Parabolic Boundary Value Problems

ISBN-13: 9783034897655 / Angielski / Miękka / 2012 / 300 str.

Samuil D. Eidelman; Nicolae V. Zhitarashu

Opis

The present monograph is devoted to the theory of general parabolic boundary value problems. The vastness of this theory forced us to take difficult decisions in selecting the results to be presented and in determining the degree of detail needed to describe their proofs. In the first chapter we define the basic notions at the origin of the theory of parabolic boundary value problems and give various examples of illustrative and descriptive character. The main part of the monograph (Chapters II to V) is devoted to a the detailed and systematic exposition of the L -theory of parabolic 2 boundary value problems with smooth coefficients in Hilbert spaces of smooth functions and distributions of arbitrary finite order and with some natural appli- cations of the theory. Wishing to make the monograph more informative, we included in Chapter VI a survey of results in the theory of the Cauchy problem and boundary value problems in the traditional spaces of smooth functions. We give no proofs; rather, we attempt to compare different results and techniques. Special attention is paid to a detailed analysis of examples illustrating and complementing the results for- mulated. The chapter is written in such a way that the reader interested only in the results of the classical theory of the Cauchy problem and boundary value problems may concentrate on it alone, skipping the previous chapters.

Spis treści

I Equations and Problems.- I.1 Equations.- I.1.1 Introduction.- I.1.2 Systems parabolic in the sense of Petrovski?.- I.1.3 Systems parabolic in the sense of Solonnikov.- I.2 Initial and boundary value problems.- I.2.1 Introduction.- I.2.2 The Cauchy problem. The initial value problem.- I.2.3 Parabolic boundary value problems.- I.2.4 Particular cases. Examples.- I.2.5 Parabolic conjugation problems.- I.2.6 Nonlocal parabolic boundary value problems.- II Functional Spaces.- II.1 Spaces of test functions and distributions.- II.1.1 Definition and basic roperties of distributions. Spaces D(?) and D’(?).- II.1.2 Differentiation of distributions. Multiplication of distributions by smooth functions.- II.1.3 Distributions with compact supports. The spaces ?(?) and ?’(?), S(?) and S’(?).- II.1.4 Convolution and direct product of distributions.- II.1.5 Fourier and Laplace transformations of distributions.- II.2 The Hilbert spaces Hs and ?s.- II.2.1 The isotropic spaces Hs(?n) and H+s(?n).- II.2.2 The isotropic spaces Hs(?+n) and their duals.- II.2.3 Restriction to a hyperplane.- II.2.4 The anisotropic Sobolev-Slobodetski? spaces ?s.- II.2.5 The anisotropic spaces ?s on ?+n+1 and E+n+1, ?++n+1 and E+n.- II.2.6 The dual spaces ?s.- II.2.7 Traces and continuation of functions in the anisotropic spaces ?s.- II.2.8 Weighted anisotropic spaces; basic properties.- II.2.9 Embeddings, traces and continuation of functions in weighted anisotropic spaces.- II.2.10 Equivalent norms in ?s,r(?n+1?), ?s,?(En+1,?), ?s (?n,?).- II.2.11 The spaces Hs(G) and Hs(?).- II.2.12 The spaces ?s(S+,?) and ?s(?+,?).- II.3 Banach spaces of Hölder functions.- II.3.1 The spaces Cs(G) and Cs(F).- II.3.2 The spaces Cs (?) and Cs (S).- III Linear Operators.- III.1 Operators of potential type.- III.2 Operators of multiplication by a function.- III.2.1 Roundedness of truncation operators.- III.2.2 Boundedness of the operators of multiplication by smooth functions.