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ISRN Mathematical Analysis
Volumeย 2012ย (2012), Article IDย 634316, 9 pages
http://dx.doi.org/10.5402/2012/634316
Research Article

On Maximum Principles for ๐‘š-Metaharmonic Equations

Department of Mathematics and Computer Science, Pennsylvania State University, Middletown, PA 17057, USA

Received 27 November 2011; Accepted 29 December 2011

Academic Editor: E.ย Beretta

Copyright ยฉ 2012 A. Mareno. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study homogeneous linear elliptic partial differential equations of even order. Several maximum principle results are deduced for such equations as well as a priori bounds for certain boundary value problems.

1. Introduction

The ๐‘ƒ-function technique for obtaining subharmonic functions defined on the solution of certain partial differential equations of order โ‰ฅ4 is well established. In [1] Miranda shows that the functional ๐‘ƒ=๐‘ข,๐‘–๐‘ข,๐‘–โˆ’๐‘ขฮ”๐‘ข is subharmonic where ๐‘ข is a solution to the biharmonic equation ฮ”2๐‘ข=0. Later, in [2], Payne uses functionals containing the square of the second gradient of the solution to semilinear equations of the formฮ”2๐‘ข=๐‘“(๐‘ข)(1.1) to deduce integral bounds on (ฮ”๐‘ข)2.

Other works such as [3, 4] develop maximum principle results for the sixth-order equations of the formฮ”3๐‘ขโˆ’๐‘Žฮ”2๐‘ข+๐‘ฮ”๐‘ขโˆ’๐‘๐‘ข=0(1.2) with constant coefficients. In [5, 6], ๐‘ƒ-functions containing the squares of terms of the form ฮ”๐‘–๐‘ข are used to obtain a priori that bounds for solutions to the constant coefficient ๐‘š-metaharmonic equationฮ”๐‘š๐‘ข+๐‘Ž๐‘šโˆ’1ฮ”๐‘šโˆ’1๐‘ข+โ‹…โ‹…โ‹…+๐‘Ž0๐‘ข=0.(1.3)

Most recently the authors in [7] obtain maximum principles results for the more general variable coefficient ๐‘š-metaharmonic equationฮ”๐‘š๐‘ขโˆ’๐‘Ž๐‘šโˆ’1(๐‘ฅ)ฮ”๐‘šโˆ’1๐‘ข+๐‘Ž๐‘šโˆ’2(๐‘ฅ)ฮ”๐‘šโˆ’2๐‘ขโˆ’โ‹…โ‹…โ‹…+(โˆ’1)๐‘š๐‘Ž0(๐‘ฅ)๐‘ข=0(1.4) using ๐‘ƒ-functions containing terms of the form (ฮ”๐‘–๐‘ข)2. For three special cases, namely, when ๐‘š=3, ๐‘š=4, and when (1.4) reduces to the equation ฮ”๐‘š๐‘ข=0 for any integer ๐‘šโ‰ฅ2, a more complicated class of ๐‘ƒ-functions containing the squares of certain gradient terms is used. An open question arising from [7] is whether maximum principle results for (1.4) can be obtained for say any integer ๐‘š>4 for the latter class of ๐‘ƒ-functions. In this work, we establish such results by requiring that certain bounds and differential inequalities for the the coefficient functions ๐‘Ž0(๐‘ฅ),โ€ฆ,๐‘Ž๐‘šโˆ’2(๐‘ฅ) hold. Then we obtain integral bounds on various gradient terms.

