The best constant of Sobolev inequality corresponding to a bending problem of a beam on an interval
The authors consider the boundary value problem \[ \text{BVP}(\alpha,\beta): u^{(4)}=f(x)\text{ for }x\in (-1,1) \] with boundary conditions \[ u^{(\alpha _{j})}(-1)=u^{(\beta _{j})}(1)=0\text{ for }j=0,1 \] associated to a bending problem of a beam on the interval \(I:=(-1,1)\), where \(f\) is a bounded continuous function on \(I\) and \((\alpha,\beta):=(\alpha_0,\alpha_1,\beta_0,\beta_1)\) takes one of the following ten values: Case I [\((0,1,0,1), (0,1,0,2), (0,1,1,3), (0,1,2,3), (0,2,0,2), (0,2,1,3)\)]; Case II [\((0,2,2,3),(1,3,1,3),(1,3,2,3)\)]; and Case III [\((2,3,2,3)\)]. They prove that BVP(\(\alpha,\beta\)) has a unique classical solution for the parameters in Case I. The parameters in Cases II and III have eigenfunctions \(\varphi\) corresponding to \(\lambda=0\) for the eigenvalue problem \(u^{(4)}(x)=\lambda u(x)\) with \(x\in I\), \(u^{(\alpha _{j})}(-1)=u^{(\beta _{j})}(1)=0,\) \(j=0,1.\) Then, if the conditions \(\int_{-1}^{1}f(y)\varphi(y)\,dy=0\) (solvability condition) and \(\int_{-1}^{1}u(y)\varphi(y)\,dy=0\) (orthogonality condition) are added to Cases II and III, BVP(\(\alpha,\beta\)) presents again a unique classical solution. In all cases, the solutions are given in terms of Green functions \(G(x,y)\) which are explicitly described. In the second part of the paper, it is proved that \(G(x,y)\) is the reproducing kernel for a suitable Hilbert space \(H\) and an inner product \((\cdot,\cdot)_H\). In particular, the authors show that \(G(y,y)=\int_{-1}^{1}|\partial^2_{x}G(x,y)|\,dx\) which allows them to prove that the Sobolev inequality \[ \left(\sup_{|y|\leq 1}|u(y)|\right)^2\leq C\int_{-1}^{1}|u''(x)|\,dx \] holds for any function \(u\) of \(H\) and a positive constant \(C\), independent of \(u\). Even more, they realize that the best constant \(C(\alpha,\beta)\) is the maximum of \(G(y,y),\) \(y\in [-1,1],\) and compute it in all cases.
scientific article; zbMATH DE number 5712999