- III.3 Commutators. Green formulas.- III.3.1 The operators Jn, J0.- III.3.2 Formulas for calculating P(x, D)v+ (?) and b(?’, t, D’, Dt)?+(?’, t).- III.3.3 Formulas for calculating l(x, t, D, Dt)u++(x, t).- III.3.4 Corollaries. Commutation formulas for differential and truncation operators. Green formulas.- III.4 On equivalent norms in ?s(?+n+1,?), ?s(E+n+1,?), and Hs(?+n), s ? 0.- III.4.1 Equivalents norms defined by truncations.- III.4.2 Restriction of distributions to an open half-space.- III.5 The spaces $${{\tilde}^}$$ and $${{\tilde{\mathcal}}^}$$.- III.5.1 The spaces $$\tilde_{{(K)}}^(G)$$ and $$\tilde{\mathcal}_{{(Q)}}^{{s,r}}(E_{ + }^{{n + 1}},\gamma )$$.- III.5.2 The spaces $$\tilde{\mathcal}_{{(\kappa ,\tau ,P)}}^(\Omega )$$ and $$\tilde{\mathcal}_{{(\kappa ,\tau ,P)}}^({{\Omega }_{ + }},\gamma )$$.- III.6 Differential operators in the space $${{\tilde{\mathcal}}^}$$.- III.6.1 Definition of the operator lu in $${{\bar{\Omega }}_{ + }}$$; its boundedness.- III.6.2 Definition of the operator $$lu{{|}_{{{{{\bar}}_{ + }}}}}$$; its boundedness.- III.6.3 Definition of the operators $$lu{{|}_{{\bar}}},lu{{|}_{\Gamma }}$$; their boundedness.- IV Parabolic Boundary Value Problems in Half-Space.- IV.1 Non-homogeneous systems in the space ?++s(?n+1,?).- IV.1.1 Non-homogeneous systems in $$\bar_{ + }^{{n + 1}}$$.- IV.1.2 Non-homogeneous systems in $$\bar{\mathbb}_{ + }^{{n + 1}}$$.- IV.2 Initial value and Cauchy problems for parabolic systems in spaces ?s.- IV.2.1 Formulation of the initial value problem in the spaces of distributions ?s.- IV.2.2 The Cauchy problem in $${{\tilde{\mathcal}}^}$$ for a system parabolic in in the sense of Petrovski?.- IV.2.3 Theorem on the solvability of the general initial value problem.- IV.3 Model parabolic boundary value problems in $$\bar{\mathbb}_{{ + + }}^{{n + 1}}$$.- IV.3.1 Formulation of the model boundary value problem in the spaces $${{\tilde{\mathcal}}_}$$ for a system parabolic in the sense of Petrovski?.- IV.3.2 Reduction of the boundary value problem to a system of linear algebraic equations.- IV.3.3 Analysis of the algebraic system; construction of a solution analytic in p.- IV.3.4 Theorem on well-posedness of the model parabolic boundary value problem in the spaces $${{\tilde{\mathcal}}^},s < - + \tfrac$$.- IV.3.5 Analysis of the model boundary value problem in $${{\tilde{\mathcal}}^}$$ with data compatible with zero at t = 0.- IV.3.6 Equivalence of Condition IV.1 and the Lopatinski? condition.- IV.4 The model boundary value problemin in $$\bar{\mathbb}_{{ + + }}^{{n + 1}}$$ for general parabolic systems.- IV.4.1 Formulation of the boundary value problem.- IV.4.2 The model boundary value problem with data compatible with zero.- IV.5 The model parabolic conjugation problem in classes of smooth functions.- IV.5.1 Formulation of the problem; the compatible covering condition.- IV.5.2 Reduction of the model conjugation problem to an equivalent boundary value problem for a block-diagonal system.- IV.6 Boundary value problem in $$\tilde{\mathcal}_{ + }^(\bar{\mathbb}_{{ + + }}^{{n + 1}},\gamma )$$ for operators in which the coefficients of the highest-order derivatives are slowly varying functions.- IV.7 Conjugation problem for operators in which the coefficients of the highest-order derivatives are slowly varying.- V Parabolic Boundary Value Problems in Cylindrical Domains.- V.1 Boundary value problems in a semi-infinite cylinder.