the INSTITUTE JA rill 1. A FA 1984 ISSUE 1 of 4 ISSN 0081-5438 Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order Translation of TPYA61 opaeHa ."IeHHH2 MATEMATMECKOFO HHCTFITYTA HMeHH B. A. CTEKJIOBA Tom 160 (1982) AMERICAN MATHEMATICAL SOCIETY PROVIDENCE RHODE ISLAND Proceedings of the STEKLOV INSTITUTE OF MATHEMATICS 1984, ISSUE 1 Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order by A. V. Ivanov AKAI[EMHSI HAYK CO103A COBETCKIIX COUYIAJI11CTIILIECK11X PECfYlJIHK TPYAbI oprleua J1eHHHa MATEMATI44ECKOFO 14HCT 14TYTA HMeHH B. A. CTEKJIOBA CLX A. B. I'IBAHOB KBA3HJIHHEI1HbIE BbIPO)KAAIOIII,HECA H HEPABHOMEPHO 3JUIHrITHtIECKHE H rlAPABOJIHLIECKHE YPABHEH14A BTOPOr'O fOPARKA OTBeTCTBCHHbII peaaKTOp (Editor-in-chief) aKaaeMHK C. M. HHKOJIbCKH0 (S. M. Nikol'skii) 3aMecTHTeJib oTBeTCTBeHHoro pe.aaKTopa (Assistant to the editor-in-chief) npo4leccop E. A. BOJIKOB (E. A. Volkov) H3.aaTefbCTBO "HayKa" AenHHrpan 1982 Translated by J. R. SCHULENBERGER Library of Congress Cataloging in Publication Data lvanov, A. V. Quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order. (Proceedings of the Steklov Institute of Mathematics; 1984, issue 1) Translation of: Kvazilineinye vyrozhdaiushchiesfa i naravnomerno ellipticheskie i para- boliticheskie uravneniia vtorogo poriadka. Bibliography: p. 1. Differential equations, Elliptic-Numerical solutions. 2.Differential equations, Parabolic- Numerical solutions. 3. Boundary value problems-Numerical solutions.I. Title. II. Series: Trudy ordena Lenina Matematicheskogo instituta imeni V. A. Steklova. English; 1984, issue 1. QAI.A413 1984, issue 1 )QA377) 510s 1515.3'531 84-12386 ISBN 0-8218-3080-5 March 1984 Translation authorized by the All-Union Agency for Author's Rights, Moscow Information on Copying and Reprinting can be found at the back of this journal. The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability. Copyright 0 1984, by the American Mathematical Society PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS IN THE ACADEMY OF SCIENCES OF THE USSR TABLE OF CONTENTS Preface 1 Basic Notation 3 PART I. QUASILINEAR, NONUNIFORMLY ELLIPTIC AND PARA- BOLIC EQUATIONS OF NONDIVERGENCE TYPE 7 CHAPTER 1. The Dirichlet Problem for Quasilinear, Nonuniformly Elliptic Equations 13 §1. The basic characteristics of a quasilinear elliptic equation 13 §2. A conditional existence theorem 15 §3. Some facts about the barrier technique 16 §4. Estimates of IVul on the boundary ail by means of global barriers 18 §5. Estimates of jVul on the boundary by means of local barriers. 22 §6. Estimates of maxnIVul for equations with structure described in terms of the majorant £1 27 §7. The estimate of maxn I VuI for equations with structure described in terms of the majorant E2 31 §8. The estimate of maxolVul for a special class of equations 34 §9. The existence theorem for a solution of the Dirichlet problem in the case of an arbitrary domain 11 with a sufficiently smooth boundary 38 §10. Existence theorem for a solution of the Dirichlet problem in the case of a strictly convex domain fl 40 CHAPTER 2.The First Boundary Value Problem for Quasilinear, Nonuniformly Parabolic Equations 43 §1. A conditional existence theorem 43 §2. Estimates of IVul on r 46 §3. Estimates of maxQIVul 49 §4. Existence theorems for a classical solution of the first boundary value problem 54 §5. Nonexistence theorems 56 iii iv CONTENTS CHAPTER 3. Local Estimates of the Gradients of Solutions of Quasi- linear Elliptic Equations and Theorems of Liouville Type 60 §1. Estimates of IVu(xo)I in terms of maxK,(xo)IuI 60 §2. An estimate of jVu(xo)I in terms of maxK,(xo)u (minK,(so)u). I larnack's inequality 67 §3. Two-sided Liouville theorems 71 §4. One-sided Liouville theorems 74 PART II. QUASILINEAR (A, b)-ELLIPTIC EQUATIONS 77 CHAPTER 4. Some Analytic Tools Used in the Investigation of Solv- ability of Boundary Value Problems for (A, b)-Elliptic Equations 85 §1. Generalized A-derivatives 85 §2. Generalized limit values of a function on the boundary of a domain 89 §3. The regular and singular parts of the boundary 31 95 §4. Some imbedding theorems 98 §5. Some imbedding theorems for functions depending on time 102 §6. General operator equations in a Banach space 106 §7. A special space of functions of scalar argument with values in a Banach space 112 CHAPTER 5. The General Boundary Value Problem for (A, b, m, m)- Elliptic Equations 118 §1. The structure of the equations and the classical formulation of the general boundary value problem 118 §2. The basic function spaces and the operators connected with the general boundary value problem for an (A, b, m, m)-elliptic equation 128 §3. A generalized formulation of the general boundary value prob- lem for (A, b, m, m)-elliptic equations 137 §4. Conditions for existence and uniqueness of a generalized solution of the general boundary value problem 139 §5. Linear (A, b)-elliptic equations 146 CHAPTER 6. Existence Theorems for Regular Generalized Solutions of the First Boundary Value Problem for (A, b)-Elliptic Equations. 149 §1. Nondivergence (A,b)-elliptic equations 149 §2. Existence and uniqueness of regular generalized solutions of the first boundary value problem 152 §3. The existence of regular generalized solutions of the first bound- ary value problem which are bounded in 11 together with their partial derivatives of first order 163 CONTENTS V PART III. (A, 0)-ELLIPTIC AND (A, 0)-PARABOLIC EQUATIONS 173 CHAPTER 7. (A, 0)-Elliptic Equations 177 § 1. The general boundary value problem for (A, 0, m, m)-elliptic equa- tions 177 §2. (A, 0)-elliptic equations with weak degeneracy 179 §3. Existence and uniqueness of A -regular generalized solutions of the first boundary value problem for (A, 0)-elliptic equations 191 CHAPTER 8. (A, 0)-Parabolic Equations 203 §1. The basic function spaces connected with the general boundary value problem for (A, 0, m, m)-parabolic equations 203 §2. The general boundary value problem for (A, 0, m, m)-parabolic equations 216 §3. (A, 0)-parabolic equations with weak degeneracy 222 §4. Linear A-parabolic equations with weak degeneracy 238 PART IV. ON REGULARITY OF GENERALIZED SOLUTIONS OF QUASILINEAR DEGENERATE PARABOLIC EQUATIONS. 