Some Extremal Problems for Solutions of the Modified Helmholtz Equation in the Half-Space

Representations for the sharp coefficients in pointwise estimates involving the gradient of the solution to the modified Helmholtz equation (Δ- c²) u= 0 in the half-space R+n are described. It is assumed that the boundary data of the Dirichlet and Neumann problems in R+n belong to the space L^p. Each of these representations includes an extremal problem with respect to a vector parameter inside of an integral over the unit sphere in Rⁿ. Explicit formulas for solutions to the extremal problems are indicated for p∈ [ 2, ∞] and p∈ [ 2, (n+ 2 ) / 2 ] in the cases of Dirichlet and Neumann boundary data, respectively.