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					SCIENTIA
Series A: Mathematical Sciences, Vol. 15 (2007), 17–22
               e
Universidad T´cnica Federico Santa Mar´ıa
Valpara´ıso, Chile
ISSN 0716-8446
 c Universidad T´cnica Federico Santa Mar´ 2007
                   e                      ıa




           Maximum Principles For Some Elliptic Problems

                                      M.M. Al-Mahameed

         Abstract. In this paper we introduce a maximum principle for some semilinear
         elliptic equations subject to mixed boundary conditions which may be used to
         deduce bounds on important quantities in physical problems of iterest.




                                        1. Introduction
                                                                          2
         In [6],maximum principles for the functions p = g(u) | u| + h(u) and
                             u
                      2
    q = g(u) | u| + c f (s)g(s)ds, c ∈ R, which are defined on solutions of the
                             0
semilinear partial differential equation ∆u + f (u) = 0 in some region Ω ⊂ Rn are
found using the classical maximum principle [3]. In [7], a maximum principle for the
function q at a critical point of u under some conditions on ∂Ω is introduced. In [2],
the following result is proved : Let u ∈ C 3 (Ω) be a solution of


                                 ∆u + f (u) = 0             in Ω ⊂ Rn ,

                                      u=0                   on ∂Ω.
If the boudary ∂Ω has a nonnegative mean curvature, then the function

                                                        u
                                             2
                                  Φ = | u| + 2              f (s)g(s)ds
                                                    0

assumes its maximum at a point where u = 0. In [3], maximum principles are derived
for certain functions defined for solutions of the equation


                                       ∆u + λρ(x)f (u) = 0

    2000 Mathematics Subject Classification. Primary 35B50, 35J60.
    Key words and phrases. Maximum Principles, elliptic equations.
                                                  17
18                                       M.M. AL-MAHAMEED


in some region Ω ⊂ R2 subject to a mixed boundary condition.
    In this paper we derive maximum principles for functions defined for solutions of
the semilinear equation
(1.1)                                       ∆u + f (x, u) = 0
     in some region Ω ⊂ Rn subject to a mixed boudary condition.
     In order to motivate our work, let us first look at the one dimensional problem

(1.2)                                       uxx + f (x, u) = 0.

If we multiply (1.2) by ux we get

                                    1 2
                                     (u )x + f (x, u)ux = 0,
                                    2 x
that is

                                    u
                         1 2
(1.3)                     u +           f (x, s)ds − H(x, u) = constant,
                         2 x
                                0

          where H(x, u) satisfies:

                                                                u

                                    Hx (x, u) =                     fx (x, s)ds.
                                                            0
Thus we conclude that the function

                                                   u

(1.4)                       p = u2 + 2
                                 x                     f (x, s)ds − 2H(x, u)
                                               0

is a constant, where u is a solution of (1.2). It is obvious that p satisfies a maximum
principle.
     Let u be a solution of (1.1) . We look for a function p of the form

                                                       u
                                        2
(1.5)                     p = | u| + 2                     f (x, s)ds − 2H(x, u),
                                                   0

where H(x, u) satisfies:

                                                               u

                                    H,i(x,u) =                     f,i (x, s)ds.
                                                           0
                MAXIMUM PRINCIPLES FOR SOME ELLIPTIC PROBLEMS                           19


     Our goal is to find conditions such that (1.5) satisfies a maximum principle.
     Let us first give the following lemma.
     Lemma. Let u be a C 3 (Ω) solution of (1.1) with f ∈ C 1 (Ω × R), Ω ⊂ Rn ,
n 2.Then the function p defined by (1.5) takes its maximum either on ∂Ω or at a
critical point of u.
     Proof. By differentiating (1.5) we obtain

(1.6)                        p,i = 2u,j u,ij + 2f u,i



(1.7)             ∆p = p,ii = 2u,ij u,ij + 2u,j u,iij + 2f ∆u + 2f,i u,i .
    Now we have

(1.8)                                      ∆u = −f,


(1.9)                                    u,iij = −f,j .
    This allows us to rewrite (1.7) as

(1.10)                              ∆p = 2u,ij u,ij − 2f 2 .
  From (1.6) and Schwarz’s inequality, it follows that

                                                                                  2
(1.11)        (p,i − 2f u,i )(p,i − 2f u,i ) = 4u,ji u,j u,ki u,k   4u,ij u,ij | u| .
    Consequently, by (1.10) and (1.11) , we can write

                                              Lk p,k
(1.12)                                ∆p +         2      0,
                                              | u|
where

                                              1
                               Lk = 2f u,k − p,k .
                                              2
Hopf’s first maximum principle [4] implies the lemma.

