Approximation by Analytic and Harmonic Functions, Incompressible Vector Fields, and Temperature Distributions

In this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field ξ is Killing. We also show that a nontrivial incompressible vector field ξ on a compact Riemannian manifold is a Jacobi-type vector field if, and only if, ξ is Killing. Finally, we show that a nontrivial incompressible vector field ξ on a connected Riemannian manifold has a certain lower bound on the Ricci curvature in the direction of ξ, and if ξ is also a geodesic vector field, it necessarily implies that ξ is Killing.

Given bounded vector field \({b : {\mathbb{R}^{d}} \to {\mathbb{R}^{d}}}\), scalar field \({u : {\mathbb{R}^{d}} \to {\mathbb{R}}}\), and a smooth function \({\beta : {\mathbb{R}} \to {\mathbb{R}}}\), we study the characterization of the distribution \({{\rm div}(\beta(u)b)}\) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions), such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on the so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular, we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions), in the case when d = 2, we provide complete characterization of div(\({\beta(u)b}\)) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on \({{\mathbb{R}^{2}}}\) obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique, we obtain new sufficient conditions when any bounded weak solution u of \({\partial_t u + b \cdot \nabla u=0}\) is renormalized, that is when it also solves \({\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}\) for any smooth function \({\beta \colon{\mathbb{R}} \to {\mathbb{R}}}\). As a consequence, we obtain new a uniqueness result for this equation.

CHAPTER V - VARIOUS GENERALIZED BOUNDARY VALUE PROBLEMS

Electromigration of intergranular voids in metal films for microelectronic interconnects

Harmonic maps between pseudo-Riemannian surfaces

This survey paper deals with the unitary treatment of some classes of linear partial differential equations on Klein surfaces, which are the most general two-manifolds that support harmonic functions. We are mainly concerned with the study of harmonic functions with Dirichlet or Neumann boundary conditions. In such a way, the present paper extends several classical results to the abstract setting of dianalytic manifolds of complex dimension 1. We bring together systematically and concisely the Green and Neumann functions, the harmonic kernel function, and the harmonic measure on Klein surfaces. The technique to extend these concepts is to apply the classical methods to the complex double of a Klein surface, which is a Riemann surface endowed with an antianalytic involution. The existence of a harmonic function on a Riemann surface is given by the property that a harmonic function remains invariant under biholomorphic mappings. Similar problems on Riemann surfaces were developed by L.V. Ahlfors and L. Sario. The analysis developed in this paper offers perspectives to the qualitative analysis of other classes of linear or nonlinear elliptic equations on Klein surfaces.

In this article, we study an inverse problem (IP) for a fourth-order hyperbolic equation with nonlocal boundary conditions. This IP is reduced to the not self-adjoint boundary value problem (BVP) with corresponding boundary condition. Then, we use the separation of variables method, to reduce the not self-adjoint BVP to an integral equation. The existence and uniqueness of the integral equation are established by the contraction mappings principle and it is concluded that this solution is unique for a not-adjoint BVP. The existence and uniqueness of a nonlocal BVP with integral condition is proved. In addition, the fourth-order hyperbolic PDE is discretized using a collocation technique based on the quintic B-spline (QnB-spline) functions and reformed by the Tikhonov regularization function. The noise and analytical data are considered. The numerical outcome for a standard numerical example is discussed. Furthermore, the stability of the discretized system is also analyzed. The rate of convergence (ROC) of the method is also obtained.

A transition is typically required in irrigation canals, laboratory flumes and many other waterways. The warped type transition (WTT) is the preferred link structure between a small rectangular and relatively large trapezoidal channel section. However, the best method of determining the WTT geometry still requires evaluation. This paper provides an analytical function for the three-dimensional (3D) WTT geometry. This was achieved by solving a Dirichlet problem for Laplace's equation. The boundary conditions in the problem were chosen based on guidelines and recommendations from earlier studies of transitions. Solving the problem analytically yields a harmonic function. The geometry given by this function guarantees a streamlined rectangular-trapezoidal link and avoids sharp corners. The streamlined transition would work to reduce flow separation and head loss. This paper contributes to the development of a new method for fabricating WTT with precision and repeatability. The harmonic function can generate geometrical data as essential input for laboratory-scale and field-scale WTTs and can aid in the construction of a three-dimensional model for use in computational fluid dynamic models.

