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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