Initial-irregular oblique derivative problems for nonlinear parabolic complex equations of second order with measurable coefficients
The authors study the parabolic equation \(F(x,t,u,Du,D^2 u)=u_t\) in a cylinder, such that the cross-section of the cylinder is an \((N+1)\)-connected bounded domain in \(\mathbb R^2\). On each boundary component, they prescribe the combination \[ \frac {\partial u}{\partial \nu} +\sigma u, \tag{*} \] where \(\nu\) is a sufficiently smooth vector field (which may be tangential at some points), and \(\sigma\) is a sufficiently smooth scalar field. A crucial assumption is that the product \((\nu\cdot \gamma)\sigma\) should be nonnegative, where \(\gamma\) is the outer normal. In addition, a value is prescribed for \(u\) at any point where \(\nu\cdot \gamma\) and \(\sigma\) simultaneously vanish. (This situation includes prescribed Dirichlet or Neumann data as well as more general boundary conditions.) Finally, initial data is prescribed which satisfies the boundary condition (*). Assuming that \(F\) is uniformly parabolic and that the derivatives of \(F\) with respect to \(Du\) and \(u\), along with \(F(x,t,0,0,0)\), lie in suitable \(L^p\) spaces, they show that this initial boundary value problem has a unique weak solution, that is, a continuous function with continuous spatial gradient and \(D^2 u\) and \(u_t\) in \(L^2\),which satisfies the differential equation almost everywhere and the boundary and initial conditions everywhere. Furthermore, this solution has Hölder continuous spatial gradient, and an a priori estimate for the Hölder norm is given. A key element of the proof is to write the equation in complex form (that is, in terms of the complex variable \(z=x_1+ix_2\)). From that point, the proof relies on results previously derived by the first author.
scientific article; zbMATH DE number 1419632