On mixed boundary value problems for parabolic equations in singular domains (Q1074768)

Consider a simply connected bounded domain \(G\subset R^ 2\) whose boundary consists of q curves \(\Gamma_ k\), \(k=1,...,q\) of class \(C^{m+\alpha}\), \(m\geq 2\), \(\alpha\in (0,1)\), meeting at points \(x_ k\) where an angle \(\gamma_ k\) is formed. Consider the problem \[ \sum^{2}_{i,j=1}a_{ij}(x,t)(\partial^ 2/\partial x_ i\partial x_ j)u(x,t)+\sum^{2}_{i=i}b_ i(x,t)(\partial /\partial x_ i)u(x,t)+ \] \[ a(x,t)u-(\partial u/\partial t)(x,t)=f(x,t) \] in \(G\times [0,T)\); \(u(x,0)=0\) in \(\bar G,\) \(\eta_ ku+(1-\eta_ k)\partial u/\partial n=0\) on \(\Gamma_ k\times [0,T)\) and \(\eta_ k\) is either 0 or 1 (with the constraint that \(\eta\) changes at every corner point). It is assumed that f and the coefficients of the operator are \(C^{m-2+\alpha}.\) Author proves that any bounded solution of the problem belongs to \(C^{\gamma}(\bar G\times [0,T])\) where \(\nu =\min (m+\alpha,\pi /\beta -\epsilon)\) where \(\epsilon >0\) arbitrarily small and \(\beta\) is related to the geometry of the image of G under the transformation bringing the differential operator to the canonical form.