Abstract
The need to study boundary value problems for elliptic parabolic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory, quantum physics.Let H (dimH ≥ 1) – a finite-dimensional real Hilbert space with inner product ⟨·,·⟩ and norm ∥ · ∥. We will study the equation of the following form u + L (u) = g ∈ H, where L(·) is a non-linear continuous transformation, g is an element of the space H, u is the required solution of the problem from H.In this paper, we obtain two theorems on a priori estimates for solutions of nonlinear equations in a finite-dimensional Hilbert space. The work consists of four items. The conditions of the theorems are such that they can be used in the study of a certain class of initial-boundary value problems to obtain strong a priori estimates. This is the meaning of these theorems.