2. Assumptions and Results

Throughout this work we assume that ฮฉ is a bounded domain in ๐‘๐ง whose boundary ๐œ•ฮฉ is sufficiently smooth, that the integer ๐‘›โ‰ฅ2 is even (without loss of generality), and that the integer ๐‘š>4. We identify the products of the first, second, and third gradients of the functions ๐‘ฃ and ๐‘ค as follows:๐‘ฃ,๐‘–๐‘ค,๐‘–โ‰กโˆ‡๐‘ฃโ‹…โˆ‡๐‘ค,๐‘ฃ,๐‘–๐‘—๐‘ค,๐‘–๐‘—โ‰กโˆ‡2๐‘ฃโˆถโˆ‡2๐‘ค,๐‘ฃ,๐‘–๐‘—๐‘˜๐‘ค,๐‘–๐‘—๐‘˜โ‰กโˆ‡3๐‘ฃโ‹ฎโˆ‡3๐‘ค,(2.1) where the commas denote partial differentiation.

For functions ๐‘Ž๐‘–(๐‘ฅ)โˆˆ๐ถ2(ฮฉ) we impose the boundedness conditions๐‘šโˆ’3๎“๐‘–=0๐‘Ž2๐‘–โ‰ค๐›ฝ,๐‘šโˆ’1๎“๐‘–=0โˆ‡๐‘Ž๐‘–โ‹…โˆ‡๐‘Ž๐‘–โ‰ค๐›พ,(2.2) for constants ๐›พโ‰ฅ0, ๐›ฝ>0.

Finally, ฮ” denotes the Laplace operator, ฮ”๐‘šโ‰กฮ”(ฮ”๐‘šโˆ’1), and ฮ”0โ‰ก๐ผ.

Now we derive two general maximum principle results for (1.4).

Theorem 2.1. Suppose that ๐‘ขโˆˆ๐ถ2๐‘š+1(ฮฉ)โˆฉ๐ถ2๐‘šโˆ’1(ฮฉ) is a solution of (1.4). Furthermore for ๐‘›>4 one defines ๐‘ƒ=โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโ‹…โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡mโˆ’1๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’12โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’32โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’3๐‘ข๎€ธ+๐‘šโˆ’2๎“๐‘–=0๐œ™๐‘–(ฮ”๐‘–๐‘ข)2+๐‘Ž๐‘šโˆ’2๎‚€๐‘›โˆ’4๎‚๐‘›+2(ฮ”๐‘šโˆ’2๐‘ข)2โˆ’(4โˆ’๐‘›)2(๐‘›+2)(ฮ”๐‘šโˆ’1๐‘ข)2,(2.3) where the functions ๐œ™0(๐‘ฅ),โ€ฆ,๐œ™๐‘šโˆ’2(๐‘ฅ)โˆˆ๐ถ2(ฮฉ)โˆฉ๐ถ0(ฮฉ) satisfy โˆ‘๐‘šโˆ’2๐‘–=0๐œ™2๐‘–+1โ‰ค๐›ผ for some positive constant ๐›ผ.
Additionally, one imposes the conditions ๐œ™๐‘–โ‰ฅ๐›ฝ2๐‘Ž(๐‘–=0,โ€ฆ,๐‘šโˆ’2),๐‘šโˆ’2๎‚€๐‘Žโ‰ฅ1,๐‘šโˆ’1โˆ’12๎‚โ‰ฅ๐›พ2๎‚€๐‘›+2๎‚,๐‘›โˆ’4ฮ”๐‘Ž๐‘–2โˆ’โˆ‡๐‘Ž๐‘šโˆ’๐‘–โ‹…โˆ‡๐‘Ž๐‘šโˆ’๐‘–๐‘Ž๐‘šโˆ’๐‘–โ‰ฅ0(๐‘–=1,3),ฮ”๐‘Ž๐‘šโˆ’2โˆ’4โˆ‡๐‘Ž๐‘šโˆ’2โ‹…โˆ‡๐‘Ž๐‘šโˆ’2๐‘Ž๐‘šโˆ’2โ‰ฅ0,ฮ”๐œ™๐‘–3๎‚ป๐›ฝโ‰ฅmax2๎‚€๐‘›โˆ’4๎‚+๐›พ๐‘›+22,๐›ผ,4โˆ‡๐œ™๐‘–โ‹…โˆ‡๐œ™๐‘–๐œ™๐‘–๎‚ผ(๐‘–=0,โ€ฆ,๐‘šโˆ’2).(2.4)
Then, ๐‘ƒ is subharmonic in ฮฉ.

