Partial differential equations: time-periodic solutions by Otto VejvodaPartial differential equations: time-periodic solutions by Otto Vejvoda

Partial differential equations: time-periodic solutions

byOtto VejvodaOtherL. Herrmann, V. Lovicar

Paperback | October 12, 2011

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As far as the number of new results and quoted papers is concerned the present book may be considered a monograph. However, it also has some features of a textbook. Firstly, it proceeds from concrete problems to abstract ones, and secondly, all considerations and procedures are presented in much detail when met for the first time (such very elementary expositions can be found especially at the beginning of Chapters III and V). Finally, the authors focus their attention on elementary problems which can be dealt with by relatively simple methods. The authors hope that all this will make it possible also for an applied or technical research worker with some mathematical training to read this book. Naturally, the reader is supposed to be familiar with some basic notions from mathematical analysis, functional analysis and theory of partial differential equations. Also, the arguments and procedures which are repeated in the book are presented more briefly when met again, the reader being expected to become gradually more thoroughly acquainted with them. The authors have tried to provide a complete bibliography of all relevant publications (their number reaches about 500) from the theory of time-periodic solutions to non-linear partial and abstract differential equations whose origin may be put in the early thirties of this century.
Title:Partial differential equations: time-periodic solutionsFormat:PaperbackPublished:October 12, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9400976747

ISBN - 13:9789400976740

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Table of Contents

I Preliminaries from Functional Analysis.- § 1. Linear Spaces and Operators.- 1.1. Basic Notation.- 1.2. Banach Spaces.- 1.3. Hilbert Spaces.- 1.4. Fourier Series.- 1.5. Self-Adjoint Operators.- 1.6. Spectral Properties of Operators; Compact Operators.- 1.7. Embeddings; Negative Norms.- 1.8. Differentials.- §2. Function Spaces.- 2.1. Spaces of Smooth Functions.- 2.2. Spaces of Integrable Functions.- 2.3. Sobolev Spaces. (Spaces of Differentiable Functions).- 2.4. Periodic Functions.- 2.5. Representation of Vector-Valued Functions.- 2.6. Periodic Functions of Two Arguments.- 2.7. Embedding Theorems.- 2.8. Traces of Functions.- 2.9. The Substitution Operators.- § 3. Existence Theorems for Operator Equations.- 3.1. Theorems of a "Metric" Character.- 3.2. Theorems of a Topological Character.- 3.3. Equations with Monotone Operators.- 3.4. Equations Depending on a Parameter; Local Theorems.- 3.5. Equations Depending on a Parameter; Global Theorems.- II Preliminaries from the Theory of Differential Equations.- § 1. Boundary-Value and Eigenvalue Problems for Elliptic and Ordinary Differential Operators.- 1.1. Elliptic operators.- 1.2. Boundary-value problems for ordinary differential equations.- 1.3. Eigenvalue problems for elliptic operators.- § 2. The Wave and the Telegraph Equations.- 2.1. The Cauchy Problem for the Wave Equation in R.- 2.2. The Initial Boundary-Value Problem with the Dirichlet Boundary Conditions.