Harnack Inequalities and Nonlinear Operators : Proceedings of the INdAM conference to celebrate the 70th birthday of Emmanuele DiBenedetto / edited by Vincenzo Vespri, Ugo Gianazza, Dario Daniele Monticelli, Fabio Punzo, Daniele Andreucci.

Format:

Book

Contributor:

Vespri, Vincenzo, editor.

Series:

Springer INdAM Series, 2281-5198 ; 46

Language:

English

Subjects (All):

Mathematical analysis.

Global analysis (Mathematics).

Manifolds (Mathematics).

Mathematical optimization.

Calculus of variations.

Potential theory (Mathematics).

Analysis.

Global Analysis and Analysis on Manifolds.

Calculus of Variations and Optimization.

Potential Theory.

Local Subjects:

Analysis.

Global Analysis and Analysis on Manifolds.

Calculus of Variations and Optimization.

Potential Theory.

Physical Description:

1 online resource (202 pages)

Edition:

1st ed. 2021.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2021.

Summary:

The book contains two contributions about the work of Emmanuele DiBenedetto and a selection of original papers. The authors are some of the main experts in Harnack’s inequalities and nonlinear operators. These papers are part of the contributions presented during the conference to celebrate the 70th birthday of Prof. Emmanuele DiBenedetto, which was held at “Il Palazzone” in Cortona from June 18th to 24th, 2017. The papers are focused on current research topics regarding the qualitative properties of solutions, connections with calculus of variations, Harnack inequality and regularity theory. Some papers are also related to various applications. Many of the authors have shared with Prof. DiBenedetto an intense scientific and personal collaboration, while many others have taken inspiration from and further developed his field of research. The topics of the conference are certainly of great interest for the international mathematical community.

Contents:

Intro

Contents

What I Learnt from Emmanuele DiBenedetto

1 Introduction

2 Linear Equations: Preliminary Results

2.1 Local Energy and Logarithmic Estimates

2.2 Preliminary Technical Results

2.3 Critical Mass Lemma and Shrinking Lemma

2.4 Expansion of Positivity Lemmata

3 Linear Equations: Regularity Results and Harnack Estimates

3.1 Regularity of the Solutions

3.2 Harnack Estimates

4 The p-Laplacean Equation

4.1 Statement of the Results: Regularity and Harnack Estimates

4.2 Differences between the Linear Case and the Degenerate One

References

A Short Presentation of Emmanuele's Work

1 The Beginning

1.1 Master Degree and an Encounter with Carlo Pucci

1.2 The Ph.D.

2 The First Works

2.1 The Stefan Problem ...

2.2 ...and the Porous Medium Equation

3 A Brief Overview

3.1 DeGiorgi's Classes, Both Elliptic and Parabolic

3.2 The Method of Intrinsic Scaling

4 A More Specific (but Rather Quick) Look

4.1 A Semilinear Elliptic Equation

4.2 The ∞

4.3 A Theorem by DiBenedetto, Friedman and ...Newton

5 Applications to Biology

5.1 Other Topics

6 There is Much More

6.1 Few Numbers

7 Some Personal Words by Ugo Gianazza

7.1 What have I Learnt?

7.2 Just One Final Picture

7.3 A Final Remark

Evolutionary Problems in Non-Cylindrical Domains

1 History of the Problem

2 Gradient Flows in Non-Cylindrical Domains

3 Existence and Regularity in Non-Decreasing Domains

4 Variational Solutions in General Variable Domains

4.1 Existence of Variational Solutions

4.2 Regularity of Variational Solutions

4.3 Uniqueness of Variational Solutions

4.4 Weak Solutions of the Associated Differential Equation

A Compactness Result for the Sobolev Embeddingvia Potential Theory

1 Introduction.

2 Continuous and Compact Embeddings: The Euclidean Case

2.1 Some Preliminary Results

2.2 The Sobolev and Morrey Embedding Theorems

2.3 A More General Compactness Result

3 Continuous and Compact Embeddings for Degenerate Kolmogorov Equations

3.1 Dilation and Translation Groups Associated to L

3.2 Preliminary Results on Convolutions in Homogeneous Lie Groups

3.3 Compactness Estimates for Convolutions with Homogeneous Kernels

3.4 Proof of Our Main Result

4 Representation Formulas

4.1 Fundamental Solution to L0 and Representation Formula

5 Conclusion

Mathematical Modeling of the Rod Phototransduction Process

2 Mathematical Description of the Rod Phototransduction Process

2.1 The Activation Cascade

2.2 The Cytosol Cascade

3 Homogenization and Concentration of Capacity

3.1 Computing the Homogenization Limit

The Interior Limit

Limit on the Activated Disc

Limit in the Outer Shell

3.2 Variational Formulation

4 A Finite Element Formulation

5 Numerical Results and Comparison with Experiments

5.1 Reproducing SPR Measurements

5.2 Investigating of the Role of Incisures

5.3 Investigating the Variability of the SPR

Existence, Decay Time and Light Yield for a Reaction-Diffusion-Drift Equation in the Continuum Physics of Scintillators

2 The Evolution Equation for Scintillators

2.1 The Excitation Carrier Density Vector

2.2 The Reaction-Diffusion-Drift Equation

3 Existence, Decay Time Estimates and Light Yield

3.1 Global Existence

3.2 Decay Time

3.3 Light Yield

Boundary Harnack Type Inequality and Regularity for Quasilinear Degenerate Elliptic Equations

2 Strong A∞ Weights and Function Spaces

3 Harnack Inequality

4 Boundary Harnack Inequality

References.

Monotonicity of Positive Solutions to -p u+a(u)|u|q=f(u) in the Half-Plane in the Case p2

2 Preliminaries

3 Proof of Theorem 1.1

Geometric Tangential Analysis and Sharp Regularity for Degenerate pdes

2 Geometric Tangential Analysis

3 The Parabolic p-Poisson Equation

4 The Porous Medium Equation

5 The Doubly Nonlinear Equation

Complete List of Mathematical Papers Authored by Emmanuele DiBenedetto

Complete List of Monographs and Textbooks Authored or Edited by Emmanuele DiBenedetto.

ISBN:

3-030-73778-0

OCLC:

1255179425