Operator Theory

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 Holder 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 $${{tilde{H}}^{s}}$$ and $${{tilde{mathcal{H}}}^{s}}$$.- III.5.1 The spaces $$tilde{H}_{{(K)}}^{s}(G)$$ and $$tilde{mathcal{H}}_{{(Q)}}^{{s,r}}(E_{ + }^{{n + 1}},gamma )$$.- III.5.2 The spaces $$tilde{mathcal{H}}_{{(kappa ,tau ,P)}}^{s}(Omega )$$ and $$tilde{mathcal{H}}_{{(kappa ,tau ,P)}}^{s}({{Omega }_{ + }},gamma )$$.- III.6 Differential operators in the space $${{tilde{mathcal{H}}}^{s}}$$.- III.6.1 Definition of the operator lu in $${{bar{Omega }}_{ + }}$$; its boundedness.- III.6.2 Definition of the operator $$lu{{|}_{{{{{bar{S}}}_{ + }}}}}$$; its boundedness.- III.6.3 Definition of the operators $$lu{{|}_{{bar{G}}}},lu{{|}_{Gamma }}$$; 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 $$bar{E}_{ + }^{{n + 1}}$$.- IV.1.2 Non-homogeneous systems in $$bar{mathbb{R}}_{ + }^{{n + 1}}$$.- 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 $${{tilde{mathcal{H}}}^{s}}$$ 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 $$bar{mathbb{R}}_{{ + + }}^{{n + 1}}$$.- IV.3.1 Formulation of the model boundary value problem in the spaces $${{tilde{mathcal{H}}}_{s}}$$ 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 $${{tilde{mathcal{H}}}^{s}},s < - {{t}_{m}} + tfrac{1}{2}$$.- IV.3.5 Analysis of the model boundary value problem in $${{tilde{mathcal{H}}}^{s}}$$ 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 $$bar{mathbb{R}}_{{ + + }}^{{n + 1}}$$ 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 $$tilde{mathcal{H}}_{ + }^{s}(bar{mathbb{R}}_{{ + + }}^{{n + 1}},gamma )$$ 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 $${{tilde{mathcal{H}}}^{s}}({{Omega }_{ + }},gamma )$$.- 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 $${{tilde{mathcal{H}}}^{s}}$$.- V.2.3 The nonlocal boundary value problem in $${{tilde{mathcal{H}}}^{s}}$$ 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.