Existence and smoothness for certain degenerate parabolic boundary value problems (Q1106406)

This paper examines a class of parabolic initial-boundary value problems of the form, \[ (1)\quad \partial u/\partial t=\sum^{n}_{i,j=1} a_{ij}(x,t)\partial^ 2u/(\partial x_ i\partial x_ j)+\sum^{n}_{i,j=1} b_{ij}(x,t)\partial u/\partial x_ i+c(x,t)u \] in \(\Omega \times R_+\) with \(\Omega \subset R^ n\) and \[ (2)\quad \alpha (x,t)(\partial u/\partial n)(x,t)+\beta (x,t)u(x,t)=f(x,t) \] on \(\partial \Omega \times R_+\) where \(\partial /\partial n\) is the derivation in the direction of the outer co-normal and \[ (3)\quad u(x,0)=u_ 0(x)\quad for\quad x\in \Omega. \] The boundary condition given in expression (2) can have coefficient functions, \(\alpha\) (x,t), \(\beta\) (x,t), relaxed to be complex valued functions. The technique implemented to obtain existence, uniqueness and smoothness of solutions is the method of reducing boundary value problems to pseudo-differential problems on the boundary. A component of this technique requires a weighted class of vector weight functions enjoying several specialized bound conditions. This approach differs from the elliptic case where the boundary operator in expression (2) is invertible in a specific function space. The function space implemented in this paper is a modification of several well known generalized function test spaces. Specifically the function space, \({\mathbb{S}}^ m_{\lambda,p,d}(H\times R)\), has derivative growth conditions controlled by \[ (4)\quad | \partial^{\beta}_{\xi}\partial_{\tau}^{\alpha_{d-1}}D_ x^{\beta}D_ t^{\beta_{d+1}}\alpha (x,t,\xi,\tau)| \leq C\lambda^{m-| \alpha | \rho -2\alpha_{d+1}\rho +| \beta | \delta +\alpha \beta_{d+1}\delta} \] where \(\rho\) and \(\delta\) are fixed constants satisfying \(1\geq \rho \geq \delta \geq 0.\) Several carefully developed and lengthy computations implementing derivative growth conditions of the form given in expression (4) proves the hypoellipticity of certain degenerate pseudo-differential operators of weighted order 1 on open subsets of cylindrical manifolds. The calculations are extremely delicate and completed with precision. A careful development now includes the unique solution, f, to the problem \(f+\phi Pf=g\). The paper concludes with several examples illustrating the technique developed therein.