Amann / Arendt / Neubrander Functional Analysis and Evolution Equations

The Günter Lumer Volume
1. Auflage 2008
ISBN: 978-3-7643-7794-6
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 637 Seiten, eBook

Gunter Lumer was an outstanding mathematician whose works have great influence on the research community in mathematical analysis and evolution equations. He was at the origin of the breath-taking development the theory of semigroups saw after the pioneering book of Hille and Phillips from 1957. This volume contains invited contributions presenting the state of the art of these topics and reflecting the broad interests of Gunter Lumer.

Zielgruppe

Research

Autoren/Hrsg.

Amann, Herbert

Arendt, Wolfgang

Hieber, Matthias

Neubrander, Frank

Nicaise, Serge

Below, Joachim

Inhaltsverzeichnis

1;Contents;5
2;Life and Work of Günter Lumer;9
3;In Remembrance of Günter Lumer;18
4;Expansions in Generalized Eigenfunctions of the Weighted Laplacian on Star-shaped Networks;20
4.1;1. Introduction;20
4.2;2. Data and functional analytic framework;24
4.3;3. Expansion in generalized eigenfunctions;24
4.4;4. Application of Stone’s formula and limiting absorption principle;27
4.5;5. A Plancherel-type formula and a functional calculus for the operator;30
4.6;References;34
5;Diffusion Equations with Finite Speed of Propagation;36
5.1;1. Introduction;36
5.2;2. The Cauchy problem for a strongly degenerate quasi-linear equation;39
5.3;3. The evolution of the support of the solutions of the relativistic heat equation;49
5.4;References;51
6;Subordinated Multiparameter Groups of Linear Operators: Properties via the Transference Principle;54
6.1;1. Introduction;54
6.2;2. The transference principle and generator formulas;56
6.3;3. Examples;66
6.4;References;68
7;An Integral Equation in AeroElasticity;70
7.1;1. Introduction;70
7.2;2. The Possio Equation;71
7.3;3. Possio Integral: Time domain version;73
7.4;4. Special case;76
7.5;5. The general case;79
7.6;6. Generalization;83
7.7;References;84
8;Eigenvalue Asymptotics Under a Nondissipative Eigenvalue Dependent Boundary Condition for Second- order Elliptic Operators;85
8.1;1. Introduction;85
8.2;2. Eigenvalue asymptotics for constant dynamical coeffcient;88
8.3;3. Dynamical coeffcient of constant sign;92
8.4;4. Dynamical coeffcient changing sign;93
8.5;References;98
9;Feynman-Kac Formulas, Backward Stochastic Differential Equations and Markov Processes;100
9.1;1. Introduction;100
9.2;2. Preliminary results and auxiliary notation;102
9.3;3. A probabilistic approach: weak solutions;113
9.4;4. Existence and uniqueness of solutions to BSDEs;116
9.5;5. Backward stochastic di.erential equations and Markov processes;123
9.6;References;127
10;Generation of Cosine Families on Lp(0, 1) by Elliptic Operators with Robin Boundary Conditions;129
10.1;1. Introduction;129
10.2;2. Preliminaries and results;131
10.3;3. The case of the Laplacian;133
10.4;4. Proof of the main results;139
10.5;5. Second-order elliptic operators on unbounded intervals;142
10.6;References;145
11;Global Smooth Solutions to a Fourth-order Quasilinear Fractional Evolution Equation;147
11.1;1. Introduction;147
11.2;2. Preliminaries;149
11.3;3. Local well-posedness;149
11.4;4. A priori estimates and global well-posedness;153
11.5;5. Proof of Lemma;160
11.6;References;161
12;Positivity Property of Solutions of Some Quasilinear Elliptic Inequalities;163
12.1;1. Introduction;163
12.2;2. Main result;164
12.3;3. Some extensions of the main result;169
12.4;References;171
13;On a Stochastic Parabolic Integral Equation;172
13.1;1. Introduction;172
13.2;2. The stochastic machinery;173
13.3;3. Existence of solutions;176
13.4;4. Additional time-regularity;179
13.5;References;183
14;Resolvent Estimates for a Perturbed Oseen Problem;185
14.1;1. Introduction;185
14.2;2. Notations, de.nitions and main result;186
14.3;3. Convolutions of E; estimates of PB . F);189
14.4;4. Solving a perturbed Oseen system;191
14.5;5. Some resolvent estimates for a perturbed Oseen system;193
14.6;6. Estimate of the semigroup;195
14.7;References;200
15;Abstract Delay Equations Inspired by Population Dynamics;201
15.1;1. Introduction;201
15.2;2. The abstract setting;203
15.3;3. Delay equations as abstract integral equations;206
15.4;4. A model involving cannibalistic behaviour;207
15.5;5. Conclusions;211
15.6;References;212
16;Weak Stability for Orbits of C0-semigroups on Banach Spaces;215
16.1;1. Introduction;215
16.2;2. The result;217
16.3;References;221
17;Contraction Semigroups on L8(R);223
17.1;1. Introduction;223
17.2;2. Preliminaries;224
17.3;3. Extension properties;226
17.4;4. Examples;231
17.5;5. Volume doubling;234
17.6;References;235
18;On the Curve Shortening Flow with Triple Junction;236
18.1;1. Introduction;236
18.2;2. Local existence;239
18.3;References;250
19;The Dual Mixed Finite Element Method for the Heat Diffusion Equation in a Polygonal Domain, I;252
19.1;1. Introduction;252
19.2;2. Regularity of the solution of the heat diffusion equation;253
19.3;3. The dual mixed formulation for the heat diffusion equation;255
19.4;4. Semi-discrete solution of the dual mixed method for the heat diffusion equation in a polygonal domain of R2;257
19.5;5. A priori error estimates for the semi-discrete solution of the dual mixed method for the heat diffusion equation;260
19.6;References;268
20;Maximal Regularity of the Stokes Operator in General Unbounded Domains of Rn;270
20.1;1. Introduction;270
20.2;2. Preliminaries;274
20.3;3. Proof of Theorem 1.4;280
20.4;References;284
21;Linear Control Systems in Sequence Spaces;286
21.1;1. Introduction;286
21.2;2. The first example;289
21.3;3. The maximum principle and optimal controls;290
21.4;4. The second example;295
21.5;5. The time optimal problem;297
21.6;6. Hypersingular controls;298
21.7;7. Singular functionals;301
21.8;8. Conclusions and new questions;302
21.9;References;302
22;On the Motion of Several Rigid Bodies in a Viscous Multipolar Fluid;304