243 CHAPTER 9. Investigation of the Properties of Generalized Solutions 245 §1. The structure of the equations and their generalized solutions. 245 §2. On regularity of generalized solutions in the variable t 250 §3. The energy inequality 253 §4. Functions of generalized solutions 255 §5. Local estimates in LA PO 262 §6. Global estimates in LP,P° 268 §7. Exponential summability of generalized solutions 270 §8. Local boundedness of generalized solutions 272 §9. Boundedness of generalized solutions of the boundary value prob- lem 275 §10. The maximum principle 277 Bibliography 281 [...]... study of questions of solvability of main boundary value problems for degenerate and nonuniformly elliptic and parabolic equations of second order and to the investigation of differential and certain qualitative properties of the solutions of such equations The study of various questions of variational calculus, differential geometry, and the mechanics of continuous media leads to quasilinear degenerate. .. arbitrary parabolic equations of the form (2) Chapter 3, which concludes Part I, is devoted to obtaining local estimates of the gradients of solutions of quasilinear elliptic equations of the form (1) and their application to the proof of certain qualitative properties of solutions of these equations In the case of uniformly elliptic equations local estimates of the gradients of solutions of equations of. .. possibility of using these and similar estimates to study quasilinear degenerate elliptic and parabolic equations Parts II and III are devoted to the construction of a theory of solvability of the main boundary value problems for large classes of quasilinear equations with a nonnegative characteristic form In Part II the class of quasilinear, so-called (A, b )elliptic equations is introduced Special cases of. .. I: EQUATIONS OF NONDIVER(iENCE TYPE question of the validity of a priori estimates of the maximum moduli of the gradients of solutions for quasilinear elliptic and parabolic equations is the key question, since the basic restrictions on the structure of such equations arise precisely at this stage Nonuniformly elliptic equations of the form (1) are considered in Chapter 1 It is known (see [83] and. .. Euler equations for many variational problems are quasilinear, degenerate or nonuniformly elliptic equations With regard to the character of the methods applied, this monograph is organi- cally bound with the monograph of 0 A Ladyzhenskaya and N N Ural'tseva, Linear and quasilinear equations of elliptic type, and with the monograph of 0 A Ladyzhenskaya, V A Solonnikov, and N N Ural'tseva, Linear and quasilinear. .. quasilinear equations of parabolic type In particular, a theory of solvability of the basic boundary value problems for quasilinear, nondegenerate and uniformly elliptic and parabolic equations was constructed in those monographs The monograph consists of four parts In Part I the principal object of investigation is the question of classical solvability of the first boundary value problem for quasilinear, nonuniformly. .. equations- (A,0 )elliptic and so-called (A, 0) -parabolic equations, which are more immediate generali- zations of classical elliptic and parabolic quasilinear equations All the conditions under which theorems on the existence and uniqueness of a generalized solution (of energy type) of the general boundary value problem are established for (A, 0, m, m )elliptic and (A, 0, m, m) -parabolic equations are of easily verifiable character... nonuniformly elliptic and parabolic equations of nondivergence form A priori estimates of the gradients of solutions in a closed domain are established for large classes of such equations; these estimates lead to theorems on the existence of solutions of the problem in question on the basis of the well-known results of Ladyzhenskaya and Ural'tseva In this same part qualified local estimates of the gradients of. .. first second, and third parts of the monograph are numbered in special fashion Here the numbering reflects only the number of the formula within the given introduction There are no references to these formulas outside the particular introduction PART I QUASILINEAR, NONUNIFORMLY ELLIPTIC AND PARABOLIC EQUATIONS OF NONDIVERGENCE TYPE Boundary value problems for linear and quasilinear elliptic and parabolic. .. depending on the parameter in In view of the results of Ladyzhenskaya and Ural'tseva the problem of solvability of a boundary value problem for a nonuniformly elliptic or parabolic equation reduces to the question of constructing a priori estimates of the maximum nioduli of the gradients of solutions for a suitable one-parameter family of similar equations Much of Part I of the present monograph is devoted . questions of solvability of basic boundary value problems for quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order, and also to the investigation of differential and. study of questions of solvability of main boundary value problems for degenerate and nonuniformly elliptic and parabolic equations of second order and to the investigation of differential and certain qualitative. SCHULENBERGER Library of Congress Cataloging in Publication Data lvanov, A. V. Quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order. (Proceedings of the Steklov Institute of