Proof. A straightforward calculation yields ฮ”๐‘ƒ=2โˆ‡3๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโ‹ฎโˆ‡3๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆ’๎‚€6๎‚โˆ‡๎€ทฮ”๐‘›+2๐‘šโˆ’1๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’1๐‘ข๎€ธ๎€ทฮ”โˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธโ‹…โˆ‡(ฮ”๐‘š๎‚€๐‘ข)โˆ’4โˆ’๐‘›๎‚ฮ”๐‘›+2๐‘šโˆ’1๐‘ขฮ”๐‘š๐‘ข+๐‘Ž๐‘šโˆ’1โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’1๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’3โˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+๐‘šโˆ’2๎“๐‘–=0๎‚€ฮ”๐œ™๐‘–๎€ทฮ”๐‘–๐‘ข๎€ธ2๎€ท๐œ™+4โˆ‡๐‘–๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธฮ”๐‘–๐‘ข+2๐œ™๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธ๎‚+๐‘Ž๐‘šโˆ’2๎‚€๐‘›โˆ’4๎‚๎€บโˆ‡๎€ทฮ”๐‘›+2๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+ฮ”๐‘šโˆ’2๐‘ขฮ”๐‘šโˆ’1๐‘ข๎€ป+๎‚€๐‘›โˆ’4๎‚๐‘›+2ฮ”๐‘Ž๐‘šโˆ’2(ฮ”๐‘šโˆ’2๐‘ข)2๎‚€+4๐‘›โˆ’4๎‚๐‘›+2โˆ‡๐‘Ž๐‘šโˆ’2๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธฮ”๐‘šโˆ’2๐‘ข+12ฮ”๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+12ฮ”๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’3๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ+๐‘šโˆ’2๎“๐‘–=02๐œ™๐‘–ฮ”๐‘–๐‘ขฮ”๐‘–+1๐‘ข.(2.5) Using the well-known inequality (see [2]) โˆ‡3๐‘คโ‹ฎโˆ‡33๐‘คโ‰ฅ๐‘›+2โˆ‡(ฮ”๐‘ค)โ‹…โˆ‡(ฮ”๐‘ค),forallfunctions๐‘คโˆˆ๐ถ3(ฮฉ),(2.6) and (1.4) we deduce the inequality ๎€ทฮ”ฮ”๐‘ƒโ‰ฅโˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธโ‹…๎ƒฌ๐‘šโˆ’1๎“๐‘–=0(โˆ’1)๐‘–+1โˆ‡๎€ท๐‘Ž๐‘–๎€ธฮ”๐‘–๐‘ข+๐‘šโˆ’1๎“๐‘–=0(โˆ’1)๐‘–+1๐‘Ž๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎ƒญ+๎‚€๐‘›โˆ’4๎‚ฮ”๐‘›+2๐‘šโˆ’1๐‘ข๎ƒฌ๐‘šโˆ’1๎“๐‘–=0(โˆ’1)๐‘–+1๐‘Ž๐‘–ฮ”๐‘–๐‘ข๎ƒญ+๐‘Ž๐‘šโˆ’1โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’1๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’3โˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+๐‘šโˆ’2๎“๐‘–=0๎‚€ฮ”๐œ™๐‘–(ฮ”๐‘–๐‘ข)2๎€ท๐œ™+4โˆ‡๐‘–๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธฮ”๐‘–๐‘ข+2๐œ™๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธ๎‚+๐‘Ž๐‘šโˆ’2๎‚€๐‘›โˆ’4๎‚๎€บโˆ‡๎€ทฮ”๐‘›+2๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+ฮ”๐‘šโˆ’2๐‘ขฮ”๐‘šโˆ’1๐‘ข๎€ป+๎‚€๐‘›โˆ’4๎‚๐‘›+2ฮ”๐‘Ž๐‘šโˆ’2(ฮ”๐‘šโˆ’2๐‘ข)2๎‚€+4๐‘›โˆ’4๎‚๐‘›+2โˆ‡๐‘Ž๐‘šโˆ’2๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธฮ”๐‘šโˆ’2๐‘ข+12ฮ”๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+12ฮ”๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’3๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ+๐‘šโˆ’2๎“๐‘–=02๐œ™๐‘–ฮ”๐‘–๐‘ขฮ”๐‘–+1๐‘ข.