A sharp trace result on a thermo-elastic plate equation with coupled hinged/Neumann boundary conditions
The partial differential equations of a linear thermo-elastic thin plate in the bounded domain \(\Omega\) of \(\mathbb{R}^2\) with coupled hinged mechanical and Neumann (Robin) thermal boundary conditions are studied. The considered initial-boundary value problem is as follows: \[ w_{tt}- \gamma_\Delta w_{tt}+ \Delta^2w+\Delta\theta= 0\quad\text{in } (0,T]\times \Omega= Q,\;\gamma>0;\tag{1} \] \[ \theta_t-\Delta\theta- \Delta w_t=0\quad\text{ in }Q;\tag{2} \] \[ w(0,\cdot)= w_0,\;w_t(0,\cdot)= w_1,\;\theta(0,\cdot)= \theta_0\quad\text{in } \Omega;\tag{3} \] \[ w|_\Sigma= 0,\;\Biggl[{\partial\theta\over\partial\nu}+ b\theta\Biggr]_\Sigma= 0,\;b\geq 0\quad\text{in }\Sigma= (0,T]\times \partial\Omega;\tag{4} \] \[ [\Delta w+\theta]_\Sigma= 0\quad\text{in }\Sigma.\tag{5} \] Let \(A_D\) be the positive, selfadjoint operator on \(L_2(\Omega)\) defined by \[ A_Dh= -\Delta h,\;D(A_D)= H^2(\Omega)\cap H^1_0(\Omega),\;D(A^{1/2}_D)= H^1_0(\Omega)\tag{6} \] and let \(Y_\gamma\) be \[ Y_\gamma\equiv D(A_D)\times D((I+ \gamma D_D)^{1/2})\times L_2(\Omega)\equiv [H^2(\Omega)\cap H^1_0(\Omega)]\times H^1_0(\Omega)\times L_2(\Omega).\tag{7} \] The coupling between the deflection \(w\) and the temperature \(\theta\) in the boundary condition (5) yields some complications. The main result of the paper can be formulated as follows: for the initial-boundary value problem \((1),(2),\dots(5)\), where \(Y_0= [w_0,w_1,\theta_0]\in Y_\gamma\) in (7), we have, continuously in \(Y_0\) \[ {\partial w_t\over\partial\nu}\in L_2(0, T,L_2(\partial\Omega))= L_2(\partial\Omega).\tag{8} \] (8) gives a sharp result on the Neumann trace of the mechanical velocity, which is ``\({1\over 2}\)'' shaper in the space variable than one would obtain by a formal application of trace theory on the optimal interior regularity for \(w_t\). This fact of ``hidden regularity'' is in agreement with the general regularity property of a Petrowsky type partial differential equation. Two proofs based on energy methods are presented.
scientific article; zbMATH DE number 1492593