22.1;1. Introduction;304
22.2;2. Variational formulation;308
22.3;3. Global existence – main results;310
22.4;4. Approximate problems;311
22.5;5. Uniform estimates;312
22.6;6. Convergence;313
22.7;References;317
23;On the Stokes Resolvent Equations in Locally Uniform Lp Spaces in Exterior Domains;319
23.1;1. Introduction;319
23.2;2. Preliminaries;321
23.3;3. The Stokes operator in Lp spaces in exterior domains;323
23.4;References;325
24;Generation of Analytic Semigroups and Domain Characterization for Degenerate Elliptic Operators with Unbounded Coeffcients Arising in Financial Mathematics. Part II;327
24.1;1. Introduction;327
24.2;2. Preliminary material and notation;329
24.3;3. Generation of analytic semigroups on Lp(Rd);333
24.4;References;341
25;Numerical Approximation of Generalized Functions: Aliasing, the Gibbs Phenomenon and a Numerical Uncertainty Principle;343
25.1;1. Introduction;343
25.2;2. Basics of the theory of distributions;346
25.3;3. Approximating families;350
25.4;4. Convergence;355
25.5;5. Interpolation;358
25.6;6. Oscillations in the wave equations;363
25.7;7. Conclusion;367
25.8;References;368
26;No Radial Symmetries in the Arrhenius– Semenov Thermal Explosion Equation;369
26.1;1. Introduction;369
26.2;2. No symmetries in the Arrhenius–Semenov equation;372
26.3;3. Conclusions and discussion;379
26.4;References;381
27;Mild Well-posedness of Abstract Differential Equations;383
27.1;1. Introduction;383
27.2;2. Preliminaries;384
27.3;3. Mild-well-posedness and Lp-multipliers;386
27.4;4. Mild solutions for second-order equations;389
27.5;5. Fractional di.erentiation and well-posedness;393
27.6;6. Application to semi-linear equations in Hilbert spaces;396
27.7;References;398
28;Backward Uniqueness in Linear Thermoelasticity with Time and Space Variable Coeffcients;400
28.1;1. Introduction;400
28.2;2. Main result;401
28.3;3. The energy estimates;404
28.4;4. Carleman estimates for parabolic equations;408
28.5;5. Carleman estimates for thermoelastic system;411
28.6;6. Completion of the proof;413
28.7;References;414
29;Measure and Integral: New Foundations after One Hundred Years;415
29.1;1. The two abstract theories of the 20th century;416
29.2;2. The generation of measures in the two previous theories;420
29.3;3. The origin of the new systematization;424
29.4;4. The new theory;425
29.5;5. The further development in a few examples;427
29.6;References;431
30;Post-Widder Inversion for Laplace Transforms of Hyperfunctions;433
30.1;Introduction;433
30.2;1. Hyperfunctions with compact support;434
30.3;2. Hyperfunctions on [0,8);437
30.4;3. Post-Widder inversion for general hyperfunctions;440
30.5;References;441
31;On a Class of Elliptic Operators with Unbounded Time-and Space-dependent Coeffcients in Rn;442
31.1;1. Introduction;442
31.2;2. Main assumptions and preliminaries;447
31.3;3. The case of continuous coeffcients independent of the space variables;449
31.4;4. The general case (when the coeffcients are continuous in (t, x));455
31.5;5. The case when the coeffcients are only measurable;460
31.6;References;464
32;Time-dependent Nonlinear Perturbations of Analytic Semigroups;466
32.1;1. Introduction;467
32.2;2. A linear theory;470
32.3;3. Fractional powers of non-densely defined closed linear operators;474
32.4;4. Nonlinear perturbations of analytic semigroups;475
32.5;5. Uniqueness and regularity of mild solutions;481
32.6;6. Generation of nonlinear evolution operator U in Y;483
32.7;7. Discrete local multiple Laplace transforms;491
32.8;8. Characterization of nonlinearly perturbed analytic semigroups;497
32.9;9. Applications to convective reaction-diffusion systems;500
32.10;References;509
33;A Variational Approach to Strongly Damped Wave Equations;512
33.1;1. Introduction;512
33.2;2. First well-posedness results;514
33.3;3. Interpolation spaces and nonlinear problems;519
33.4;References;522
34;Exponential and Polynomial Stability Estimates for the Wave Equation and Maxwell’s System with Memory Boundary Conditions;524
34.1;1. Introduction;524
34.2;2. The wave equation;526
34.3;3. Maxwell’s equations;531
34.4;4. Examples;534
34.5;References;538
35;Maximal Regularity for Degenerate Evolution Equations with an Exponential Weight Function;540
35.1;1. Introduction;540
35.2;2. Parametric symbols;542
35.3;3. The evolution equation;549
35.4;4. Examples;551
35.5;References;554
36;An Analysis of Asian options;555
36.1;1. Introduction;555
36.2;2. The Black-Scholes approach;557
36.3;3. Well-posedness of the problem;560
36.4;4. The call-put parity;564
36.5;References;567
37;Linearized Stability and Regularity for Nonlinear Age-dependent Population Models;568
37.1;1. Introduction;568
37.2;2. Linearized stability and regularity for (ADP);570
37.3;3. Proofs of Theorems 2.2 and 2.3;572
37.4;4. Appendix;580
37.5;References;583
38;Space Almost Periodic Solutions of Reaction Diffusion Equations;584
38.1;0. Introduction;584
38.2;1. Notation;585
38.3;2. Spaces of almost periodic functions;585
38.4;3. Slow instable manifolds;588
38.5;4. Outlook;594
38.6;Appendix;598
38.7;References;600
39;On the Oseen Semigroup with Rotating Effect;602
39.1;1. Introduction and main results;602
39.2;2. Analysis in R3;605
39.3;3. Rough ideas of proofs of Theorems 1.1 and 1.3;607
39.4;4. On some new treatment of the pressure term;610
39.5;5. The idea of proofs of Theorems 3.1 and 3.2;614
39.6;6. Remark on the stability theorem;616
39.7;References;618
40;Exact Controllability in L2(O) of the Schrödinger Equation in a Riemannian Manifold with L2(S1)-Neumann Boundary Control;619
40.1;1. Introduction. Problem statement. Assumptions;619
40.2;2. The adjoint problem and the equivalent COI under the working assumption R,. = 0 on G1 (resp. on G);624
40.3;3. Proof of the COI (2.14) under (A.5);627
40.4;4. Proof of Theorem 3.1;631
40.5;5. Proof of Theorem 1.2: Removal of Assumption (A.5) = (2.4);635
40.6;6. Illustrations and examples;636
40.7;References;640
41;List of Authors;643