                            2. The Result and its Proof
    We give our result by the following theorem.

    Theorem. Let u be a C 3 (Ω) solution of the problem
        ∆u + f (x, u) = 0      in Ω ,
                                         ∂u
                       u=0       on Γ1 , ∂n = 0 , on Γ2 , Γ1 ∪ Γ2 = ∂Ω ,
20                               M.M. AL-MAHAMEED

                                                             ∂u
where f ∈ C 1 (Ω ×R), Ω is a convex domain in R2 and ∂n denotes the outward
normal derivative. Then the function p defined by (1.5) takes its maximum at a
critical point of u.
     Proof. We will show that p cannot attain its maximum on ∂Ω unless it is
attained at a critical point of u which is on Γ2 .
      Suppose that p takes its maximum at a point M ∈ Γ1 . Then M cant be a critical
                                                   ∂u
point of u. Since u = 0 on Γ1 , we have| u| = ∂n and

                              ∂p
(2.1)                            = 2un unn + 2f un ,
                              ∂n
  Where un denotes the outward normal derivative.By introducing normal coordinates
in the neighbourhood of the boundary , we can write

(2.2)                          ∆u = unn + kun = −f,
 where k denotes the curvature of the boundary.Thus it follows that

                                    ∂p
(2.3)                                  = −2ku2 ,
                                             n
                                    ∂n
                           ∂p
   and since Ω is convex , ∂n 0 at M . This contradicts Hopf’s second maximum
principle [5].
     We now suppose that p takes its maximum at M ∈ Γ2 and that M is not a
critical point of u. Since ∂n = 0 on Γ2 , we have | u| = ∂u and
                           ∂u
                                                         ∂t


                                    ∂p
(2.4)                                   = 2ut utn ,
                                    ∂n
where ut denotes the tangential derivative of u. In terms of normal coordinates in the
neighbourhood of the boundary , we have

(2.5)                             utn = unt − kut ,
so that

                              ∂p
                                 = −2ku2 on Γ2 .
                                          t
                              ∂n
Thus we again have a contradiction of the second maximum principle when Ω is
convex. The lemma, and our calculations , gives the theorem.
    Example. Let u ∈ C 3 (Ω) be a positive solution of the problem

                 ∆u + 4u − (x2 + x2 ) exp(α2 − x2 − x2 ) = 0
                             1    2             1    2          inΩ,



                                 u=0        on ∂Ω,
                 MAXIMUM PRINCIPLES FOR SOME ELLIPTIC PROBLEMS                                  21


                                                                                            where

                                            Ω = {x = (x1 , x2 )\ |x| < α}

and

                          f (x, u) = 4u − (x2 + x2 ) exp(α2 − x2 − x2 ),
                                            1    2             1    2

it follows from the theorem (2.1) that

                                  u
                  2
              | u| + 2                4s − (x2 + x2 ) exp(α2 − x2 − x2 )ds − 2H(x, u)
                                             1    2             1    2
                              0


                                 u
                                                                                        

               max 2                 4s − (x2 + x2 ) exp(α2 − x2 − x2 )ds − 2H(x, u) 
                                             1    2             1    2
              Ω∪ ∂Ω
                              0




or


                          2
                 | u|                     max 4u2 − 2(x2 + x2 ) exp(α2 − x2 − x2 )u
                                                       1    2             1    2
                                      Ω∪ ∂Ω



                              − 4u2 − 2(x2 + x2 ) exp(α2 − x2 − x2 )u
                                         1    2             1    2

.
From the above inequality , we get

                                      2                          2
                      | u|                  4(u2 − u2 ) + 2α2 eα uM ,
                                               M


where uM is the maximum of u in Ω ∪ ∂Ω.


                                            3. Concluding Remarks

        1. One can prove the result of the lemma for the function
                                  u
                  2
      p = g(u) | u| + 2               f (x, s)g(s)ds − 2H(x, u) with suitable assumptions on g(u)
                              0
as in [5].
     2. Theorem 2.1 is also valid for n > 2, [5].
     3.One may give an extention of the maximum principle for a uniformly elliptic
equation Lu + f (x, u) = 0 under suitable assumptions, [5].
22                                    M.M. AL-MAHAMEED


                                          References
[1] A. Greco,A Maximum principle for some second order elliptic semilinear
    equations.Rend. Sem. Fac. Sci. Univ. Cagliari. 59(2)(1989), 147-154.
[2] L.E. Payne and I.Stakgold, Nonlinear problems in nuclear reactor analysis.Lecture Notes in math.
    322(1972) 298-307.
[3] P.w.Schaefer and R.P. Sperb.Maximum principles and bounds in some inhomogeneous elliptic
    boundary value problems.SIAM J.Math. Anal.8(1977), 871-878.
[4] M.H.Protter and Weinberger, Maximum principles in diffefential equations, Prentice-
    Hall,Englewood Cliffs, N.J.,1984.
[5] M.H.Protter and Weinberger, Maximum principles in diffefential equations, Prentice-
    Hall,Englewood Cliffs, N.J.,1984.
[6] M.M.Al-Mahameed, Maximum principles for semilinear elliptic partial differential equations, Far
    East. j. applied Math.3(3)(1999), 287-292.
[7] M.M.Al-Mahameed ,A maximum principle for the P-function at a point where grad u = 0 in some
    region Ω ⊂ Rn .J .Ins.Math.and Comp.Sci..14(1)(2001), 65-71.


                                                           Received 28 06 2006, revised 06 03 2007



Department of Mathematics,
Irbid National University,
P.O.Box 2600,Irbid,
Jordan
     E-mail address: al mahameed2000@yahoo.com

				
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