کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4619595 1339441 2010 17 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Universal bounds for eigenvalues of the biharmonic operator
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
Universal bounds for eigenvalues of the biharmonic operator
چکیده انگلیسی

In this paper, we study the eigenvalues of the clamped plate problem:{Δ2u=λu,in D,u|∂D=∂u∂ν|∂D=0, where D is a bounded connected domain in an n-dimensional complete minimal submanifold of a unit m  -sphere Sm(1)Sm(1) or of an m  -dimensional Euclidean space RmRm. Let 0<λ1<λ2⩽⋯⩽λk⩽⋯0<λ1<λ2⩽⋯⩽λk⩽⋯ be the eigenvalues of the above problem. We obtain universal bounds on λk+1λk+1 in terms the first k eigenvalues independent of the domains. For example, when D is contained in an n  -dimensional complete minimal submanifold of Sm(1)Sm(1), we show thatλk+1−1k∑i=1kλi⩽1kn{∑i=1k(λk+1−λi)1/2((2n+4)λi1/2+n2)}1/2⋅{∑i=1k(λk+1−λi)1/2(4λi1/2+n2)}1/2, from which one can obtain a more explicit upper bound on λk+1λk+1 in terms of λ1,…,λkλ1,…,λk (see Corollary 1). When D is contained in a complete n  -dimensional minimal submanifold of RmRm, we prove the inequalityλk+1⩽1k∑i=1kλk+(8(n+2)n2)1/21k∑i=1k(λi(λk+1−λi))1/2 which generalizes the main theorem in Cheng and Yang (2006) [10] that states that the same estimate holds when D   is a connected and bounded domain in RnRn.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Mathematical Analysis and Applications - Volume 364, Issue 1, 1 April 2010, Pages 1–17
نویسندگان
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