This paper discusses the minimal eigenvalue lambda (1)(p,q,Omega) and its best estimate of the following nonlinear eigenvalue problem -Delta (p)u = lambda \u \ (q-2)u, u \ (partial derivative Omega) = 0. Where Omega i...This paper discusses the minimal eigenvalue lambda (1)(p,q,Omega) and its best estimate of the following nonlinear eigenvalue problem -Delta (p)u = lambda \u \ (q-2)u, u \ (partial derivative Omega) = 0. Where Omega is a bounded smooth domain in R-N, Delta (p) is the p-laplace operator, p greater than or equal to 2, q is an element of [p, Np/N-p). [p, NP). By this result, the famous Opial inequality in R-N are deduced.展开更多
设:f(x)∈AC[o,A),并f(0)＝f(h)＝0.则有integral from n=0 to h(|f(x)f(x)|dx)≤h/4 integral from n=0 to h(|f'(x)|~2dx)这个不等式叫做Opial不等式.许多数学家对它曾进行过研究.在此我们给予有意义的改进:integral from n=0 to ...设:f(x)∈AC[o,A),并f(0)＝f(h)＝0.则有integral from n=0 to h(|f(x)f(x)|dx)≤h/4 integral from n=0 to h(|f'(x)|~2dx)这个不等式叫做Opial不等式.许多数学家对它曾进行过研究.在此我们给予有意义的改进:integral from n=0 to h (|ff'|dx)≤1/2(h/2)^(2/Q)(integral from n=0 to h(|f'|~pdx))^((2/p)-(2/Q)){(integral from n=0 to h(|f'|~pdx))~2-1/4(integral from n=0 to h(|f'|~pcos(2πx/h)dx)~2)}((?)/Q)其中I<P≤2,Q＝p/(P—1)．(2)显然比(1)优秀,实际上我们已证得更一般的结果.展开更多
基金The research supported by Natural Science Fundation of China Excellent Teachers Fundation of Ministry of Education of China
文摘This paper discusses the minimal eigenvalue lambda (1)(p,q,Omega) and its best estimate of the following nonlinear eigenvalue problem -Delta (p)u = lambda \u \ (q-2)u, u \ (partial derivative Omega) = 0. Where Omega is a bounded smooth domain in R-N, Delta (p) is the p-laplace operator, p greater than or equal to 2, q is an element of [p, Np/N-p). [p, NP). By this result, the famous Opial inequality in R-N are deduced.
文摘设:f(x)∈AC[o,A),并f(0)＝f(h)＝0.则有integral from n=0 to h(|f(x)f(x)|dx)≤h/4 integral from n=0 to h(|f'(x)|~2dx)这个不等式叫做Opial不等式.许多数学家对它曾进行过研究.在此我们给予有意义的改进:integral from n=0 to h (|ff'|dx)≤1/2(h/2)^(2/Q)(integral from n=0 to h(|f'|~pdx))^((2/p)-(2/Q)){(integral from n=0 to h(|f'|~pdx))~2-1/4(integral from n=0 to h(|f'|~pcos(2πx/h)dx)~2)}((?)/Q)其中I<P≤2,Q＝p/(P—1)．(2)显然比(1)优秀,实际上我们已证得更一般的结果.
