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Steepest descent and the least C for Sobolev’s inequality. (English) Zbl 0585.46026
A theorem is proved concerning use of steepest descent in Sobolev space to solve differential equations with nonlinear boundary conditions of the form \[ B(u)=\int^{1}_{0}K(x,u(x))dg(x), \] where \(u\in H^ n[0,1]\), \(K\in C^{(1)}([0,1]\times {\mathbb{R}},{\mathbb{R}})\), and g: [0,1]\(\to {\mathbb{R}}\) is of bounded variation. As a corollary one shows that the least constant \(C_ n\) for the imbedding of \(H^ n[0,1]\to C[0,1]\) is given by \[ C_ n=\{\frac{2}{n+1}\sum^{n}_{k=1}\frac{\sin^ 3(k\theta)}{\tanh (\sin (k\theta)\quad)}\}^{1/2}, \] where \(\theta =n/(n+1)\).

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
90C99 Mathematical programming
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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