Historical milestones and directions of scientic research. To the 100th anniversary of the Oles Honchar Dnipro National University and the 52nd anniversary of the Department of Dierential Equations.
We study an optimal control problem for mixed Dirichlet-Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and L1-nonlinearity in its right-hand side. A distribution u acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution yd 2 L2( ) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After dening a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. We derive also conditions when the set of feasible solutions has a nonempty intersection with the space of bounded distributions L1( ).
We study a Dirichlet-Navier optimal design problem for a quasi-linear mono-tone p-biharmonic equation with control and state constraints. The coecient of the p-biharmonic operator we take as a design variable in BV ( )\L1( ). In order to handle the inherent degeneracy of the p-Laplacian and the pointwise state constraints, we use regularization and relaxation approaches. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted p-biharmonic operator and Henig approximation of the ordering cone. Further we discuss the asymptotic behaviour of the solutions to regularized problem on each ("; k)-level as the parameters tend to zero and innity, respectively.
The representation u(x) = F2(x)Qm-2(x)+Qm(x) for the solution to the Dirichlet problem for the Laplace equation in a disk: F2(x) = jx - x0j2 - c2 6 0, is proved using the Poisson integral; Qm(x) being the polynomial boundary function of degree m, Qm-2(x) being the uniquely determined polynomial of degree m - 2.
The inverse problems for dierential equations are investigated, the solutions of which do not use information about the exact characteristics of the physical process. Such inverse problems have not yet become widespread, but they are of great practical importance. Some approaches to solving inverse problems of this type are suggested.
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