- 2.3. The Telegraph Equation.- 2.4. The Fourier Method.- § 3. The Heat Equation.- 3.1. The Cauchy Problem.- 3.2. The Initial Boundary-Value Problems.- 3.3. The Fourier Method.- III The Heat Equation.- § 1. The (t, s)-Fourier Method.- 1.1. The Linear Ease.- 1.2. The weakly Non-Linear Case.- 1.3. The n-dimensional Case.- § 2. The t-Fourier and s-Fourier Methods.- 2.1. The t-Fourier Method: 0 ? x ? ?.- 2.2. The t-Fourier Method: ? ? <_20_3f_.-20_2.3.20_the20_s-fourier20_method.-20_c2a7_20_3.20_the20_poincarc3a9_20_method.-20_3.1.20_the20_linear20_case.-20_3.2.20_the20_weakly20_non-linear20_case.-20_c2a7_20_4.20_supplements20_and20_comments20_on20_the20_linear20_and20_weakly20_non-linear20_heat20_equation.-20_4.1.20_the20_adjoint20_problem20_method.-20_4.2.20_comments20_on20_other20_results.-20_c2a7_20_5.20_comments20_on20_strongly20_non-linear20_parabolic20_equations.-20_5.1.20_results20_of20_prodi2c_20_vaghi20_and20_bange.20_28_differential20_inequality20_techniques29_.-20_5.2.20_results20_of20_kolesov2c_20_klimov2c_20_amann2c_20_tsai2c_20_deuel20_and20_hess.20_28_use20_of20_upper20_and20_lower20_solutions29_.-20_5.3.20_results20_of20_smulev2c_20_fife2c_20_kusano2c_20_kruzkov2c_20_gaines20_and20_walter.20_28_a20_priori20_estimates20_techniques29_.-20_5.4.20_results20_of20_smulev2c_20_mal27_27_cev2c_20_walter20_and20_knolle.20_28_methods20_of20_discretizations29_.-20_5.5.20_results20_of20_palmieri2c_20_nakao2c_20_nanbu2c_20_biroli20_and20_zecca.20_28_problems20_with20_special20_type20_of20_non-linearity29_.-20_5.6.20_results20_of20_brc3a9_zis20_and20_nirenberg2c_20_s3f_astnovc3a1_20_and20_fu3f_c3ad_k.20_28_critical20_cases29_.-20_5.7.20_results20_of20_klimov2c_20_krasnosel27_27_skic3ad_20_and20_sobolevskic3ad_.20_28_eigenvalue20_problem29_.-20_5.8.20_results20_of20_other20_authors.-20_5.9.20_comments20_on20_papers20_of20_an20_applied20_character.-20_c2a7_20_6.20_comments20_on20_the20_navier-stokes20_equations20_and20_related20_problems.-20_6.1.20_the20_non-autonomous20_navier-stokes20_equations.-20_6.2.20_the20_autonomous20_navier-stokes20_equations.-20_6.3.20_related20_problems.-20_iv20_the20_telegraph20_equation.-20_c2a7_20_1.20_the20_fourier20_methods.-20_1.1.20_the20_28_t2c_20_s29_-fourier20_method3b_20_the20_linear20_case.-20_1.2.20_the20_weakly20_non-linear20_case.-20_1.3.20_the20_t-fourier20_method.-20_1.4.20_the20_s-fourier20_method.-20_c2a7_20_2.20_the20_poincarc3a9_20_method.-20_2.1.20_the20_linear20_case3a_20_3f_20_3f_20_gt3b_20_x20_gt3b_20_3f_.-20_2.2.20_the20_linear20_case3a_20_020_3f_20_x20_3f_20_3f_.-20_2.3.20_the20_weakly20_non-linear20_case.-20_c2a7_20_3.20_singularly20_perturbed20_problems.-20_3.1.20_general20_considerations.-20_3.2.20_the20_main20_theorems.-20_c2a7_20_4.20_supplements20_and20_comments20_on20_linear20_and20_weakly20_non-linear20_problems.-20_4.1.20_the20_adjoint20_problem20_method.-20_4.2.20_the20_ficken-fleishman20_method.-20_4.3.20_results20_of20_rabinowitz.-20_4.4.20_the20_telegraph20_equation20_with20_a20_3d_20_0.-20_c2a7_20_5.20_comments20_on20_strongly20_non-linear20_problems.-20_5.1.20_results20_of20_prodi2c_20_prouse2c_20_krylovc3a1_2c_20_buzzetti2c_20_lions2c_20_tousck2c_20_biroli20_and20_nakao.20_28_the20_wave20_equation20_with20_strongly20_non-linear20_damping29_.-20_5.2.20_results20_of20_mawhin2c_20_fucc3ad_k2c_20_brc3a9_zis2c_20_nirenberg2c_20_biroli2c_20_horc3a1_sek20_and20_zecca.20_28_problems20_with20_non-linear20_forcing20_term29_.