(2.7) The right side of (2.7) is =๐‘Ž๐‘šโˆ’2๎‚€1+๐‘›โˆ’4๎‚โˆ‡๎€ทฮ”๐‘›+2๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’1๎‚€๐‘›โˆ’4๎‚๎€ทฮ”๐‘›+2๐‘šโˆ’1๐‘ข๎€ธ2+๐‘Ž๐‘šโˆ’1โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’3โˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ+๐‘šโˆ’2๎“๐‘–=0๎‚€ฮ”๐œ™๐‘–๎€ทฮ”๐‘–๐‘ข๎€ธ2๎€ท๐œ™+4โˆ‡๐‘–๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธฮ”๐‘–๐‘ข+2๐œ™๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธ๎‚+๎‚€๐‘›โˆ’4๎‚ฮ”๐‘›+2๐‘šโˆ’1๐‘ข๐‘šโˆ’3๎“๐‘–=0(โˆ’1)๐‘–+1๐‘Ž๐‘–ฮ”๐‘–๎€ทฮ”๐‘ขโˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธโ‹…๐‘šโˆ’1๎“๐‘–=0(โˆ’1)๐‘–+1โˆ‡๎€ท๐‘Ž๐‘–๎€ธฮ”๐‘–๐‘ข+๎‚€๐‘›โˆ’4๎‚๐‘›+2ฮ”๐‘Ž๐‘šโˆ’2(ฮ”๐‘šโˆ’2๐‘ข)2๎‚€+4๐‘›โˆ’4๎‚๐‘›+2โˆ‡๐‘Ž๐‘šโˆ’2๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธฮ”๐‘šโˆ’2๐‘ข+12ฮ”๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+12ฮ”๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’3๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธโ‹…๐‘šโˆ’4๎“๐‘–=0(โˆ’1)๐‘–+1๐‘Ž๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ+๐‘šโˆ’2๎“๐‘–=02๐œ™๐‘–ฮ”๐‘–๐‘ขฮ”๐‘–+1๐‘ข.(2.8) Subsequently, we obtain ๎‚ƒ๐‘Žฮ”๐‘ƒโ‰ฅ๐‘šโˆ’2๎‚€1+๐‘›โˆ’4๎‚๎‚„โˆ‡๎€ทฮ”๐‘›+2โˆ’1๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’1โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’3โˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ+๎‚ƒ๐‘Ž๐‘šโˆ’1โˆ’12๎‚„๎‚€๐‘›โˆ’4๎‚๐‘›+2(ฮ”๐‘šโˆ’1๐‘ข)2โˆ’12๎ƒฉ๐‘šโˆ’1๎“๐‘–=0(โˆ’1)๐‘–+1โˆ‡๎€ท๐‘Ž๐‘–๎€ธฮ”๐‘–๐‘ข๎ƒช2โˆ’12๎‚ƒ๐‘›โˆ’4๎‚„๎ƒฉ๐‘›+2๐‘šโˆ’3๎“๐‘–=0(โˆ’1)๐‘–+1๐‘Ž๐‘–๎€ทฮ”๐‘–๐‘ข๎€ธ๎ƒช2โˆ’12๎ƒฉ๐‘šโˆ’4๎“๐‘–=0(โˆ’1)๐‘–+1๐‘Ž๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎ƒช2+๐‘šโˆ’2๎“๐‘–=0ฮ”๐œ™๐‘–๎€ทฮ”๐‘–๐‘ข๎€ธ2๎€ท๐œ™+4โˆ‡๐‘–๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธฮ”๐‘–๐‘ข+2๐œ™๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธ+๎‚€๐‘›โˆ’4๎‚๐‘›+2ฮ”๐‘Ž๐‘šโˆ’2(ฮ”๐‘šโˆ’2๐‘ข)2๎‚€+4๐‘›โˆ’4๎‚๐‘›+2โˆ‡๐‘Ž๐‘šโˆ’2๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธฮ”๐‘šโˆ’2๐‘ข+12ฮ”๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+12ฮ”๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’3๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’3โˆ‡๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’3๐‘ข๎€ธ+๐‘šโˆ’2๎“๐‘–=02๐œ™๐‘–ฮ”๐‘–๐‘ขฮ”๐‘–+1๐‘ข.(2.9) We now state a series of inequalities to demonstrate that ฮ”๐‘ƒโ‰ฅ0. First we note ๐‘šโˆ’2๎“๐‘–=02๐œ™๐‘–ฮ”๐‘–๐‘ขฮ”๐‘–+1๎ƒฉ๐‘ขโ‰ฅโˆ’๐‘šโˆ’2๎“๐‘–=0๎€ท๐œ™๐‘–๎€ธ2๎ƒช+1๐‘šโˆ’2๎“๐‘–=0(ฮ”๐‘–๐‘ข)2.