Vol 239, No 1 (2019)

It is established that any homeomorphism f of the Sobolev class \( {W}_{\mathrm{loc}}^{1,1} \) with outer dilatation \( {K}_O\left(x,f\right)\in {L}_{\mathrm{loc}}^{n-1} \) is the so-called lower Q-homeomorphism with Q(x) = K_O(x, f) and also a ring Q-homeomorphism with \( Q(x)={K}_O^{n-1}\left(x,f\right) \). This allows us to apply the theory of boundary behavior of ring and lower Q-homeomorphisms. In particular, we have found the conditions imposed on the outer dilatation K_O(x, f) and the boundaries of domains under which any homeomorphism of the Sobolev class \( {W}_{\mathrm{loc}}^{1,1} \) admits continuous or homeomorphic extensions to the boundary.

We obtain the exact values of the best approximations, basic widths and Kolmogorov widths for some sets of images of multipliers in the modular Orlicz spaces l_M: We give a description of the space S_M,N of all multipliers from the space l_M to l_N.

We have introduced new functional spaces of the Lizorkin–Triebel–Morrey type, and a Sobolev-type inequality is proved. We have also shown that the generalized derivatives of functions from this spaces satisfy the generalized Hölder condition.

We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω × (0; 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ Ω [0; T]; Σ ⊂ ????Ω of the lateral surface of the cylinder Ω × (0; T). The domain of observation is Σ × [0; 2T]; and the pressure on another part (????ΩnΣ) × [0; 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H¹(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave u^f such that u^f (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincaré inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound.

The correspondence between a monogenic function in an arbitrary finite-dimensional commutative associative algebra and a finite collection of monogenic functions in a special commutative associative algebra is established.