Oblique derivative problem for a second-order parabolic equation in a domain with edges (Q915984)

Let \(\Omega =\{x=[x_ 1,...,x_ N]\in {\mathbb{R}}^ N\); \(x_ 1\geq 0\), \(x_ 2\geq 0\}\), \(T_ 1=\{x\in \Omega\); \(x_ 1=0\}\), \(T_ 2=\{x\in \Omega\); \(x_ 2=0\}\). The author studies the problem \[ \partial u(x,t)/t-Lu(x,t)=f(x,t)\text{ on } \Omega \times <0,T>, \] \[ \partial u/\partial \ell_ 1=g_ 1(x,t)\text{ on } T_ 1\times <0,T>,\quad \partial u/\partial \ell_ 2=g_ 2(x,t)\text{ on } T_ 2\times <0,T>,\quad u(x,0)=\phi (x)\text{ on } \Omega. \] Here L is an elliptic second order operator with smooth coefficients depending on t, \(\ell_ 1\) and \(\ell_ 2\) are smooth vector fields on \(T_ 1\times <0,T>\) and \(T_ 2\times <0,T>\) (depending on t in general) which are not tangent to \(T_ 1\times <0,T>\) and \(T_ 2\times <0,T>\), respectively. An a-priori estimate and an existence result for this problem are given.