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Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space L-P. Derivation of the coefficients is based on solving certain optimization problems with respect to a vector parameter inside of an integral over the unit sphere.

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on f(n-1) is obtained under the assumption that f belongs to L-p. It is assumed that the kernel of the integral depends on the parameters alpha and beta. The explicit formulas for the sharp coefficients are found for the cases p = 1, p = 2 and for some values of alpha, beta in the case p = infinity. Conditions ensuring the validity of some analogues of the Khavinsons conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space., Funding Agencies|Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies are obtained for linear systems without zero order term on bounded domains and quasilinear systems of special form on bounded domains and on a class of unbounded domains. These conditions are formulated in algebraic form. They describe relation between the geometry of the invariant convex body and the coefficients of the system. Next, necessary conditions, which are also sufficient, for the invariance of some convex bodies are found for elliptic homogeneous systems with constant coefficients in a half-space. The necessary conditions are derived by using a criterion on the invariance of convex bodies for normalized matrix-valued integral transforms also obtained in the paper. In contrast with the previous studies of invariant sets for elliptic systems, no a priori restrictions on the coefficient matrices are imposed. © 2018, Hebrew University Magnes Press.