Expansions in Generalized Eigenfunctions of the Weighted Laplacian on Star-shaped Networks.- Diffusion Equations with Finite Speed of Propagation.- Subordinated Multiparameter Groups of Linear Operators: Properties via the Transference Principle.- An Integral Equation in AeroElasticity.- Eigenvalue Asymptotics Under a Non-dissipative Eigenvalue Dependent Boundary Condition for Second-order Elliptic Operators.- Feynman-Kac Formulas, Backward Stochastic Differential Equations and Markov Processes.- Generation of Cosine Families on L p (0,1) by Elliptic Operators with Robin Boundary Conditions.- Global Smooth Solutions to a Fourth-order Quasilinear Fractional Evolution Equation.- Positivity Property of Solutions of Some Quasilinear Elliptic Inequalities.- On a Stochastic Parabolic Integral Equation.- Resolvent Estimates for a Perturbed Oseen Problem.- Abstract Delay Equations Inspired by Population Dynamics.- Weak Stability for Orbits of C 0-semigroups on Banach Spaces.- Contraction Semigroups on L ?(R).- On the Curve Shortening Flow with Triple Junction.- The Dual Mixed Finite Element Method for the Heat Diffusion Equation in a Polygonal Domain, I.- Maximal Regularity of the Stokes Operator in General Unbounded Domains of ? n .- Linear Control Systems in Sequence Spaces.- On the Motion of Several Rigid Bodies in a Viscous Multipolar Fluid.- On the Stokes Resolvent Equations in Locally Uniform L p Spaces in Exterior Domains.- Generation of Analytic Semigroups and Domain Characterization for Degenerate Elliptic Operators with Unbounded Coefficients Arising in Financial Mathematics. Part II.- Numerical Approximation of Generalized Functions: Aliasing, the Gibbs Phenomenon and a Numerical Uncertainty Principle.- No Radial Symmetries in the Arrhenius-Semenov ThermalExplosion Equation.- Mild Well-posedness of Abstract Differential Equations.- Backward Uniqueness in Linear Thermoelasticity with Time and Space Variable Coefficients.- Measure and Integral: New Foundations after One Hundred Years.- Post-Widder Inversion for Laplace Transforms of Hyperfunctions.- On a Class of Elliptic Operators with Unbounded Time- and Space-dependent Coefficients in ? N .- Time-dependent Nonlinear Perturbations of Analytic Semigroups.- A Variational Approach to Strongly Damped Wave Equations.- Exponential and Polynomial Stability Estimates for the Wave Equation and Maxwell’s System with Memory Boundary Conditions.- Maximal Regularity for Degenerate Evolution Equations with an Exponential Weight Function.- An Analysis of Asian options.- Linearized Stability and Regularity for Nonlinear Age-dependent Population Models.- Space Almost Periodic Solutions of Reaction Diffusion Equations.- On the Oseen Semigroup with Rotating Effect.- Exact Controllability in L 2(?) of the Schrödinger Equation in a Riemannian Manifold with L 2(?1)-Neumann Boundary Control.