(2.10) Similarly, we obtain โˆ’12๎ƒฉ๐‘šโˆ’1๎“๐‘–=0(โˆ’1)๐‘–+1โˆ‡๎€ท๐‘Ž๐‘–๎€ธฮ”๐‘–๐‘ข๎ƒช21โ‰ฅโˆ’2๎ƒฉ๐‘šโˆ’1๎“๐‘–=0๎€บโˆ‡๎€ท๐‘Ž๐‘–๎€ธ๎€ท๐‘Žโ‹…โˆ‡๐‘–๎ƒช๎€ธ๎€ป(ฮ”๐‘šโˆ’1๐‘ข)2โˆ’12๎ƒฉ๐‘šโˆ’1๎“๐‘–=0๎€บโˆ‡๎€ท๐‘Ž๐‘–๎€ธ๎€ท๐‘Žโ‹…โˆ‡๐‘–๎ƒช๎€ธ๎€ป๐‘šโˆ’2๎“๐‘–=0ฮ”๐‘–๐‘ขฮ”๐‘–๐‘ข,(2.11)โˆ’12๎‚ƒ๐‘›โˆ’4๎‚„๎ƒฉ๐‘›+2๐‘šโˆ’3๎“๐‘–=0(โˆ’1)๐‘–+1a๐‘–ฮ”๐‘–๐‘ข๎ƒช21โ‰ฅโˆ’2๎‚ƒ๐‘›โˆ’4๎‚„๐‘›+2๐‘šโˆ’3๎“๐‘–=0๐‘Ž2๐‘–๐‘šโˆ’3๎“๐‘–=0(ฮ”๐‘–๐‘ข)2,(2.12)โˆ’12๎ƒฉ๐‘šโˆ’4๎“๐‘–=0(โˆ’1)๐‘–+1๐‘Ž๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎ƒช21โ‰ฅโˆ’2๐‘šโˆ’4๎“๐‘–=0๐‘Ž2๐‘–๐‘šโˆ’4๎“๐‘–=0โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธ,(2.13)+๐‘šโˆ’2๎“๐‘–=0๎€ทฮ”๐œ™๐‘–๎€ธ(ฮ”๐‘–๐‘ข)2๎€ท๐œ™+4โˆ‡๐‘–๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธฮ”๐‘–๐‘ข+๐œ™๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธโ‰ฅ๐‘šโˆ’2๎“๐‘–=0(ฮ”๐‘–๐‘ข)2๎ƒฌฮ”๐œ™๐‘–3โˆ’๎€ท๐œ™4โˆ‡๐‘–๎€ธ๎€ท๐œ™โ‹…โˆ‡๐‘–๎€ธ๐œ™๐‘–๎ƒญ.(2.14) We also have that 12ฮ”๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ+2โˆ‡๐‘Ž๐‘šโˆ’1โˆ‡๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ+๐‘Ž๐‘šโˆ’1โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‰ฅโˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎‚ธฮ”๐‘Ž๐‘šโˆ’12โˆ’โˆ‡๐‘Ž๐‘šโˆ’1โ‹…โˆ‡๐‘Ž๐‘šโˆ’1๐‘Ž๐‘šโˆ’1๎‚น.(2.15) Additional inequalities analogous to (2.15) can be developed for the remaining terms in (2.9) involving ฮ”๐‘Ž๐‘šโˆ’2 and ฮ”๐‘Ž๐‘šโˆ’3.
From (2.2), (2.10)โ€“(2.15) and upon inclusion of the aforementioned additional inequalities we deduce that๎ƒฉ๎‚ƒ๐‘Žฮ”๐‘ƒโ‰ฅ๐‘šโˆ’1โˆ’12๎‚„๎‚€๐‘›โˆ’4๎‚โˆ’1๐‘›+22๐‘šโˆ’1๎“๐‘–=0โˆ‡๎€ท๐‘Ž๐‘–๎€ธ๎€ท๐‘Žโ‹…โˆ‡๐‘–๎€ธ๎ƒช๎€ทฮ”๐‘šโˆ’1๐‘ข๎€ธ2+๐‘šโˆ’2๎“๐‘–=0๎‚ธฮ”๐œ™๐‘–3โˆ’๐›ฝ2๎‚€๐‘›โˆ’4๎‚โˆ’๐›พ๐‘›+22๎‚น๎€ทฮ”๐‘–๐‘ข๎€ธ2+๐‘šโˆ’2๎“๐‘–=0๎‚ธ๐œ™๐‘–โˆ’๐›ฝ2๎‚น๎€ทฮ”๐‘–๐‘ข๎€ธ2+๐‘šโˆ’2๎“๐‘–=0๎‚ธฮ”๐œ™๐‘–3๎‚น๎€ทฮ”โˆ’๐›ผ๐‘–๐‘ข๎€ธ2+๐‘šโˆ’2๎“๐‘–=0๎€ทฮ”๐‘–๐‘ข๎€ธ2๎ƒฌฮ”๐œ™๐‘–3โˆ’๎€ท๐œ™4โˆ‡๐‘–๎€ธ๎€ท๐œ™โ‹…โˆ‡๐‘–๎€ธ๐œ™๐‘–๎ƒญ+๎€ท๐‘Ž๐‘šโˆ’2๎€ธโˆ‡๎€ทฮ”โˆ’1๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”+โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎‚ธฮ”๐‘Ž๐‘šโˆ’12โˆ’โˆ‡๐‘Ž๐‘šโˆ’1โ‹…โˆ‡๐‘Ž๐‘šโˆ’1๐‘Ž๐‘šโˆ’1๎‚น๎€ทฮ”+โˆ‡๐‘šโˆ’3๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’3๐‘ข๎€ธ๎‚ธฮ”๐‘Ž๐‘šโˆ’32โˆ’โˆ‡๐‘Ž๐‘šโˆ’3โ‹…โˆ‡๐‘Ž๐‘šโˆ’3๐‘Ž๐‘šโˆ’3๎‚น+๎‚€๐‘›โˆ’4๎‚๐‘›+2(ฮ”๐‘šโˆ’2๐‘ข)2๎‚ธฮ”๐‘Ž๐‘šโˆ’2โˆ’4โˆ‡๐‘Ž๐‘šโˆ’2โ‹…โˆ‡๐‘Ž๐‘šโˆ’2๐‘Ž๐‘šโˆ’2๎‚น.(2.16) Utilizing (2.4) the conclusion follows.