- V.1.1 Formulation of the boundary value problem in $${{\tilde{\mathcal}}^}({{\Omega }_{ + }},\gamma )$$.- V.1.2 Boundary value problem in $${{\bar{\Omega }}_{ + }} = G \times [0,\infty )$$ with the data compatible with zero at t = 0. Regularizer.- V.1.3 Boundary value problem in ?+ in the general case.- V.2 Nonlocal boundary value problems. Conjugation problems.- V.2.1 Problem setting in classes of smooth functions. Conditions on operators.- V.2.2 The nonlocal boundary value problem in the spaces $${{\tilde{\mathcal}}^}$$.- V.2.3 The nonlocal boundary value problem in $${{\tilde{\mathcal}}^}$$ with data compatible with zero at t = 0.- V.2.4 The nonlocal boundary value problem in ?+ in the general case.- V.2.5 Parabolic conjugation problems.- V.3 Boundary value problems in cylindrical domains of finite height.- V.4 Solvability of the parabolic boundary value problems for right-hand sides with regular singularities.- V.4.1 Anisotropic regularizations of divergent integrals.- V.4.2 The main solvability theorem.- V.5 Green formula, boundary and initial values of weak generalized solutions.- V.5.1 Preliminary considerations. Notation.- V.5.2 The main theorem on boundary and initial values.- V.5.3 Limit values of weak generalized solutions on the boundary of the domain.- VI The Cauchy Problem and Parabolic Boundary Value Problems in Spaces of Smooth Functions.- VI.1 Fundamental solutions of the Cauchy problem.- VI.1.1 Introduction.- VI.1.2 Systems with bounded coefficients.- VI.1.3 Systems with growing coefficients.- VI.1.4 Second-order parabolic equations.- VI.1.5 Estimates for fundamental solutions of parabolic systems in ?+n +1 and elliptic systems generated by parabolic systems.- VI.2 The Cauchy problem.- VI.2.1 Introduction.- VI.2.2 Well-posedness.- VI.2.3 Existence of a solution for systems with growing coefficients.- VI.2.4 Uniqueness.- VI.2.5 Initial values for solutions of parabolic systems. Integral representation of solutions.- VI.3 Schauder theory of parabolic boundary value problems.- VI.3.1 Introduction.- VI.3.2 The well-posedness theorem.- VI.3.3 On the proof of the well-posedness theorem.- VI.3.4 Solution of the model parabolic boundary value problem.- VI.3.5 Necessity of the parabolicity condition.- VI.3.6 General boundary value problems. Well-posedness theorem.- VI.4 Green functions.- VI.4.1 Introduction.- VI.4.2 Green functions. Homogeneous Green functions.- VI.4.3 The Green function for conjugation problems.- VII Behaviour of Solutions of Parabolic Boundary Value Problems for Large Values of Time.- VII.1 Asymptotic representations and stabilization of solutions of model problems.- VII.1.1 Formulation of the problem.- VII.1.2 Poisson kernels of an elliptic boundary value problem with a parameter.- VII.1.3 Asymptotic representation of Poisson kernels of an elliptic boundary value problem with a parameter.- VII.1.4 Definition of the class of boundary functions used here.- VII.1.5 Asymptotic representation of solutions.- VII.1.6 Necessary and sufficient conditions of stabilization.- VII.1.7 The case of a single equation and of boundary data whose mean values have a limit.- VII.1.8 The case of a single space variable.- VII.1.9 Examples.- VII.2 Tikhonov’s problem.- VII.2.1 Statement of the problem. Notation. Conditions.- VII.2.2 Lemmas.- VII.2.3 Study of the Poisson kernel.- VII.2.4 Stabilization theorem.- VII.2.5 Examples.- VII.2.6 Necessity.- VII.2.7 Discussion.- VII.2.8 Heat and mass exchange equations.- VII.2.9 A model equation of higher order.- Comments.- References.