Leseproben

Feynman-Kac Formulas, Backward Stochastic Differential Equations and Markov Processes (p. 81-82)

Jan A. Van Casteren

This article is written in honor of G. Lumer whom I consider as my semi-group teacher

Abstract. In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic Differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial Differential equations of parabolic type. Some emphasis is put on the fact that the whole theory generalizes Feynman-Kac formulas. A new method of proof of the existence of solutions is given. All the existence arguments are based on rather precise quantitative estimates.

1. Introduction

Backward stochastic Differential equations, in short BSDEs, have been well studied during the last ten years or so. They were introduced by Pardoux and Peng [20], who proved existence and uniqueness of adapted solutions, under suitable squareintegrability assumptions on the coeffcients and on the terminal condition. They provide probabilistic formulas for solution of systems of semi-linear partial Differential equations, both of parabolic and elliptic type. The interest for this kind of stochastic equations has increased steadily, this is due to the strong connections of these equations with mathematical finance and the fact that they provide a generalization of the well-known Feynman-Kac formula to semi-linear partial differential equations. In the present paper we will concentrate on the relationship between time-dependent strong Markov processes and abstract backward stochastic Differential equations. The equations are phrased in terms of a martingale type problem, rather than a strong stochastic Differential equation. They could be called weak backward stochastic Differential equations. Emphasis is put on existence and uniqueness of solutions. The paper in [27] deals with the same subject, but it concentrates on comparison theorems and viscosity solutions.

The notion of squared gradient operator is implicitly used by Bally at al in [4]. The latter paper was one of the motivations to write the present paper with an emphasis on the squared gradient operator. In addition, our results are presented in such a way that the state space of the underlying Markov process, which in most of the other papers on BSDEs is supposed to be Rn, can be any diffusion with an abstract state space, which throughout our text is denoted by E. In fact in the existing literature the underlying Markov process is a (strong) solution of a (forward) stochastic Differential equation: see, e.g., [4], [8] and [7] and [19]. For more on this see Remark 2.9 below. In particular our results are applicable in case the Markov process under consideration is Brownian motion on a Riemannian manifold. Our condition on the generator (or coefficient) of the BSDE f in terms of the squared gradient is very natural. In the Lipschitz context it is more or less optimal. Moreover, our proof of existence is not based on standard regularization methods by using convolution products with smooth functions, but on a homotopy argument due to Crouzeix [11], which seems more direct than the classical approach. We also obtain rather precise quantitative estimates. Only very rudimentary sketches of proofs are given, details will appear elsewhere.