Now we handle the case where ๐‘›โ‰ค4 in the following theorem.

Theorem 2.2. Suppose that ๐‘ขโˆˆ๐ถ2๐‘š+1(ฮฉ)โˆฉ๐ถ2๐‘šโˆ’1(ฮฉ) is a solution of (1.4) where ๐‘›โ‰ค4. For ๐œ™0(๐‘ฅ),โ€ฆ,๐œ™๐‘šโˆ’1(๐‘ฅ)โˆˆ๐ถ2(ฮฉ)โˆฉ๐ถ0(ฮฉ)satisfying the inequality โˆ‘๐‘šโˆ’2๐‘–=0๐œ™2๐‘–+1โ‰ค๐›ผ for some positive constant ๐›ผ, one defines ๐‘ƒ=โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโˆถโˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’1๐‘ข๎€ธ+๐‘šโˆ’1๎“๐‘–=0๐œ™๐‘–๎€ทฮ”๐‘–๐‘ข๎€ธ2+12๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ2.(2.17) Then if ๐œ™๐‘–โ‰ฅ๐›ผ2,for๐‘–=0,โ€ฆ,๐‘šโˆ’1,ฮ”๐œ™๐‘šโˆ’๐‘–3โˆ’12โ‰ฅ0,for๐‘–=1,2,ฮ”๐œ™๐‘–3โˆ’4โˆ‡๐œ™๐‘–โ‹…โˆ‡๐œ™๐‘–๐œ™๐‘–โ‰ฅ0,for๐‘–=0,โ€ฆ,๐‘šโˆ’1,(2.18)๐‘ƒis subharmonic in ฮฉ.

