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Umfang: VIII, 190 p. 18 illus., 13 illus. in color. , online resource.

Ausgabe: 1st ed. 2024.

ISBN: 9783031567322

Serie: Research Perspectives Ghent Analysis and PDE Center, 4

Inhalt: The principal aim of the volume is gathering all the contributions given by the speakers (mini courses) and some of the participants (short talks) of the summer school "Modern Problems in PDEs and Applications" held at the Ghent Analysis and PDE Center from 23 August to 2 September 2023. The school was devoted to the study of new techniques and approaches for solving partial differential equations, which can either be considered or arise from the physical point of view or the mathematical perspective. Both sides are extremely important since theories and methods can be developed independently, aiming to gather each other in a common objective. The aim of the summer school was to progress and advance in the problems considered. Note that real-world problems and their applications are classical study trends in physical or mathematical modelling. The summer school was organised in a friendly atmosphere and synergy, and it was an excellent opportunity to promote and encourage the development of the subject in the community.

Anmerkung: Part I Mini-courses -- The Hardy constant: a review -- Some harmonic analysis in a general Gaussian setting -- Introduction to hypercomplex analysis -- Some norm bounds for the spectral projections of the Heisenberg sublaplacian -- Boundary value problems for elliptic operators satisfying Carleson condition -- An elementary computation of heat trace invariants -- Can we divide vectors? - Geometric calculus in Science and Engineering -- Maximal regularity as a tool for partial differential equations -- Recent existence results for some critical subelliptic problems -- Introduction to the Analysis on Manifolds with Conical Singularities -- Elliptic systems of phase transition type -- Cordes condition, Campanato nearness and beyond -- Part II Short talks -- Global hypoellipticity on homogeneous vector bundles: Necessary and sufficient conditions -- On inverse source problems for space-dependent sources in thermoelasticity -- Symmetric properties of eigenvalues and eigenfunctions of uniform beams with axial loads -- Cylindrical and horizontal extensions of critical Sobolev type inequalities and identities -- Very weak solution of the wave equation for Sturm-Liouville operator -- Some new multidimensional Cochran-Lee and Hardy type inequalities.

In: Springer Nature eBook

Weitere Ausg.: Printed edition: ISBN 9783031567315

Weitere Ausg.: Printed edition: ISBN 9783031567339

Weitere Ausg.: Printed edition: ISBN 9783031567346

Sprache: Englisch

DOI: 10.1007/978-3-031-56732-2

URL: https://doi.org/10.1007/978-3-031-56732-2

URL: Volltext  (URL des Erstveröffentlichers)

UID:

Umfang: 1 online resource (187 pages)

Ausgabe: 1st ed.

ISBN: 9783031567322

Serie: Trends in Mathematics Series ; v.4

Anmerkung: Intro -- Preface -- Contents -- Part I Mini-Courses -- The Hardy Constant: A Review -- 1 Introduction -- 2 The Hardy Constant -- 2.1 Domains with Critical Hardy Constant -- 3 The Hardy Constant in Two Dimensions -- 3.1 Sectors in R2 -- References -- Some Harmonic Analysis in a General Gaussian Setting -- 1 Introduction -- 2 Main Results -- References -- Introduction to Hypercomplex Analysis -- 1 Introduction -- 2 Universal Clifford Algebras -- 2.1 Quadratic Spaces -- 2.2 Clifford Algebras -- 2.3 Representations of Orthogonal Groups -- 2.4 Function Spaces -- 3 Inner and Outer Spherical Monogenics -- 3.1 Euler and Gamma Operators -- 3.2 Cauchy-Kovalevskaya Theorem for Monogenic Functions -- 3.3 Inner and Outer Spherical Monogenics -- 3.4 Fischer Decomposition and Sommen-Weyl Relations -- 4 Spectral Theory of a Clifford Vector -- References -- Some Norm Bounds for the Spectral Projections of the Heisenberg Sublaplacian -- 1 Introduction -- 2 Restriction Operator and Spectral Decomposition of the Laplacian -- 2.1 The Restriction Conjecture -- 3 The Heisenberg Group -- 4 Improving on Müller Bounds -- References -- Boundary Value Problems for Elliptic Operators Satisfying Carleson Condition -- 1 Introduction -- 2 Motivation and Statements of Main Results -- 3 More General Operators and Open Problems -- References -- An Elementary Computation of Heat Trace Invariants -- 1 Introduction -- 2 Computation of the Resolvent Symbols -- 3 Computation of the Heat Coefficients -- References -- Can We Divide Vectors?-Geometric Calculus in Science and Engineering -- 1 Classic Vector Calculus vs. Clifford Algebras -- 2 Möbius Transformations and the Conformal Model -- 3 Outlook-Morphological Calculus by F. Sommen -- References -- Maximal Regularity as a Tool for Partial Differential Equations -- 1 Introduction -- 2 Maximal Regularity -- 2.1 Analytic Semigroup. , 2.2 Maximal Regularity in Banach Spaces-Weighted Maximal Regularity -- 3 Semi-linear Equations in Critical Spaces -- 3.1 Mild Solutions -- 3.2 The Nonlinear Heat Equation: Existence and Uniqueness of Solutions -- 3.3 Navier-Stokes Equations -- References -- Recent Existence Results for Some Critical Subelliptic Problems -- 1 Introduction -- 2 Main Results and Discussion -- References -- Introduction to the Analysis on Manifolds with Conical Singularities -- 1 Introduction -- 2 Manifolds with Conical Singularities -- 3 Cone Differential Operators -- 4 The Mellin Transform -- 4.1 The Weighted Mellin Transform -- 5 Cone Sobolev Spaces -- 6 Symbols -- 7 The Cone Calculus -- 7.1 Ellipticity -- 7.2 Back to the Beginning -- 8 Historical Remarks -- References -- Elliptic Systems of Phase Transition Type -- 1 Some Basic Facts on the Allen-Cahn Equation -- 2 The Connecting Orbit Problem -- 3 The Heteroclinic Double Layers Problem -- 4 The Triple Junction Problem -- References -- Cordes Condition, Campanato Nearness and Beyond -- 1 Introduction -- 2 Part I: Cordes Condition -- 2.1 The Functional Setting -- 2.2 The Cordes Condition -- 2.3 Interpretation of Cordes Condition -- 3 Part 2: Campanato Nearness -- 4 Part 3: Adequate Nearness -- References -- Part II Short Talks -- Global Hypoellipticity on Homogeneous Vector Bundles: Necessary and Sufficient Conditions -- 1 Introduction -- 2 Preliminaries -- 2.1 The Group Fourier Transform -- 2.2 The Quantisation Formula on Compact Lie Groups -- 2.3 Vector Bundles -- 2.4 Homogeneous Vector Bundles -- 2.5 The Quantisation Formula on Homogeneous Vector-Bundles -- 2.6 Additional Remarks -- 3 Main Results -- References -- On Inverse Source Problems for Space-Dependent Sources in Thermoelasticity -- 1 Introduction and Literature Overview -- 2 Inverse Problems of Determining a Space-Dependent Load Source. , 3 Inverse Problem of Determining a Space-Dependent Heat Source -- 4 Combined Problems -- 5 Discussion Conditions and Outlook -- References -- Symmetric Properties of Eigenvalues and Eigenfunctions of Uniform Beams with Axial Loads -- 1 Introduction -- 2 Formulation of the Problem and Main Result -- References -- Cylindrical and Horizontal Extensions of Critical Sobolev Type Inequalities and Identities -- 1 Introduction -- 2 Cylindrical Extension of Critical Sobolev Type Inequality and Identity on Rk Rn-k -- 3 Critical Sobolev Type Inequality and Identity on Stratified Lie Groups -- References -- Very Weak Solution of the Wave Equation for Sturm-LiouvilleOperator -- 1 Introduction -- 2 Homogeneous Equation Case -- 3 Non-homogeneous Equation Case -- 4 Very Weak Solution -- References -- On Cochran-Lee and Hardy-type Inequalities in Some Classical and Homogeneous Group Cases -- 1 Introduction -- 2 Main Results -- 3 On Hardy-Type Inequalities on Homogeneous Groups -- References.

