 in О and for simplicity let us denote Фь0 =  Фо and A^ o = Ao∙ Since Lu =  L qu+ ^div(eM2) it follows that L0o = Ao0o, 0o = 0 on <90, 0o > 0 in O, i.e. 0o is the first eigenfunction of L, 0o = Фь and Xl = X о =  Al0∙Suppose that 0o = 0l ∙ An easy calculations give us the identityХьфо = Хьфь = Ьфь = Ьф0 = Lo0o + т -div (сф1) = Х0ф0 + -^-div (сф1)Фо Фоi.e. (AL — Ао) 0о = div(c0Q) in О. Integrating the above expression in О we get immediately that A^  = Aq = A^0 and div(c0g) = div(c0^ ) = 0 in O.On principal eigenvalue for linear second order elliptic equations 169Finally, let us suppose that =  Ao∙ By integration by parts we haveJ  фоЬофо dx =  Ao = Xl =  J  фь^фь dx = J  фь^^офь dxn n nand from theorem 2 in section 6.5 in [5], it follows that фь = Фо-To prove (20) Ü, let us suppose that Cj  = akjZxk for some £ E C 1(0). Since the operator ezL(ue~z) =  L\u  has the same first eigenvalue as the operator L we have Al  =  A^. Moreover, if ф^  is the first eigenfunction of L\ then Фь =  егфЬ1 , фь* = е~гфЬх and г =  ln{фь/фь*)-The rest of the proof of (20)^ follows by means of (14) and the special choice(16) of in the proof of theorem 2. □Using theorem 3 we will give some partial results about the monotonicity ofAL with respect to the matrix |a ^  j .  For this purpose we introduce the operator M u  =  -  ( m kj U Xk + (gj -  cj ) u)j +  (gj +  cj ) uXj +  b°u.Proposition 2. Let the operators L and M  satisfy (2) and (3) and >Suppose that one of the following assumptions is satisfied:i) L and M  are symmetric operators;ii) Am = Am0; M q =  - ( M  +  M*);iii) cJ =  a!j zXk for some z E C0,1(0) and ркс^ск =  ak c^ck for a.e. x E Öwhere j  ∕iv) +  kl,  к =  const > 0 , /  is the unit matrix and fik c^ck <к (сonj  |0 |)n 2^ for a.e. x E Ö, where u)n is the volume of the unit ball in R n and|0| =  messft.Then the inequality Xl > Am holds.As for the monotonicity of A  ^ with respect to gi and cJ , it is trivially to prove that Al increases when div g decreases. However, the monotonicity of A^  with respect to cJ is not clear. For convenience we will denote the operator L with Lc and with Ac and фс the first eigenvalue and the first eigenfunction of Lc, respectively, when the coefficients ak, b° are fixed and cJ vary.Proposition 3. Let the operator Lc satisfy (2) and (3). Then the following inequality holds:(20) Atc > Ac for every \t\ > 1 where Xtc =  Ac Asc = Aq for every s E R.170 A. Fabricant, N. Kutev, T. RangelovAs for the concavity of Ac with respect to the coefficients с we have such result only in fixed directions t c , t E R. In different directions с , с a similar result is true with a correction term. More precisely, we get the following result.P roposition  4. Let the operators Lc and L t satisfy (2) and (3). Then Act is a concave function of t2. If  сф  с then for every 0 < t < 1 the inequalityAs ≥ (1 — t)^Lc+ t \b 0 holds, where Su  =  L(1_^c+^ + ^ ( l  — t)ak(cP — &)(ck — ck)u.R E F E R E N C E S[1] S. A g m o n . On positivity and decay of solutions of second-order elliptic equations on Riemannian manifolds. In: Methods of Functional Analysis and Theory of Elliptic Equations (Ed. D. Greco), Lignori Ed. Napoli, 1983, 19-52.[2] H. B e r e s t y c k i , L. N ir e n b e r g , S. R. S. Va r a d h a n . The principal eigen­value and maximum principle for second order elliptic operators in general domains. Comm. Pure Appl. Math., 47, (1994), 47-92.[3] M. D o n s k e r , S. R. S. Va r a d h a n . On the principal eigenvalue of second- order elliptic differential operators. Comm. Pure Appl. Math., 29, (1976), 595-621.[4] J. D o u g l a s , T. D u p o n t , J. Se r r in . Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form. Arch. Rat. Mech. Appl., 42, (1971), 157-168.[5] L. E v a n s . Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, AMS, Providence, Rhode Island, 1998.[6] D. G il b a r g , N. T r u d in g e r . Elliptic Partial Differential Equations of Sec­ond Order. Springer, Berlin , 1983.[7] M. P r o t t e r . Lower bounds for the first eigenvalue of elliptic equations, Annals of Math., 71, (1960), 423-444.[8] M. P r o t t e r , H. W e in b e r g e r . Maximum Principle in Differential Equa­tions. Providence Hall, 1967.A. Fabricant, N. Kutev, T. Rangelov Institute of Mathematics and Informatics Bulgarian Academy of Sciences 8 G Bonchev s t r 1113 Sofia, BULGARIA

On Principle Eigenvalue for Linear Second Order Elliptic Equations in Divergence Form

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On Principle Eigenvalue for Linear Second Order Elliptic Equations in Divergence Form

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