On the Hölder regularity for parabolic quasivariational inequalities of impulse control (Q1920684)

In the paper, a question of the regularity of solutions to a nonlinear parabolic quasivariational inequality (QVI) of impulsive control for the operator \(Pu = {{\partial u}\over {\partial t}} - {{\partial}\over {\partial x_i}} a_i(x,t,u,u_x) + a(x,t,u,u_x)\) is posed. Given a convex domain \(\Omega\) of class \(C^2\) satisfying suitable regularity conditions, \(0 < T < \infty\) and \(Q = \Omega \times (0,T)\), the QVI of impulsive control is formulated as follows: find a function \(u = u(x,t)\) such that \(u(x,t) - Mu(x,t) \leq 0\) in \(Q\), \(u(0) = 0\) and \[ \displaystyle \int_{Q} \left[ u_t (v-u) + a_i(x,t,u,u_x){(v-u)}_{x_i} + a(x,t,u,u_x)(v-u) \right] dxdt \geq 0 \tag{1} \] \noindent for all test functions \(v\) satisfying \(v - Mv \leq 0\) in \(Q\). Here \(M\) denotes the nonlinear operator \(u \mapsto M(u)\) which transforms measurable bounded functions into itself and satisfies some properties (the most important of which is the increasing property: \(u_1 \leq u_2\) implies \(M (u_1) \leq M (u_2)\)). In this paper, the problem (1) with the prototype of the operator \(M\) of the form \(M (u(x,t)) = 1 + \text{ess} \inf \{ u(x+\xi, t) \colon \xi \geq 0, x + \xi \in \Omega \} \) is studied. First, a local estimate of the Hölder norms for an arbitrary solution to (1) with the Dirichlet boundary condition is provided. Then, the author gives the estimate on the Hölder norm of a solution to (1) with the Neumann boundary condition, near the boundary of the cylinder \(Q\). As a consequence it is shown that the solution of this problem lies in \(H^{\alpha,\alpha/2}(Q)\). The proofs of both above results are rather long and technical.