Weitere Ausg.: ISBN 9783031567315

Sprache: Englisch

UID:

Umfang: 1 Online-Ressource (VIII, 190 p. 18 illus., 13 illus. in color.)

ISBN: 9783031567322

Serie: Research Perspectives Ghent Analysis and PDE Center 4

Inhalt: Part I Mini-courses -- The Hardy constant: a review -- Some harmonic analysis in a general Gaussian setting -- Introduction to hypercomplex analysis -- Some norm bounds for the spectral projections of the Heisenberg sublaplacian -- Boundary value problems for elliptic operators satisfying Carleson condition -- An elementary computation of heat trace invariants -- Can we divide vectors? - Geometric calculus in Science and Engineering -- Maximal regularity as a tool for partial differential equations -- Recent existence results for some critical subelliptic problems -- Introduction to the Analysis on Manifolds with Conical Singularities -- Elliptic systems of phase transition type -- Cordes condition, Campanato nearness and beyond -- Part II Short talks -- Global hypoellipticity on homogeneous vector bundles: Necessary and sufficient conditions -- On inverse source problems for space-dependent sources in thermoelasticity -- Symmetric properties of eigenvalues and eigenfunctions of uniform beams with axial loads -- Cylindrical and horizontal extensions of critical Sobolev type inequalities and identities -- Very weak solution of the wave equation for Sturm-Liouville operator -- Some new multidimensional Cochran-Lee and Hardy type inequalities.

Inhalt: The principal aim of the volume is gathering all the contributions given by the speakers (mini courses) and some of the participants (short talks) of the summer school "Modern Problems in PDEs and Applications" held at the Ghent Analysis and PDE Center from 23 August to 2 September 2023. The school was devoted to the study of new techniques and approaches for solving partial differential equations, which can either be considered or arise from the physical point of view or the mathematical perspective. Both sides are extremely important since theories and methods can be developed independently, aiming to gather each other in a common objective. The aim of the summer school was to progress and advance in the problems considered. Note that real-world problems and their applications are classical study trends in physical or mathematical modelling. The summer school was organised in a friendly atmosphere and synergy, and it was an excellent opportunity to promote and encourage the development of the subject in the community.

Weitere Ausg.: ISBN 9783031567315

Weitere Ausg.: ISBN 9783031567339

Weitere Ausg.: ISBN 9783031567346

Weitere Ausg.: Erscheint auch als Druck-Ausgabe Modern problems in PDEs and applications Cham : Birkhäuser, 2024 ISBN 9783031567315

Weitere Ausg.: ISBN 9783031567339

Weitere Ausg.: ISBN 9783031567346

Sprache: Englisch

DOI: 10.1007/978-3-031-56732-2

URL: lizenzpflichtig

Mehr zum Autor: Ruzhansky, Michael 1972-