-20_5.3.20_results20_of20_v.20_wahl2c_20_clements2c_20_kakita20_and20_sowunmi.20_28_problems20_with20_non-linear20_elliptic20_part20_and20_other20_problems29_.-20_v20_the20_wave20_equation.-20_c2a7_20_1.20_the20_dirichlet20_boundary20_conditions3b_20_the20_poincarc3a9_20_method.-20_1.1.20_the20_linear20_case3b_20_general20_considerations.-20_1.2.20_the20_case20_3f_20_3d_20_23f_n.-20_1.3.20_the20_case20_3f_20_3d_20_23f_p2f_q.-20_1.4.20_the20_non-linear20_case.-20_c2a7_20_2.20_the20_dirichlet20_boundary20_conditions3b_20_the20_gc3bc_nzlcr20_method.-20_2.1.20_the20_linear20_case3b_20_general20_considerations.-20_2.2.20_the20_case20_3f_20_3d_20_220_3f_.-20_2.3.20_the20_case20_3f_20_3d_20_220_3f_20_p2f_q.-20_2.4.20_the20_non-linear20_case.-20_2.5.20_the20_case20_of20_monotone20_perturbations.-20_c2a7_3.20_examples.-20_3.1.20_the20_problem20_28_p3f_23f_29_20_with20_f28_u2c_20_3f_29_20_28_t2c_20_x29_20_3d_20_h28_t2c_20_x29_20_2b_20_3f_20_u28_t2c_20_x29_20_2b_20_2b_20_3f_20_u328_t2c_20_x29_20_28_the20_poincarc3a9_20_method29_.-20_3.2.20_the20_problem20_28_p3f_23f_29_20_with20_f28_u2c_20_3f_29_20_28_t2c_20_x29_20_3d_20_h28_t2c_20_x29_20_2b_20_3f_20_u28_t2c_20_x29_20_2b_20_2b_20_3f_20_u328_t2c_20_x29_20_28_the20_gc3bc_nzler20_method29_.-20_c2a7_20_4.20_the20_newton20_and20_combined20_boundary20_conditions3b_20_the20_gc3bc_nzler20_method.-20_4.1.20_general20_considerations.-20_4.2.20_the20_case20_3f_20_3d_20_12c_20_3f_20_3d_20_23f_.-20_4.3.20_the20_case20_3f_20_3d_20_02c_20_3f_128_t29_20_3d_20_12c_20_3f_20_3d_20_220_3f_.-20_c2a7_20_5.20_entrainment20_of20_frequency.-20_5.1.20_introduction.-20_5.2.20_the20_existence20_of20_28_23f_20_2b_20_3f_3f_29_-periodic20_solutions.-20_5.3.20_equations20_with20_monotone20_right-hand20_sides.-20_c2a7_20_6.20_the20_fourier20_method.-20_6.1.20_introductory20_remarks.-20_6.2.20_the20_case20_3f_20_3d_20_23f_p2f_q.-20_6.3.20_the20_case20_3f_20_3d_20_23f_3f_.-20_6.4.20_the20_weakly20_non-linear20_case.-20_6.5.20_the20_28_t2c_20_s29_-fourier20_method3b_20_the20_newton20_and20_combined20_boundary20_conditions.-20_6.6.20_problems20_with20_variable20_coefficients20_and20_problems20_in20_several20_variables.-20_c2a7_20_7.20_the20_wave20_equation20_in20_an20_unbounded20_domain.-20_7.1.20_a20_general20_existence20_theorem.-20_7.2.20_applications.-20_c2a7_20_8.20_supplements20_and20_comments20_on20_non-autonomous20_hyperbolic20_equations.-20_8.1.20_the20_adjoint20_problem20_method.-20_8.2.20_averaging20_methods.-20_8.3.20_comments20_on20_other20_papers.-20_8.4.20_comments20_on20_papers20_of20_an20_applied20_character.-20_c2a7_20_9.20_comments20_on20_autonomous20_hyperbolic20_equations.-20_9.1.20_comments20_on20_periodic20_solutions20_of20_boundary-value20_problems.-20_9.2.20_comments20_on20_periodic20_solutions20_in20_unbounded20_domains.-20_9.3.20_comments20_on20_papers20_of20_an20_applied20_character.-20_vi20_the20_beam20_equation20_and20_related20_problems.-20_c2a7_20_1.20_the20_equations20_of20_a20_beam20_and20_of20_a20_thin20_plate.-20_1.1.20_equations20_without20_damping.-20_1.2.20_equations20_with20_a20_damping20_term.-20_c2a7_20_2.20_supplements20_and20_comments.-20_2.1.20_the20_adjoint20_problem20_method.-20_2.2.20_results20_of20_krylovc3a1_2c_20_vejvoda2c_20_kopc3a1_3f_kovc3a1_2c_20_solov27_27_ev2c_20_karimov2c_20_mitrjakov20_and20_filip.20_28_results20_based20_on20_fourier20_methods29_.