Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains

ISBN-13: 9783034604765 / Angielski / Miękka / 2010 / 220 str.

Mikhail Borsuk

Opis

Thegoalofthisbookistoinvestigatethebehaviourofweaksolutionstotheelliptic transmisssion problem in a neighborhood of boundary singularities: angular and conic points or edges. We consider this problem both for linear and quasi-linear (very little studied) equations. In style and methods of research, this book is close to our monograph 14] together with Prof. V. Kondratiev. The book consists of an Introduction, seven chapters, a Bibliography and Indexes. Chapter 1 is of auxiliary character. We recall the basic de?nitions and properties of Sobolev spaces and weighted Sobolev-Kondratiev spaces. Here we recall also the well-known Stampacchia's Lemma and derive a generalization for the solution of the Cauchy problem - the Gronwall-Chaplygin type inequality. Chapter 2 deals with the eigenvalue problem for m-Laplace-Beltrami op- ator. By the variational principle we prove a new integro-di?erential Friedrichs- Wirtinger type inequality. This inequality is the basis for obtaining of precise exponents of the decreasing rate of the solution near boundary singularities. Chapter 3 deals with the investigation of the transmission problem for linear elliptic second order equations in the domains with boundary conic point. Chapter 4 is devoted to the transmission problem in conic domains with N di?erent media for an equation with the Laplace operator in the principal part. Chapters 5, 6 and 7 deal with the investigation of the transmission problem forquasi-linearellipticsecondorderequationsinthe domainswithboundaryconic point (Chapters 5-6) or with an edge at the boundary of a domain.

Spis treści

Preface.- Introduction.- 1 Preliminaries.- 2 Eigenvalue problem and integro-differential inequalities.- 3 Best possible estimates of solutions to the transmission problem for linear elliptic divergence second order equations in a conical domain.- 4 Transmission problem for the Laplace operator with N different media.- 5 Transmission problem for weak quasi-linear elliptic equations in a conical domain.- 6 Transmission problem for strong quasi-linear elliptic equations in a conical domain.- 7 Best possible estimates of solutions to the transmission problem for a quasi-linear elliptic divergence second order equation in a domain with a boundary edge.- Bibliography.- Index.- Notation Index.

Blurb

The goal of this book is to investigate the behavior of weak solutions of the elliptic transmission problem in a neighborhood of boundary singularities: angular and conic points or edges. This problem is discussed for both linear and quasilinear equations. A principal new feature of this book is the consideration of our estimates of weak solutions of the transmission problem for linear elliptic equations with minimal smooth coeciffients in n-dimensional conic domains. Only few works are devoted to the transmission problem for quasilinear elliptic equations. Therefore, we investigate the weak solutions for general divergence quasilinear elliptic second-order equations in n-dimensional conic domains or in domains with edges. The basis of the present work is the method of integro-differential inequalities. Such inequalities with exact estimating constants allow us to establish possible or best possible estimates of solutions to boundary value problems for elliptic equations near singularities on the boundary. A new Friedrichs–Wirtinger type inequality is proved and applied to the investigation of the behavior of weak solutions of the transmission problem. All results are given with complete proofs. The book will be of interest to graduate students and specialists in elliptic boundary value problems and applications.