Search

We obtain various optimal estimates for solutions of the Laplace, Lamé, and Stokes equations in multidimensional domains. We also prove new real-part theorems for analytic functions. [ABSTRACT FROM AUTHOR]

We obtain a representation for the sharp coefficient in an estimate of the modulus of the nth derivative of an analytic function in the upper half-plane $ {\mathbb{C}_{+} } $. It is assumed that the boundary value of the real part of the function on $ \partial {\mathbb{C}_{+} } $ belongs to L. This representation is specified for p = 1 and p = 2. For p = ∞ and for derivatives of odd order, an explicit formula for the sharp coefficient is found. A limit relation for the sharp coefficient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. It is assumed that the boundary value of the real part of the function belongs to L. The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in $ {\mathbb{C}_{+} } $. As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

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. As corollaries, sharp estimates for the modulus of the gradient of a harmonic function in $$ {\mathbb R}_{+}^2 $$ are deduced. Besides, a representation for the best constant in the estimate of the modulus of the gradient of a harmonic function in $$ {\mathbb R}_{+}^n $$ by the L-norm of the boundary normal derivative is given, $$ 1 \leqslant p \leqslant \infty $$. This representation is formulated in terms of an optimization problem on the unit sphere which is solved for p ∈ [1, n]. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]