-20_2.3.20_results20_of20_kurzweil2c_20_hall2c_20_petzeltovc3a1_20_and20_nakao.20_28_results20_based20_on20_monotonicity20_of20_perturbations29_.-20_2.4.20_results20_of20_st3f_drc3bd_2c_20_krylovc3a1_20_and20_krej3f_c3ad_.20_28_problems20_with20_damping29_.-20_c2a7_20_3.20_the20_dynamic20_von20_kc3a1_rmc3a1_n20_equations20_of20_thin20_plates20_involving20_rotational20_inertia20_and20_damping.-20_3.1.20_preliminaries.-20_3.2.20_the20_existence20_theorem.-20_3.3.20_comments20_on20_papers20_of20_an20_applied20_character.-20_vii20_the20_abstract20_equations.-20_c2a7_20_1.20_the20_t-fourier20_method3a_20_preliminaries.-20_1.1.20_the20_setting20_of20_the20_problem.-20_1.2.20_auxiliary20_notions20_and20_results.-20_c2a7_20_2.20_the20_t-fourier20_method3a_20_spectral20_properties20_of20_periodic20_operators.-20_2.1.20_main20_results.-20_2.2.20_examples.-20_c2a7_20_3.20_the20_t-fourier20_method3a_20_weakly20_non-linear20_problems.-20_3.1.20_general20_scheme.-20_3.2.20_a20_special20_problem.-20_c2a7_20_4.20_comments20_on20_papers20_using20_direct20_methods.-20_4.1.20_results20_of20_lions20_and20_magenes2c_20_and20_ton.-20_4.2.20_results20_of20_taam20_and20_cend.-20_4.3.20_results20_of20_da20_prato20_and20_barbu.-20_4.4.20_results20_of20_gajewski2c_20_grc3b6_ger20_and20_zacharias.-20_4.5.20_results20_of20_sova.-20_4.6.20_results20_of20_dubinski3f_.-20_4.7.20_results20_of20_herrmann.-20_4.8.20_results20_of20_straskraba.-20_4.9.20_results20_of20_moke3f_3f_ev20_and20_kopc3a1_3f_kovc3a1_.-20_4.10.20_results20_of20_crandall20_and20_rabinowitz.-20_4.11.20_results20_of20_dezin.-20_4.12.20_results20_of20_comincioli20_and20_gaultier.-20_4.13.20_results20_of20_borisovic3a9_.-20_c2a7_20_5.20_comments20_on20_papers20_using20_indirect20_methods.-20_5.1.20_results20_of20_browder.-20_5.2.20_results20_of20_brc3a9_zis2c_20_benilan2c_20_biroli2c_20_crandall2c_20_pazy2c_20_pavel20_and20_prouse.-20_5.3.20_results20_of20_vejvoda2c_20_straskraba2c_20_krylovc3a1_2c_20_sobolevski3f_20_and20_pogore-lenko.-20_5.4.20_results20_of20_simon3f_nko.-20_5.5.20_results20_of20_fink2c_20_hall20_and20_hausrath.-20_5.6.20_results20_of20_amann.-20_5.7.20_other20_results.-20_bibliography20_to20_chapter20_i.-20_bibliography20_to20_chapter20_ii.-20_bibliography20_to20_chapter20_iii.-20_bibliography20_to20_chapter20_iv.-20_bibliography20_to20_chapter20_v.-20_bibliography20_to20_chapter20_vi.-20_bibliography20_to20_chapter20_vii.-20_bibliography20_of20_papers20_on20_related20_topics.-20_addenda20_to20_bibliography.-20_list20_of20_symbols. -="" 2.3.="" the="" s-fourier="" method.-="" _c2a7_="" 3.="" _poincarc3a9_="" 3.1.="" linear="" case.-="" 3.2.="" weakly="" non-linear="" 4.="" supplements="" and="" comments="" on="" heat="" equation.-="" 4.1.="" adjoint="" problem="" 4.2.="" other="" results.-="" 5.="" strongly="" parabolic="" equations.-="" 5.1.="" results="" of="" _prodi2c_="" vaghi="" bange.="" _28_differential="" inequality="" _techniques29_.-="" 5.2.="" _kolesov2c_="" _klimov2c_="" _amann2c_="" _tsai2c_="" deuel="" hess.="" _28_use="" upper="" lower="" _solutions29_.-="" 5.3.="" _smulev2c_="" _fife2c_="" _kusano2c_="" _kruzkov2c_="" gaines="" walter.="" _28_a="" priori="" estimates="" 5.4.="" _mal27_27_cev2c_="" walter="" knolle.="" _28_methods="" _discretizations29_.-="" 5.5.="" _palmieri2c_="" _nakao2c_="" _nanbu2c_="" biroli="" zecca.