On a model problem with second derivatives with respect to geometric variables in the boundary condition for second-order parabolic equations (Q1280696)

We consider the following problem: Let \(\Omega\subset\mathbb{R}^n\), \(n\geq 3\), and let some coordinates \(\omega= (\omega_1,\dots,\omega_{n-1})\) be introduced on the boundary \(\Gamma= \partial\Omega\). Let \(\Omega_T= \Omega\times (0,T)\), and let \(\Gamma_T= \Gamma\times (0,T)\). It is required to find functions \(u(x,t)\) and \(v(\omega,t)\) satisfying the conditions \[ {\partial u\over\partial t}- Lu= f_0(x,t),\quad (x,t)\in \Omega_T, \] \[ \sum^n_{i=1} b_i(x,t){\partial u\over\partial x_i}+ L_0v= f_1(x,t),\quad u-v= 0,\quad (x,t)\in \Gamma_T,\tag{1} \] \[ u(x,0)= \varphi(x),\quad x\in\Omega,\quad v(\omega,0)= \psi(\omega),\quad \omega\in\Gamma, \] where \[ L= \sum^n_{i,j=1} a_{ij}(x,t) D^2_{x_ix_j}+ \sum^n_{i=1} a_i(x,t) D_{x_i}+ a(x,t),\;L_0= \sum^{n-1}_{i,j= 1}\gamma_{ij}(\omega, t) D^2_{\omega_i\omega_j}+ \gamma(\omega,t) \] are uniformly elliptic operators, \((\vec b(x,t),\vec n)\geq b_0> 0\), \(\vec b= (b_1,\dots, b_n)\), and \(\vec n\) is the inward (with respect to \(\Omega\)) normal on \(\Gamma\). Obviously, for \(\psi(\omega)= \varphi(x)|_\Gamma\), system (1) is equivalent to the problem of finding a single function \(u(x,t)\) with boundary conditions containing second derivatives along the tangents to the boundary. We obtain solvability results by freezing the arguments in the coefficients, discarding lower-order terms, and straightening a part of the boundary \(\Gamma\).