A Sharp Minimum Principle for the Problem of Torsional Rigidity

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1 Journal of Mathematical Analysis and Applications 33, Article ID jmaa99969, available online at on A Sharp Minimum Principle for the Problem of Torsional Rigidity Xi-nan Ma* Department of Mathematics, East China Normal Uniersity, Shanghai 0006 People s Republic of China Submitted by Colin Rogers Received May, 998 For the equation of torsional rigidity, using the unique continuation of analytic function, we get a sharp minimum principle for a combination of the solution and its gradient As an application, a lower bound for the minimum stress on the boundary is obtained 999 Academic Press INTRODUCTION In 977, Payne and Philipin 6 derived a minimum principle for the stress function už x, x of the torsional rigidity problem: u in R Ž u 0 on Ž This principle states that if is a simply connected, strictly convex bounded domain in R with smooth boundary, then the function P Ž x Du u Ž,, takes its minimum value on, where Du is the gradient of u In Philippin 5, the above minimum principle has been extended to the problem of torsional creep which is given by the following nonlinear generalization of Ž : u Ý gž Du x x in R, i under suitable hypotheses on g 99 Mathematics Subject Classification 35J5 * xnma@mathecnueducn i i X99 $3000 Copyright 999 by Academic Press All rights of reproduction in any form reserved

2 58 XI-NAN MA The aim of this paper is to extend Payne and Philippin s minimum principle to, more precisely we have the following: THEOREM Let u C be the solution of in a strictly conex smooth bounded domain, then the function PŽ x Du u attains its minimum on the boundary, unless P x is a constant on For the proof of the above Theorem we need the following two remarks: Remark If is a strictly convex domain, then the solution už x, x of Ž Ž has only one critical point in, which is a direct consequence of the fact that the convexity of implies the convexity of the level line of u Remark According to the work of Nirenberg 4 we conclude that u is an analytic function in, a feature which will be used in this paper In section we shall present the proof of the Theorem using the unique continuation of analytic functions Section 3 is devoted to an application In this paper, the summation convention over repeated indices Žfrom to will be employed, and the following abbreviations will be adopted: u u u u, u, uij, x x x x i j Remark 3 The similar minimum principle for the equation of constant mean curvature with prescribed constant contact angle boundary condition has also been obtained in 3 A MINIMUM PRINCIPLE We consider the boundary value problem Ž Ž in a strictly convex bounded domain in R with smooth boundary, and define the following function: P Ž x Du u Ž We know that P Ž x takes its imum value at the critical point for 7, and on the boundary for 9 We concentrate now our attention on,, and state the following:

3 SHARP MINIMUM PRINCIPLE 59 LEMMA 5 The function P x defined in satisfies the following elliptic differential equation: ž / ukuki ui 4u i P P 4Ž Ž Ž i Du For the proof of Lemma, we make use of the definition Ž and of Ž the following identity valid in R only : u u Du Du Ž u u u u u uu u u ij ij i ij k kj i j ij The details of the computations are omitted here since they have been given in 9 From Lemma and the Hopf imum principle 8 we conclude that P Ž x takes its minimum value either on the boundary or at the unique critical point C for, For, the second alternative has been rejected by Philippin 5 The purpose of this section is to show that even for the second alternative can also be rejected unless P Ž x is a constant in This can be achieved as a consequence of the following: THEOREM Let u C be the solution of If PŽ x Du u attains its minimum at the unique critical point C, then PŽ x is a constant on For the proof of the Theorem, we use the strong unique continuation of analytic functions, so our program is to show that all order deriaties of PŽ x anish at C To this end, we choose the origin of the coordinate axes at the critical point C, then and orient the axes x and x in such a way that From Philippin 5,we know už C už C 0, Ž 3 u C 0 4 už C 0 už C 0, Ž 5 which will be used essentially in the following proof Proof of Theorem Our proof is divided into four steps Step We show that the derivatives of PŽ x up to order vanish at C