Proof. A calculation similar to that of Theorem 2.1 yields ฮ”๐‘ƒ=2โˆ‡3๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธโ‹ฎโˆ‡3๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โˆ’โˆ‡๐‘šโˆ’1๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’1๐‘ข๎€ธ๎€ทฮ”โˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธโ‹…๐‘šโˆ’1๎“๐‘–=0(โˆ’1)๐‘–+1โˆ‡๐‘Ž๐‘–ฮ”๐‘–๎€ทฮ”๐‘ข+โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธโ‹…๐‘šโˆ’1๎“๐‘–=0๎€บ(โˆ’1)๐‘–+1๐‘Ž๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ป+ฮ”๐‘šโˆ’2๐‘ขฮ”๐‘šโˆ’1๐‘ข+๐‘šโˆ’1๎“๐‘–=0ฮ”๐œ™๐‘–(ฮ”๐‘–๐‘ข)2๎€ท๐œ™+4โˆ‡๐‘–๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธฮ”๐‘–๐‘ข+๐‘šโˆ’1๎“๐‘–=02๐œ™๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธ+2๐œ™๐‘–ฮ”๐‘–๐‘ขฮ”๐‘–+1๐‘ข.(2.19) Using (2.6), (2.10), (2.11), (2.13), and (2.14) we obtain ฮ”๐‘ƒโ‰ฅ๐‘šโˆ’1๎“๐‘–=0๎€ทฮ”๐‘–๐‘ข๎€ธ2๎‚ธฮ”๐œ™๐‘–3โˆ’4โˆ‡๐œ™๐‘–โ‹…โˆ‡๐œ™๐‘–๐œ™๐‘–๎‚น+๐‘šโˆ’1๎“๐‘–=02ฮ”๐œ™๐‘–3๎€ทฮ”๐‘–๐‘ข๎€ธ2+๐‘šโˆ’1๎“๐‘–=0๐œ™๐‘–โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธ๎€ทฮ”โˆ’โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธโˆ’12๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ2โˆ’12๎€ทฮ”๐‘šโˆ’1๐‘ข๎€ธ2๎€ทฮ”+โˆ‡๐‘šโˆ’2๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘šโˆ’2๐‘ข๎€ธโˆ’12๎ƒฉ๐‘šโˆ’1๎“๐‘–=0โˆ‡๐‘Ž๐‘–โ‹…โˆ‡๐‘Ž๐‘–๎ƒช๐‘šโˆ’1๎“๐‘–=0๎€ทฮ”๐‘–๐‘ข๎€ธ2โˆ’12๎ƒฉ๐‘šโˆ’1๎“๐‘–=0๐‘Ž2๐‘–๎ƒช๐‘šโˆ’1๎“๐‘–=0โˆ‡๎€ทฮ”๐‘–๐‘ข๎€ธ๎€ทฮ”โ‹…โˆ‡๐‘–๐‘ข๎€ธโ‰ฅ๐‘šโˆ’1๎“๐‘–=0๎€ทฮ”๐‘–๐‘ข๎€ธ2๎‚ธฮ”๐œ™๐‘–3โˆ’4โˆ‡๐œ™๐‘–โ‹…โˆ‡๐œ™๐‘–๐œ™๐‘–๎‚น+๐‘šโˆ’1๎“๐‘–=0๎‚ƒ๐œ™๐‘–โˆ’๐›ผ2๎‚„โˆ‡(ฮ”๐‘ข)โ‹…โˆ‡(ฮ”๐‘ข)+๐‘šโˆ’1๎“๐‘–=0๎‚ธฮ”๐œ™๐‘–3โˆ’๐›พ2๎‚น๎€ทฮ”๐‘–๐‘ข๎€ธ2+๎‚ตฮ”๐œ™๐‘šโˆ’23โˆ’12๎‚ถ๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ2+๎‚ตฮ”๐œ™๐‘šโˆ’13โˆ’12๎‚ถ๎€ทฮ”๐‘šโˆ’1๐‘ข๎€ธ2.(2.20) Hence by (2.18), we conclude that ฮ”๐‘ƒโ‰ฅ0.

