Pointwise Comparison Principle for clamped Timoshenko beam - Bartuzel, Grzegorz; Fryszkowski, Andrzej (Topological Methods in Nonlinear Analysis, 2012)
We present the properties of three Green functions for:\roster\item"1." general complex ``clamped beam''$$\gatherD_{\alpha ,\beta }[y] \equiv y^{\prime \prime \prime \prime}-(\alpha ^{2}+\beta ^{2}) y^{\prime \prime }+\alpha ^{2}\beta^{2}y=f,\\y(0) =y(1) =y^{\prime }(0) =y^{\prime}(1) =0. \tag{BC}\endgather$$\item"2." Timoshenko clamped beam $D_{\alpha ,\overline{\alpha }}[y] \equiv f$ with (BC).\item"3." Euler-Bernoulli clamped beam $D_{k(1+i) ,k(1-i)} [ y] \equiv f$ with (BC).\endrosterIn case 1. we represent solution via a Green operator expressed in terms ofKourensky type system of fundamental solutions for homogeneous case. Thiscondense form is, up-to our knowledge, new even for the Euler-Bernoulliclamped beam and it allows to recognize the set of $\alpha ^{\prime }s$ forwhich the Pointwise Comparison Principle for the Timoshenko beam holds. Thepresented approach to positivity of the Green function is muchstraightforward then ones known in the literature for the case 3 (see \cite{12}).