4 60 XI-NAN MA First we compute the first derivatives of PŽ x at C Since at any point x then from 3, we obtain P uiui u Ž 6 P uiui u, Ž 7 PŽ C PŽ C 0 Ž 8 Now we compute the second derivatives of P x at C From 3 6, we have at C P u u Ž 9 P u 0 Ž 0 P u u Ž The fact that P x attains its minimum at C leads to PŽ C PŽ C P Ž C 0 Ž From 5,, 9, and we obtain už C už C Ž 3 PŽ C PŽ C 0 Ž 4 Now we shall use induction to show that all order derivatives of P x at C vanish Step As a first step for induction, we shall show that the derivatives of PŽ x of order 3, 4 at C vanish First we claim 3 P k 3k x x Using 9, 4, and 3 we have Ž C 0, k 0,,,3 Ž 5 P 3Ž C 4u 3Ž C Ž 6 x x P Ž C 4u Ž C Ž 7 x x x x P Ž C 4u Ž C Ž 8 xx xx P 3Ž C 4u 3Ž C Ž 9 x x

5 SHARP MINIMUM PRINCIPLE 6 Now, by differentiating, we obtain u 3Ž C u Ž C Ž 0 x xx u Ž C u 3Ž C Ž x x x To this end, use Ž 8, Ž 0, Ž 4, and Ž 6 Ž, we expand the function PŽ x in a Taylor series in a neighborhood of C: ½ 3 3 r P 3 PŽ x, x PŽ C Ž C cos 3 cos sin 3 3! x 3 P Ž C 3 cos sin sin OŽ r, x x where r, are polar coordinates: x r cos, x r sin Suppose ' P x 3Ž C P x x Ž C 0, then PŽ x is not a constant, so we are led to the following representation of PŽ x in a neighborhood of point C: with PŽ x PŽ c A cos3 r 3 OŽ r 4, Ž ' P x 3 C P x x C A 3 3! and P 3 x Ž C cos 3, P 3Ž C P Ž C ' x x x P x xž C sin 3 P 3Ž C P Ž C ' x x x From Ž 3 we conclude that PŽ x Pc has at least three nodal lines forming equal angles at point C, using Lemma we know that PŽ x attains its minimum only on or at the critical point C, which is a contradiction Thus A Ž C 0or 3 P x x 3 3 u Ž C 0 and Ž C 0, k 0,,,3 x x k 3k k 3k Ž 4

6 6 XI-NAN MA Now we claim 4 P k x x 4k Ž C 0, k 0,,,3,4 Ž 5 From the fact that the derivative of PŽ x follows that: up to order 3 at C vanish, it P k 4k Ž C 6u k 4k Ž C, k 0,,,3,4 Ž 6 x x x x We again differentiate, and obtain u i iž C u i Ž C, i 0,, Ž 7 x x xx4i Using the similar argument as earlier, 6 holds and u k 4k Ž C 0, k 0,,,3,4 Ž 8 x x Step 3 Now we assume all order derivatives of PŽ x up to n vanish at C, where n 5 Using the same argument as in step we have the following relations u i kiž C 0, k 5,6, n, i 0,,,3, k Ž 9 xx Step 4 We show that the derivatives of PŽ x of order n vanish at C Using the values of the derivatives of u up to order n at C, we have P k nk Ž C nu k nk Ž C, k 0,,, n Ž 30 x x x x By differentiating, we have n u 0, k 0,,, n Ž 3 k nk x x If n l, l, then 30 3 leads to P Ž C P Ž C P Ž C Ž P Ž C l n n n3 4 n x n x x x x l n n 3 n x x x x xx Ž 3 P Ž C P Ž C Ž P Ž C Ž 33 We are now able to show that the derivatives of PŽ x of order n vanish at C as in step Using the induction assumption and Ž 3 Ž 33,

