A Maximum Principle Result for a General Fourth Order Semilinear Elliptic Equation

We obtain maximum principles for solutions of some general fourth order elliptic equations by modifying an auxiliary function introduced by L.E. Payne. We give a brief application of these maximum principles by deducing apriori bounds on a certain quantity of interest.

• Nonlinear
• Fourth Order
• Partial Differential Equation
• Semilinear

1. Introduction

In [1] , Payne obtains maximum principle results for the semilinear fourth order elliptic equation

by proving that certain functionals defined on the solution of (1) are subharmonic. In this work, functionals containing the terms are utilized and apriori bounds on the integral of the square of the second gradient and on the square of the gradient of the solution are deduced. Since then, many authors [2] - [11] and references therein have used this technique to obtain maximum principle results for other fourth order elliptic differential equations whose principal part is the biharmonic operator.

Other works deal with the more general fourth order elliptic operator, where and. In [12] , Dunninger mentions that functionals containing the term can be used to obtain maximum principle results for such linear equations as

A similar approach is taken in [13] for a class of nonlinear fourth order equations.

In this paper, we modify the results in [1] and a matrix result from [14] to deduce maximum principles defined on the solutions to semilinear fourth order elliptic equations of the form:

Then we briefly indicate how these maximum principles can be used to obtain apriori bounds on a certain quantity of interest.

2. Results

Throughout this paper, the summation convention on repeated indices is used; commas denote partial differentiation. Let be a symmetric matrix. Moreover let, be a uniformly elliptic operator, i.e, the symmetric matrix is positive definite and satisfies the uniform ellipticity condition: , where is a bounded domain in and.

Let u be a solution to the equation

where f is say, a function. Now we define the functional

We show that is subharmonic and note that the constants and and any constraints on f are yet to be determined.

By a straight-forward calculation, we have

Now we write

By expanding out the derivative terms in parentheses, we see that is

The terms in lines 2 and 3 above containing two or more derivatives of can be rewritten using (3) in the form, where denotes the matrix which is the inverse of the positive definite matrix. Furthermore, we use the identity to rewrite the last two terms in line 4. Hence,

Using the identity above for and the additional identity, , which can be obtained by computing, for the terms at the ends of lines 6 and 3 respectively, we obtain

To show that is nonnegative, we establish a series of inequalities based on the following one from [14] : Let be any matrix. From the inequality

One can deduce

Repeated use of (9) on terms in lines 2, 3, 4, 5 in (7) yields the following:

Furthermore, by completing the square, we obtain useful inequalities for the last two terms in line 1 and the third term in line 2 of (7):

We add (10)-(21) and label the resulting inequality, for part of, as

Now,

Since is positive definite, for a sufficiently large value of, where depends on the coefficients and their derivatives, and for a sufficiently large value of, say, where depends on the constants, , , and various derivatives of, can be made nonnegative as desired. Thus we have the following result.

Theorem 1. Suppose that is a solution of (2) and. If, where, , is a nonnegative function such that then there exists positive constants and sufficiently large such that P cannot attain its maximum value in unless it is a constant.

We note that the function satisfies the conditions stated in Theorem 1 for a solution that is bounded above.

3. Bounds

Here we give a brief application of Theorem 1.

Suppose that

By Theorem 1,

Using integration by parts on the first two terms of P yields the identity

Upon integrating both sides of the previous inequality we deduce

References

1. Payne, L.E. (1976) Some Remarks on Maximum Principles. Journal d’Analyse Mathematique, 30, 421-433. http://dx.doi.org/10.1007/BF02786729

2. Danet C.-P. ,et al. (2014)Two Maximum Principles for a Nonlinear Fourth Order Equation from Thin Plate Theory Electronic Journal of Qualitative Theory of Differential Equations 31, 1-9.

3. Danet C.P. ,et al. (2011)Uniqueness in Some Higher Elliptic Boundary Value Problems in n Dimension Domain Electronic Journal of Qualitative Theory of Differential Equations 54, 1-12.

4. Dhaigude, D.B., Dhaigude, R.M. and Lomte, G.C. (2012) Maximum Principles for Fourth Order Uniformly Elliptic Equations with Applications. International Journal of Applied Mathematical Research, 3, 248-258.

5. Goyal, S. and Goyal, V.B. (2011) On Some Fourth Order Elliptic Boundary Value Problems. Applied Mathematical Sciences, 36, 1781-1793.

6. Lomte, G.C. and Dhaigude, R.M. (2013) Maximum Principles for Fourth Order Semilinear Elliptic Boundary Value Problems. Malaya Journal of Matematik, 1, 44-48.

7. Lu, W.D. and Wang, J.H. (1981) Maximum Principles for Some Semilinear Elliptic Equations of Fourth Order, and Their Application. Sichuan Daxue Xuebao, 4, 33-45. (In Chinese)

8. Mareno, A. (2008) Maximum Principles for a Fourth Order Equation from Thin Plate Theory. Journal of Mathematical Analysis and Applications, 343, 932-937. http://dx.doi.org/10.1016/j.jmaa.2008.02.004

9. Mareno, A. (2010) Integral Bounds for von Karmans Equations. ZAMM, 90, 509-513. http://dx.doi.org/10.1002/zamm.200900325

10. Mareno, A. (2011) Maximum Principles for Some Higher Order Semilinear Elliptic Equations. Glasgow Mathematical Journal, 53, 313-320. http://dx.doi.org/10.1017/S001708951000073X

11. Zhang, H. and Zhang, W. (2002) Maximum Principles and Bounds in a Class of Fourth Order Uniformly Elliptic Equations. Journal of Physics A: Mathematical and General, 35, 9245-9250. http://dx.doi.org/10.1088/0305-4470/35/43/318

12. Dunninger, D.R. (1972) Maximum Principles for Solutions of Some Fourth Order Elliptic Equations. Journal of Mathematical Analysis and Applications, 37, 655-658. http://dx.doi.org/10.1016/0022-247X(72)90248-X

13. Nichols, E.C. and Schaefer, P.W. (1987) A Maximum Principle In Nonlinear Fourth Order Elliptic Equations. Lecture Notes in Pure and Applied Mathematics, 109, 375-379.

14. Schaefer, P.W. (1987) Pointwise Estimates in a Class of Fourth-Order Nonlinear Elliptic Equations. ZAMP, 38, 477-479. http://dx.doi.org/10.1007/BF00944964