A. A. Kuleshov

On some properties of generalized solutions of the wave equation in the classes Lp and Wp1 for p ≥ 1

For each p ≥ 1, in closed analytic form, we establish the existence of a unique generalized solution in Lp of the mixed problem for the wave equation in the rectangle [0 ≤ x ≤ 1] × [0 ≤ t ≤ T] with zero initial conditions and with boundary conditions of the first kind, one of which is homogeneous.Next, we derive necessary conditions for this solution to belong to Wp1.We present examples showing that these necessary conditions are not sufficient for any p ≥ 1.

Necessary and sufficient condition for the generalized solution of a mixed problem for the wave equation to belong to the class L p for p ≥ 1

In terms of requirements imposed on the boundary function, we obtain a necessary and sufficient condition for the generalized solution of a mixed problem for the wave equation with zero initial conditions and with boundary conditions of the first kind to belong to the class Lp.

Continuous sums of ridge functions on a convex body and the class VMO

Sums of ridge functions on convex bodies in the space ℝn are studied. It is established that, under sufficiently general constraints on the functions of one variable generating the sums, each of these sums must belong to the class VMO on each finite closed interval of its domain.

On four mixed problems for the string vibration equation with homogeneous nonlocal conditions

We consider four mixed problems for the string vibration equation with boundary conditions and homogeneous nonlocal conditions of the first or second kind and with zero initial conditions. Using recursion relations, we find the generalized solutions of the abovementioned problems.

On some properties of smooth sums of ridge functions

The following problem is studied: If a finite sum of ridge functions defined on an open subset of Rn belongs to some smoothness class, can one represent this sum as a sum of ridge functions (with the same set of directions) each of which belongs to the same smoothness class as the whole sum? It is shown that when the sum contains m terms and there are m − 1 linearly independent directions among m linearly dependent ones, such a representation exists.

Four mixed problems for the vibrating string equation with boundary and nonlocal Dirichlet and Neumann conditions

We construct weak solutions u ( x , t ) of the wave equation for four mixed problems with zero initial conditions, the boundary condition u (0, t ) = µ ( t ) or u x (0, t ) = µ ( t ) , and the nonlocal condition u ( l , t ) – α u ( x 0 , t ) = ν ( t ) or u x ( l , t ) – α u x ( x 0 , t ) = ν ( t ) , where 0 ≤ x 0 < l and α is an arbitrary constant. On the rectangle Q T = [0 ≤ x ≤ l ] × [0 ≤ t ≤ T ] , consider the class ( Q T ) introduced by Il’in in [1], namely, the class of two-variable functions u ( x , t ) that are continuous in Q T and have generalized partial derivatives u x ( x , t ) and u t ( x , t ) that belong to L 2 ( Q T ) , as well as to L 2 [0 ≤ x ≤ l ] for all t ∈ [0, T ] and to L 2 [0 ≤ t ≤ T ] for all x ∈ [0, l ] . We search for weak solutions from ( Q T ) to mixed problems for the wave equation

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class L p for p ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation utt(x, t) − uxx(x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle QT = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ Lp[0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class Lp(QT) for p ≥ 1. We construct the solution in explicit analytic form.

Mixed problems for the string vibration equation with homogeneous boundary conditions and inhomogeneous nonlocal conditions

We consider four mixed problems for the string vibration equation with homogeneous boundary and inhomogeneous nonlocal conditions of the first or second kind and with zero initial conditions. By using recursion relations, we find generalized solutions of the abovementioned problems.

Optimization of the boundary control induced by the third boundary condition

We study the boundary control by the third boundary condition on the left end of a string, the right end being fixed. An optimality criterion based on the minimization of an integral of a linear combination of the control itself and its antiderivative raised to an arbitrary power p ≥ 1 is established. A method is developed permitting one to find a control satisfying this optimality criterion and write it out in closed form. The uniqueness of the optimal control for p > 1 is proved.