SHARP POINTWISE ESTIMATES FOR SOLUTIONS OF THE MODIFIED HELMHOLTZ EQUATION

Modified Helmholtz equation (Δ - c²)u = 0, c > 0, in the half-space Rⁿ ₊ = {x = (x', xn) : X' ∈ R^n-1, xn > 0} is considered. It is assumed that the boundary data of the Dirichlet and Neumann problems in R” belong to the space L^p. Representations for the sharp coefficients in pointwise estimates involving the gradient of solution to this equation in Rⁿ ₊ are obtained. Each of these representations includes an extremal problem with respect to a vector parameter inside of an integral over the unit sphere in Rⁿ. The extremal problems are solved for p ∈ [2, ∞] and p ∈ [2, (n + 2)/2] in the cases of Dirichlet and Neumann boundary data, respectively. Besides, the explicit formula for the sharp coefficient in the pointwise estimate for the modulus of the gradient of solution to the equation (c² - Δ)^{α/ 2}u = f with α > 1 and f ∈ L^∞(Rⁿ) is found.