Singular Perturbation Problems For Second Order Elliptic Partial Differential Equations

No detailed description available for "Singularly Perturbed Differential Equations".

Singularly perturbed second order elliptic partial differential equations with Neumann boundary conditions arise in many areas of application. These problems rarely have smooth limit solutions. In this paper, the author characterize the limit solution for a wide class of such problems. They also give an abstract rate of convergence theorem and apply the abstract theorem to certain finite difference approximations. Keywords: Viscosity solution; and Viscosity inequalities.

This work aims to be of interest to those who have to work with differential equations and acts either as a reference or as a book to learn from. The authors have made the treatment self-contained.

This introduction to the singularly perturbed methods in the nonlinear elliptic partial differential equations emphasises the existence and local uniqueness of solutions exhibiting concentration property. The authors avoid using sophisticated estimates and explain the main techniques by thoroughly investigating two relatively simple but typical non-compact elliptic problems. Each chapter then progresses to other related problems to help the reader learn more about the general theories developed from singularly perturbed methods. Designed for PhD students and junior mathematicians intending to do their research in the area of elliptic differential equations, the text covers three main topics. The first is the compactness of the minimization sequences, or the Palais-Smale sequences, or a sequence of approximate solutions; the second is the construction of peak or bubbling solutions by using the Lyapunov-Schmidt reduction method; and the third is the local uniqueness of these solutions.

Singular Partial Differential Equations provides an analytical, constructive, and elementary approach to non-elementary problems. In the first monograph to consider such equations, the author investigates the solvability of partial differential equations and systems in a class of bounded functions with complex coefficients having singularities at the inner points or boundary of the domain. Using complex variable techniques, the author considers a variety of problems, including the Dirichlet, Neumann, and other problems for first order systems. He also explores applications to singular equations, degenerate, high-dimensional Beltrami systems in Cn,, and others. Singular Partial Differential Equations fills a gap in the literature on degenerate and singular partial differential equations and significantly contributes to the theory of boundary value problems for these equations and systems. It will undoubtedly stimulate further research in the field. Practical applications in analysis and physics make this important reading for researchers and students in physics and engineering, along with mathematicians.

Improperly posed Cauchy problems are the primary topics in this discussion which assumes that the geometry and coefficients of the equations are known precisely. Appropriate references are made to other classes of improperly posed problems. The contents include straight forward examples of methods eigenfunction, quasireversibility, logarithmic convexity, Lagrange identity, and weighted energy used in treating improperly posed Cauchy problems. The Cauchy problem for a class of second order operator equations is examined as is the question of determining explicit stability inequalities for solving the Cauchy problem for elliptic equations. Among other things, an example with improperly posed perturbed and unperturbed problems is discussed and concavity methods are used to investigate finite escape time for classes of operator equations.

A systematic examination of classical and non-classical problems for linear partial differential equations and systems of elliptic, hyperbolic and mixed types. Among a number of difficult problems addressed are the Dirichlet and oblique derivative problems for non- uniformly elliptic equations and non-strongly elliptic systems and the Cauchy and Darloux problems for non-strongly hyperbolic systems and hyperbolic equations with parabolic degeneracy on the boundary. Written at a level suitable for undergraduate and graduate students and researchers. Individual price, $89. Annotation copyrighted by Book News, Inc., Portland, OR