3. Applications

Here we briefly indicate how theorem 1 and theorem 2 can be used to obtain integral bounds on the square of the second gradient of ฮ”๐‘šโˆ’2๐‘ข. suppose that the hypotheses of theorem 1 are satisfied and that the ๐‘š conditionsฮ”๐‘–๐‘ข=0,forฮ”๐‘–=0,โ€ฆ,๐‘šโˆ’5,๐‘šโˆ’2ฮ”๐‘ข=๐‘šโˆ’2๐‘ข๐œ•๐‘›=0,ฮ”๐‘šโˆ’3ฮ”๐‘ข=๐‘šโˆ’3๐‘ข๐œ•๐‘›=0(3.1) hold on ๐œ•ฮฉ. Let ๐ด denote the area of ฮฉ. As a consequence of integration by parts, Theorem 2.1 implies that๎€œฮฉ||โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ||2๐ด๐‘‘๐‘ฅโ‰ค2max๐œ•ฮฉ๎‚ธ||โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ||2+๐œ™๐‘šโˆ’4๎€ทฮ”(๐‘ฅ)๐‘šโˆ’4๐‘ข๎€ธ2+๐‘›โˆ’42(n๎€ทฮ”๎€ทฮ”+2)๐‘šโˆ’2๐‘ข๎€ธ๎€ธ2๎‚น.(3.2)

Now, if the hypotheses of Theorem 2.2 hold and if we impose the ๐‘š conditionsฮ”๐‘–๐‘ข=0,forฮ”๐‘–=0,โ€ฆ,๐‘šโˆ’3,๐‘šโˆ’2ฮ”๐‘ข=๐‘šโˆ’2๐‘ข๐œ•๐‘›=0(3.3) on ๐œ•ฮฉ we can deduce the inequality๎€œฮฉ||โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ||2๐ด๐‘‘๐‘ฅโ‰ค2max๐œ•ฮฉ๎‚ƒ||โˆ‡2๎€ทฮ”๐‘šโˆ’2๐‘ข๎€ธ||2+๐œ™๐‘šโˆ’1(๎€ทฮ”๎€ทฮ”๐‘ฅ)๐‘šโˆ’2๐‘ข๎€ธ๎€ธ2๎‚„,(3.4) from Theorem 2.2.

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