In this paper methods are derived for obtaining aribitrarily close upper and lower bounds for the solutions of Dirichlet and exterior Neumann problems for general systems of second order elliptic partial differential equations definted on arbitrary domains.
In this paper pointwise a priori bounds are obtained for the solution of the Dirichlet problem associated with a rather general second order elliptic differential operator. These bounds involve only integrals of the data itself and not of its derivatives. Furthermore, the bounds obtained are applicable at any point in the domain of definition (i.e. up to the boundary of the region).
Let omega be a smooth bounded subset of (R sup N) and L be a uniformly elliptic second order differential operator with smooth coefficients. The author obtains a bound for the difference between the solution, u, to the equation Lu = f in omega with the Dirichlet boundary conditions: u = 0 on boundary omega, and the solution, u prime, to corresponding boundary value problem on a non-smooth domain omega prime which appriximates omega. The magnitude of the bound depends only on the Euclidean distance between the domains omega and omega prime. (Author).
The primary objective of this monograph is to give a comprehensive exposition of results surrounding the work of the authors concerning boundary regularity of weak solutions of second order elliptic quasilinear equations in divergence form. The book also contains a complete development of regularity of solutions of variational inequalities, including the double obstacle problem, where the obstacles are allowed to be discontinuous. The book concludes with a chapter devoted to the existence theory thus providing the reader with a complete treatment of the subject ranging from regularity of weak solutions to the existence of weak solutions.
An analysis is presented which deals with a technique for approximating the solution to a Cauchy problem for a geneal second-order elliptic patil differential equation defined in an N-dimensional region D. The method is based upon the determination of an a priori bound for the value of an arbitrary function u at a point P in D in terms of the values of u and its gradient on the cauchy surface andA FUNCTIONAL OF THE ELLIPTIC OPERATOR APPLIED TO U. (Author).
This volume is intended as an essentially self contained exposition of portions of the theory of second order quasilinear elliptic partial differential equations, with emphasis on the Dirichlet problem in bounded domains. It grew out of lecture notes for graduate courses by the authors at Stanford University, the final material extending well beyond the scope of these courses. By including preparatory chapters on topics such as potential theory and functional analysis, we have attempted to make the work accessible to a broad spectrum of readers. Above all, we hope the readers of this book will gain an appreciation of the multitude of ingenious barehanded techniques that have been developed in the study of elliptic equations and have become part of the repertoire of analysis. Many individuals have assisted us during the evolution of this work over the past several years. In particular, we are grateful for the valuable discussions with L. M. Simon and his contributions in Sections 15.4 to 15.8; for the helpful comments and corrections of J. M. Cross, A. S. Geue, J. Nash, P. Trudinger and B. Turkington; for the contributions of G. Williams in Section 10.5 and of A. S. Geue in Section 10.6; and for the impeccably typed manuscript which resulted from the dedicated efforts oflsolde Field at Stanford and Anna Zalucki at Canberra. The research of the authors connected with this volume was supported in part by the National Science Foundation.
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