7 SHARP MINIMUM PRINCIPLE 63 we expand P x in a Tayor s series in a neighborhood of point C: n r n n n ½ x ž 0 / n l n ž / ž n / n n P n x x Ž C cos sin ž / ž 3 / PŽ x PŽ C P Ž C cos Ž 34 Ž n! n n cos sin sin n n 3 cos sin ž / 5 l n n n cos sin OŽ r n As in step, we can show that the derivatives of PŽ x of order n vanish at C If n l, l 3, the same analysis as above may be used to obtain the desired result Up to now we have shown that all order derivatives of PŽ x vanish at C, according to the unique continuation theorem for analytic functions This implies that if the function PŽ x attains its minimum at C, then it must be a constant This establishes Theorem Combination of Theorem and Lemma implies the main Theorem 3 APPLICATION From Section, we know that the function PŽ x Du u attains its minimum on under the condition of the main Theorem As an application of this minimum principle for the function PŽ x, we get in this section a lower bound for the minimum value on of the stress Q defined Ž up to a constant factor as Q Du

8 64 XI-NAN MA THEOREM 3 Let už x, x be the classical solution of Ž Ž in a strictly conex bounded smooth domain, then the following inequalities hold where K Q min Q Ž 3 min K min u už C, Ž 3 K is the imum alue of the curature on Proof of Theorem 3 Assume PŽ x attains its minimum at x o According to the Hopf imum principle 8, using curvilinear coordinate system 9, we must have at x 0 P u u u 0, Ž 33 n n n n unless PŽ x is a constant on, where the boundary condition Ž has been used in the derivation of Ž 33 From the differential equation Ž evaluated on, we have u u KŽ x on, Ž 34 n n where KŽ x is the curvature of at x Combining Ž 33 Ž 34, and noting that Du u n 0on, we obtain K Ž x DuŽ x KŽ x Q Ž 35 It thus follows that and o 0 0 min Qmin, Ž 36 KŽ x K o už C Ž 37 K Ž x K If PŽ x is a constant on then a similar argument to Ž 35 leads to DuŽ x for any x, Ž 38 R už C, Ž 39 and is a disk with radius R, where R is a constant Until now we have completed the proof of Theorem 3 o R

9 SHARP MINIMUM PRINCIPLE 65 We note that 3 3 together with the complementary result 6 : leads to Q Q Q Ž 30 K min už C, Ž 3 K min Q Ž 3 K K min už C, Ž 33 k K min with equality if and only if is a disk, which has been proved by Bandle with a purely geometric result combined with the monotonicity of u with respect to ACKNOWLEDGMENT The author thanks Professor Shen Chun-Li and Professor Zhou Qing for many stimulating discussions The author also thanks the referee for his Ž her help in English and carefully reading REFERENCES C Bandle, On isoperimetric gradient bounds for poisson problems and problems of torsional creep, Z Angew Math Phys 30 Ž 979, 7375 L G Makar-Limanov, Solution of Dirichlet s problem for the equation u ina convex region, Math Notes Acad Sci USSR 9 Ž 97, X-n Ma, Sharp size estimates for capillary free surfaces without gravity, to appear in Pacific J Math 4 L Nirenberg, On nonlinear partial differential equations and Holder continuity, Comm Pure Appl Math 6 Ž 953, G A Philippin, A minimum principle for the problem of torsional creep, J Math Anal Appl 68 Ž 979, L E Payne and G A Philippin, Some remarks on the problems of elastic torsion and of torsional creep, in Some Aspects of Mechanics of Continua, Part, Jadavpur Univ, Calcutta, India, 977, pp L E Payne and G A Philippin, Some applications of the imum principle in the problem of torsional creep, SIAM J Appl Math 33 Ž 977, M H Protter and H F Weinberger, Maximum principles in differential equations, Prentice Hall, R P Sperb, Maximum principle and their applications, Academic Press, 98