Preface for the special issue in honor of David Jerison
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Guozhen Lu
To celebrate the many distinguished achievements and a remarkable career of Professor David Jerison from the Massachusetts Institute of Technology, this special issue of Advanced Nonlinear Studies is dedicated to him on the occasion of his 70th birthday.
David Jerison is an outstanding mathematician widely known for his work on partial differential equations. He is a leading figure in the areas of harmonic analysis methods to partial differential equations. He started out studying linear elliptic boundary value problems in non-smooth domains and sub-elliptic problems, especially those related to the Heisenberg group. He proved some sharp Carleman inequalities, quantitative forms of unique continuation. He then began to consider nonlinear problems. Jerison has made many fundamental and influential contributions in all these directions and his works have been followed widely by many mathematicians in the past few decades.
The classical Minkowski problem asks whether one can find a convex polyhedron given area and normal vector of each face. The corresponding inverse problem in the plane is quite easy: find the (convex) polygon given its sidelengths and angles. Jerison noticed that the Schwarz-Christoffel formula for conformal mapping solves an analogous inverse problem for harmonic measure: find the convex polygon given its angles and the harmonic measures of its sides. He generalized this to higher dimensions, solving the problem of finding a convex polyhedron given the harmonic measure and normal vector of each face. The smooth version of this problem involves a global, fully nonlinear partial differential equation relating Gauss curvature to harmonic measure. Jerison then introduced and solved a different problem that is much closer structurally to the Minkowski problem, namely, one which substitutes electrostatic capacity for volume in Minkowski’s theory. Capacity can be defined as the total flux of the equilibrium potential through the boundary, and the flux is a suitably normalized harmonic measure viewed from infinity. Equivalently, capacity is the (exterior) Dirichlet energy of the equilibrium potential. There is a parallel, technically simpler, Minkowski-type problem with volume replaced by the ground state eigenvalue, which is a kind of interior Dirichlet energy of the convex body.
Inspired by seminal papers by Alt-Caffarelli, Alt-Caffarelli-Friedman, and Caffarelli, Jerison has worked for three decades on another family of nonlinear problems, free boundary problems. A major theme of this work is the parallel between free boundaries and minimal surfaces. The element that unites it with his other work is the relevance of harmonic measure. Jerison also studied an evolution equation, the Hele-Shaw equation, in which the growth of a fluid region is governed by the boundary pressure, which is none other than harmonic measure. This led him to a discrete version of this evolution known as internal diffusion-limited aggregation, a model introduced by chemists to describe corrosion and chemical polishing. Jerison’s current main focus is on shapes of eigenfunctions and the curvature of their level sets. In this context, flux across level sets replaces harmonic measure.
Jerison’s work on the Poincaré inequalities for Hörmander’s vector fields, the CR Yamabe problem and sub-elliptic equations have influenced greatly my own research interests. My work on sharp constants and extremal functions for geometric and functional inequalities has been inspired by his pioneering work.
Jerison is a principal investigator for the Simons Collaboration on Waves in Disorder, a cross-disciplinary project based on experiments with inorganic and organic semiconductors and Bose-Einstein condensates. He serves regularly as faculty advisor to SPUR, the MIT math department’s summer research program for undergraduates, as well as RSI, a high school science summer research program for students from across the world, which takes place at MIT. He is also known for his widely viewed, on-line lectures in calculus on MIT’s Open Courseware and other platforms.
David Jerison was born in 1953 in Lafayette, Indiana. After college and before graduate school, he studied for a year at the University of Paris 11, at Orsay. After completing his PhD at Princeton University in 1980 under the direction of Elias Stein, he spent a year as an NSF Postdoctoral Fellow at the University of Chicago. He joined the MIT mathematics faculty in 1981, where he has remained for the rest of his career. He received a Sloan Research Fellowship and a Presidential Young Investigator Award in 1985 and delivered an invited address at the ICM in Zurich in 1994. He was selected as a Margaret MacVicar Faculty Fellow in 2004 and awarded the Bergman Prize in 2012 for his joint work with Jack Lee on the CR Yamabe problem. He received a Simons Fellowship in 2018 and a Guggenheim Fellowship in 2019. Jerison is a Fellow of the AMS and the American Academy of Arts & Sciences and served as Vice President of the AMS from 2017 to 2020.
In this special issue, we invited articles from both well-established mathematicians and young scholars. Many of them are leading researchers in the areas of harmonic analysis and partial differential equations, including some of the world’s most prominent mathematicians. The wide range of topics covered in this special issue also reflect the broad scope of Jerison’s research interests. I am grateful to all the contributors and reviewers for making this special issue possible.
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Funding information: The author is partly supported by Simons Collaboration Grants (519099 and 957892) and a Simons Fellowship (1030658) from the Simons Foundation.
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Conflict of interest: Author states no conflict of interest.
Guozhen Lu
Editor-in-chief
© 2023 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Research Articles
- Asymptotic properties of critical points for subcritical Trudinger-Moser functional
- The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
- On some dense sets in the space of dynamical systems
- Sharp profiles for diffusive logistic equation with spatial heterogeneity
- Generic properties of the Rabinowitz unbounded continuum
- Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
- Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
- Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
- The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
- A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
- Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
- Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
- Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
- Ground states of Schrödinger systems with the Chern-Simons gauge fields
- Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
- A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
- A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
- On a version of hybrid existence result for a system of nonlinear equations
- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
Articles in the same Issue
- Research Articles
- Asymptotic properties of critical points for subcritical Trudinger-Moser functional
- The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
- On some dense sets in the space of dynamical systems
- Sharp profiles for diffusive logistic equation with spatial heterogeneity
- Generic properties of the Rabinowitz unbounded continuum
- Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
- Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
- Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
- The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
- A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
- Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
- Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
- Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
- Ground states of Schrödinger systems with the Chern-Simons gauge fields
- Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
- A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
- A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
- On a version of hybrid existence result for a system of nonlinear equations
- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