Analytical solution of steady state heat equation in nonlinear thermal-lensing effect

The Oshima–Sekiguchi and Liouville theorems on Heintze groups

New solutions to the theories of thermoelasticity and elasticity in the Cartesian coordinate system are found in this paper. New explicit partial solutions of thermoelasticity equations, when the temperature field is defined by 3D or 2D harmonic functions, are constructed. Displacements, deformations, and stresses determined by these partial solutions are called temperature functions. A simple formula for the expression of normal temperature stresses is obtained and it is shown that their sum is zero. Separate cases when the temperature depends on the product of harmonic functions of two variables on the degree of coordinate z are also considered. Partial and general solutions are derived for them. General solutions of thermoelasticity equations (Navier’s equations) through four harmonic functions, when the temperature field is given three-dimensional or two-dimensional harmonic functions, are constructed. The thermoelastic state of the body is divided into symmetric and asymmetric stress states. It is proposed to present the solutions of the theory of elasticity, which are expressed by the product of the harmonic function of two variables to the degree of the coordinate. Polynomial solutions that depend on three coordinate variables are recorded. An example of the application of the proposed solution is given.

The general formulation of the 3D temperature static problem of the theory of elasticity is considered. A new representation of the general solution of static equations of thermoelasticity is proposed. The three-dimensional stress state of a thick or thin plate are divided into three parts: bending of the plate, symmetric compression of the plate on it’s flat end surfaces and symmetric temperature problem. Third thermoelastic stress state of the plate is studied in detail. It have been reduced to a two-dimensional state after integration the stresses components along the normal to the middle surface. The displacements and stresses in the plate are expressed in terms of two two-dimensional harmonic functions and a particular solution, which is determined by a given temperature on the flat surfaces of the plate. Only the assumption has been used that the normal stresses perpendicular to the middle surface are insignificant. The introduced harmonic functions are determined from the boundary conditions on the lateral surface of the plate. The formula for the experimental determination of the sum of normal stresses has been found through the measured deflections of the end surfaces of the plate.

The dependence of the temperature and wind distribution of the zonal mean flow in the extratropical troposphere on the gradient of pontential vorticity along isentropes is examined. The extratropics here refer to the region outside the Hadley circulation. Of particular interest is whether the distribution of temperature and wind corresponding to a constant potential vorticity (PV) along isentropes resembles the observed, and the implications of PV homogenization along isentropes for the role of the tropics. With the assumption that PV is homogenized along isentropes, it is found that the temperature distribution in the extratropical troposphere may be determined by a linear, first-order partial differential equation. When the observed surface temperature distribution and tropical lapse rate are used as the boundary conditions, the solution of the equation is close to the observed temperature distribution except in the upper troposphere adjacent to the Hadley circulation, where the troposphere with no PV gradient is considerably colder. Consequently, the jet is also stronger. It is also found that the meridional distribution of the balanced zonal wind is very sensitive to the meridional distribution of the tropopause temperature. The result may suggest that the requirement of the global momentum balance has no practical role in determining the extratropical temperature distribution. The authors further investigated the sensitivity of the extratropical troposphere with constant PV along isentropes to changes in conditions at the tropical boundary (the edge of the Hadley circulation). It is found that the temperature and wind distributions in the extratropical troposphere are sensitive to the vertical distribution of PV at the tropical boundary. With a surface distribution of temperature that decreases linearly with latitude, the jet maximum occurs at the tropical boundary and moves with it. The overall pattern of wind distribution is not sensitive to the change of the position of the tropical boundary. Finally, the temperature and wind distributions of an extratropical troposphere with a finite PV gradient are calculated. It is found that the larger the isentropic PV gradient, the warmer the troposphere and the weaker the jet.

We prove the harmonic map version of the Royden decomposition in the sense that given any bounded C1-map f with finite total energy on a complete Riemannian manifold into a Cartan-Hadamard manifold, there exists a unique bounded harmonic map with finite total energy from the manifold into the Cartan-Hadamard manifold taking the same boundary value at each harmonic boundary point as that of f.

Three-dimensional harmonic functions near termination or intersection of gradient singularity lines: A general numerical method