Maximum principles and sharp constants for solutions of elliptic and parabolic systems (Q1661298)

This interesting book deals with the maximum principle for elliptic and parabolic system. The first part of the book is devoted to elliptic equations and systems.NEWLINENEWLINEChapter 1 is introductory. Chapter 2 concerns necessary and sufficient conditions for validity of the classical maximum modulus principle for strongly elliptic systems of the second order. Chapter 3 deals with finding representations for the best constants in the Miranda-Agmon maximum principle for solutions of homogeneous strongly elliptic systems of the second order with constant coefficients in a halfspace. Also obtained are explicit formulas for sharp constants in inequalities of the Miranda-Agmon type for solutions of the Stokes and Lamé systems as well a system of viscoelasticity in a halfspace. Chapter 4 gives estimates of solutions of the Dirichlet problem for elliptic systems on a halfspace or on a ball and the boundary data from \(L^p\). Chapter 5 is devoted to sharp Miranda-Agmon estimates for derivatives of solutions of higher order elliptic equations with constant coefficients. Chapter 6 contains inequalities with sharp coefficients for the modulus of directional derivatives of harmonic functions in a half-space and in a ball. The boundary values of harmonic functions are assumed to be bounded, semibounded or \(L^p\)-integrable. Chapter~7 deals with vector-valued potentials of the double layer type. Representations are found for the norms and for the essential norms of matrix-valued double layer potentials in spaces of continuous vector-valued functions. Representations for the essential norm are specified for boundary integral operators of elasticity theory and hydrodynamics both in the planar case for a domain with angular points, and in the three-dimensional case for a domain with conic points or edges.NEWLINENEWLINE The second part of the book is devoted to parabolic systems. Chapter 8 considers systems of partial differential equations of the first order in \(t\) and of order \(2k\) in \(x\) variables, which are uniformly parabolic. It is shown that the classical maximum modulus principle is not valid in \(\mathbb R^n \times (0,T)\) for \(k\geq 2\). For \(k=1\), necessary, and separately, sufficient conditions are obtained for the classical maximum modulus principle to hold in \(\mathbb R^n \times (0,T)\) and in \(\Omega \times (0,T)\) where \(\Omega \) is a bounded subdomain of \(\mathbb R^n\). If the coefficients of the system do not depend on \(t\), these conditions coincide. Chapter 9 considers solutions to the initial boundary value problem in the cylinder \(\Omega \times (0,T)\) with zero Dirichlet data on \(\partial \Omega \times (0,T)\) for a linear second order system, which is strongly parabolic, with smooth matrix-valued coefficients depending only on \(x\). It is assumed that \(\Omega \) is a bounded subdomain of \(\mathbb R^n\) with smooth boundary. It is shown that the criterion for validity of the classical maximum modulus principle, obtained in Chapter~8 for the case of the parabolic second order system with coefficients independent of \(t\), remains necessary if one assumes a priory that the boundary data on \(\partial \Omega \times(0,T)\) are zero. Chapter~10 and Chapter~11 consider systems of partial differential equations, which contain only second derivatives in the \(x\) variables and which are uniformly parabolic. In Chapter~10 necessary and sufficient conditions are obtained for the maximum norm principle to hold in the layer \(\mathbb R^n \times (0,T)\) and in the cylinder \(\Omega \times (0,T)\), where \(\Omega \) is a bounded subdomain of \(\mathbb R^n\). The necessary and sufficient conditions coincide if the coefficients of the system do not depend on \(t\). This criterion for validity of the maximum norm principle is formulated as a number of equivalent algebraic conditions describing the relation between the geometry of the unit sphere of the given norm and coefficients of the system. Chapter~11 studies this problem under assumption that the norm is twice continuously differentiable.