Scalar boundary value problems on junctions of thin rods and plates (Q2925709)

The authors deal with the mixed boundary value problem for the Poisson equations on the union \(\varXi(h)=\varOmega_*(h)\cup\varOmega_1(h)\cup\dots\cup\varOmega_J(h)\) of a thin cylindrical plate and thin cylindrical rods with a thickness \(h>0\) of the plate. The problem has the form NEWLINE\[NEWLINE\begin{aligned} &-\varDelta_xu_0(h,x)=f_0(h,x),\;x\in \varOmega_*(h);\;-\gamma_j(h)\varDelta_xu_0(h,x)=f_0(h,x),\;x\in \varOmega_j(h);\\ &\partial_\nu u_0(h,x)=0,\;x\in \varSigma_*(h);\;\gamma_j(h)u_j(h,x)=0,\;x\in \varSigma_j(h)\cup\omega_j^h(0);\\& u_j(h,x)=0,\;x\in \omega_j^h(\ell_j);\\& u_0(h,x)=u_j(h,x),\;x\in v_j^h;\;\partial_\nu u_0(h,x)=\gamma_j(h)u_j(h,x),\;x\in v_j^h \end{aligned}NEWLINE\]NEWLINE with the Laplace operator \(\varDelta_x\) in \(x=(y,z)=(x_1,x_2,x_3)\) and the restrictions \(u_0,\;u_j\) of the function \(u\) on the subdomains \(\varOmega_*(h)\) and \(\varOmega_j(h)\). Asymptotic formulas for the solutions and asymptotic error estimates in anisotropic weighted Sobolev norms are derived.