="" _28_problems="" with="" special="" type="" _non-linearity29_.-="" 5.6.="" _brc3a9_zis="" _nirenberg2c_="" _s3f_astnovc3a1_="" _fu3f_c3ad_k.="" _28_critical="" _cases29_.-="" 5.7.="" _krasnosel27_27_skic3ad_="" _sobolevskic3ad_.="" _28_eigenvalue="" _problem29_.-="" 5.8.="" authors.-="" 5.9.="" papers="" an="" applied="" character.-="" 6.="" navier-stokes="" equations="" related="" problems.-="" 6.1.="" non-autonomous="" 6.2.="" autonomous="" 6.3.="" iv="" telegraph="" 1.="" fourier="" methods.-="" 1.1.="" _28_t2c_="" _s29_-fourier="" _method3b_="" 1.2.="" 1.3.="" t-fourier="" 1.4.="" 2.="" 2.1.="" _case3a_="" _gt3b_="" x="" 2.2.="" 0="" singularly="" perturbed="" general="" considerations.-="" main="" theorems.-="" ficken-fleishman="" 4.3.="" rabinowitz.-="" 4.4.="" equation="" a="" _prouse2c_="" _krylovc3a1_2c_="" _buzzetti2c_="" _lions2c_="" _tousck2c_="" nakao.="" _28_the="" wave="" _damping29_.-="" _mawhin2c_="" _fucc3ad_k2c_="" _brc3a9_zis2c_="" _biroli2c_="" _horc3a1_sek="" forcing="" _term29_.-="" v.="" _wahl2c_="" _clements2c_="" kakita="" sowunmi.="" elliptic="" part="" _problems29_.-="" v="" dirichlet="" boundary="" _conditions3b_="" _case3b_="" case="" _3d_="" _23f_n.-="" _23f_p2f_q.-="" _gc3bc_nzlcr="" 2="" _p2f_q.-="" 2.4.="" 2.5.="" monotone="" perturbations.-="" _c2a7_3.="" examples.-="" _28_p3f_23f_29_="" _f28_u2c_="" _x29_="" _2b_="" _u28_t2c_="" _u328_t2c_="" _method29_.-="" _gc3bc_nzler="" newton="" combined="" _12c_="" _23f_.-="" _02c_="" _28_t29_="1%2c" entrainment="" frequency.-="" introduction.-="" existence="" _28_23f_="" _29_-periodic="" solutions.-="" right-hand="" sides.-="" introductory="" remarks.-="" _23f_3f_.-="" 6.4.="" 6.5.="" conditions.-="" 6.6.="" problems="" variable="" coefficients="" in="" several="" variables.-="" 7.="" unbounded="" domain.-="" 7.1.="" theorem.-="" 7.2.="" applications.-="" 8.="" hyperbolic="" 8.1.="" 8.2.="" averaging="" 8.3.="" papers.-="" 8.4.="" 9.="" 9.1.="" periodic="" solutions="" boundary-value="" 9.2.="" domains.-="" 9.3.="" vi="" beam="" thin="" plate.-="" without="" damping.-="" damping="" term.-="" comments.-="" _vejvoda2c_="" _kopc3a1_3f_kovc3a1_2c_="" _solov27_27_ev2c_="" _karimov2c_="" mitrjakov="" filip.="" _28_results="" based="" _methods29_.-="" _kurzweil2c_="" _hall2c_="" _petzeltovc3a1_="" monotonicity="" _perturbations29_.-="" _st3f_drc3bd_2c_="" _krylovc3a1_="" _krej3f_c3ad_.="" dynamic="" von="" _kc3a1_rmc3a1_n="" plates="" involving="" rotational="" inertia="" preliminaries.-="" 3.3.="" vii="" abstract="" _method3a_="" setting="" problem.-="" auxiliary="" notions="" spectral="" properties="" operators.-="" scheme.-="" using="" direct="" lions="" _magenes2c_="" ton.-="" taam="" cend.-="" da="" prato="" barbu.-="" _gajewski2c_="" _grc3b6_ger="" zacharias.-="" 4.5.="" sova.-="" 4.6.="" _dubinski3f_.-="" 4.7.="" herrmann.-="" 4.8.="" straskraba.-="" 4.9.="" _moke3f_3f_ev="" _kopc3a1_3f_kovc3a1_.-="" 4.10.="" crandall="" 4.11.="" dezin.-="" 4.12.="" comincioli="" gaultier.-="" 4.13.="" _borisovic3a9_.-="" indirect="" browder.-="" _benilan2c_="" _crandall2c_="" _pazy2c_="" pavel="" prouse.-="" _straskraba2c_="" _sobolevski3f_="" pogore-lenko.-="" _simon3f_nko.-="" _fink2c_="" hall="" hausrath.-="" amann.-="" bibliography="" to="" chapter="" i.-="" ii.-="" iii.-="" iv.-="" v.-="" vi.-="" vii.-="" topics.-="" addenda="" bibliography.-="" list="">