Minimization of the lowest eigenvalue for a vibrating beam
Consider a fourth order measure differential equation (FOMDE) described by \[ dy^{(3)}(t)+\lambda y^{\prime \prime }\left( t\right) dt+y(t)d\mu \left(t\right) =0 \tag{1} \] in the interval [0,1] and with boundary conditions \[ y(0)=y^{\prime \prime }\left( 0\right) =y(1)=y^{\prime \prime }(1)=0.\tag{2} \] In this paper, using the well-known variational method, the authors, first, establish the minimization characterization for the lowest eigenvalue \(\lambda _{1}\left( \mu \right) \) of the FOMDE and then they construct a relation between the minimization problem of the lowest eigenvalue for the differential equation \[ y^{\prime \prime \prime \prime }(t)+\lambda y^{\prime \prime}(t)+F(t)y(t)=0,\quad t\in \left[ 0,1\right] \tag{3} \] together with the boundary conditions (2) and the one for FOMDE (1) and use this relationship to find explicit optimal lower bound of the lowest eigenvalue for the differential equation (3) arriving the following result: Theorem. Given \(r>0,\) one has \[ \mathbf{L}(r)=Q^{-1}\left( r\right) . \] Here \[ \mathbf{L}(r):=\inf \left\{ \lambda _{1}\left( F\right) :F\in B_{1}\left[ r\right] \right\} \] for \(r\in \left( 0,+\infty \right)\), \[ B_{1}\left[ r\right] :=\left\{ F\in \mathcal{L}^{1}:\left\| F\right\|_{1}\leq r\right\} \] and the invertible elementary function \(Q:\left( -\infty ,\pi ^{2}\right) \rightarrow \left( 0,+\infty \right) \) is defined as \[ Q\left( x\right) :=\frac{2x^{\frac{3}{2}}\sin x^{\frac{1}{2}}}{1-\cos x^{\frac{1}{2}}-\frac{1}{2}x^{\frac{1}{2}}\sin x^{\frac{1}{2}}}. \]
scientific article; zbMATH DE number 6919326
scientific article