Abstract
This study applies the extended L 2 Sobolev type inequality to the L p Sobolev type inequality using Hölder’s inequality. The sharp constant and best function of the L p Sobolev type inequality are found using a Green function for the nth order ordinary differential equation. The sharp constant is shown to be equal to the L p norm of the Green function and to the pth root of the value of the origin of the best function.
Highlights
The Sobolev inequality called the Sobolev embedding theorem, is often the core inequality in partial differential equations and variation calculations
We have found that the sharp constant can be obtained using the Green function, which is the reproducing kernel of a Hilbert space [3]
The sharp constant of the corresponding Sobolev inequality is expressed as the maximum diagonal value of the Green function
Summary
The Sobolev inequality called the Sobolev embedding theorem, is often the core inequality in partial differential equations and variation calculations. There have been few studies explicitly seeking the sharp (small) constant and best function calculation of the Sobolev inequality, the so-called “best evaluation of the Sobolev inequality”. In this study, the sharp constant of Sobolev inequality is primarily calculated using the Green function of various boundary value problems when p ≤ 2 and q = ∞. The sharp constant of the corresponding Sobolev inequality is expressed as the maximum diagonal value of the Green function. The real coefficients ai (i = 0, 1, · · · , n − 1) satisfy the following inequality:.
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