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We obtain various optimal estimates for solutions of the Laplace, Lame, and Stokes equations in multidimensional domains. We also prove new real-part theorems for analytic functions. Bibliography: 14 titles., 1 Introduction In the present paper, we continue our study (1) of the best constants in certain inequalities for solutions of the Laplace, Lame, and Stokes equations. We also deal [...]

Explicit formulas for sharp coefficients in estimates of the modulus of an analytic function and its derivative in the upper half-plane are found. It is assumed that the boundary values of the real part of the function are in [L.sup.p]. As corollaries, sharp estimates for the modulus of the gradient of a harmonic function in [R.sub.+.sup.2] are deduced. Besides, a representation for the best constant in the estimate of the modulus of the gradient of a harmonic function in [R.sub.+.sup.n] by the [L.sup.p]-norm of the boundary normal derivative is given, 1 [less than or equal to] p < [infinity]. This representation is formulated in terms of an optimization problem on the unit sphere which is solved for p [member of] [1, n]. Bibliography: 6 titles., UDC 517.9 Dedicated to the 70th birthday of N. Krylov 1 Introduction In this paper, we deal first with a class of analytic functions in the half-plane [C.sub.+] = {z [...]

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on ℝn−1 is obtained under the assumption that f belongs to Lp. It is assumed that the kernel of the integral depends on the parameters α and β. The explicit formulas for the sharp coefficients are found for the cases p = 1, p = 2 and for some values of α, β in the case p = ∞. Conditions ensuring the validity of some analogues of the Khavinson’s conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space. [ABSTRACT FROM AUTHOR]