Giovanni Leoni

About

Giovanni Leoni is a Professor of Mathematical Sciences in the Department of Mathematical Sciences at Carnegie Mellon University. His current research interests focus on the calculus of variations, partial differential equations, and geometric measure theory. He places special emphasis on applications of these mathematical areas to problems in continuum mechanics, materials science, and imaging. Recent research efforts have concentrated on variational problems involving material defects such as dislocations, the epitaxial growth of thin films over crystalline substrates, and phase field models for anisotropic crystalline energies.

Research topics

• Mathematics
• Mathematical analysis
• Physics
• Computer Science
• Pure mathematics
• Mechanics
• Classical mechanics
• Mathematics education
• Astronomy

Selected publications

• A First Course in Fractional Sobolev Spaces

  Graduate studies in mathematics · 2023 · 51 citations

  1st authorCorresponding
  PublisherDOI

• Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations

  Communications on Pure and Applied Mathematics · 2022 · 13 citations

  1st authorCorresponding

  Abstract In this paper we study a finite‐depth layer of viscous incompressible fluid in dimension , modeled by the Navier‐Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity‐capillary effects, we allow for a second force field in the bulk a…

  PublisherDOI

• A First Course in Fractional Sobolev Spaces

  arXiv (Cornell University) · 2023 · 9 citations

  1st authorCorresponding

  This book provides a gentle introduction to fractional Sobolev spaces, which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of one variable. It covers the definition, standard properties, extensions, embeddings, Hardy inequalities, and interpolation inequalities. The second part deals with fractional Sobolev spaces of several variables. The author studies completeness, density, homogene…

  PublisherOA PDFDOI

• Global and local energy minimizers for a nanowire growth model

  Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2022-11-04 · 4 citations

  articleOpen access

  We consider a model for vapor–liquid–solid growth of nanowires proposed in the physical literature. Liquid drops are described as local or global volume-constrained minimizers of the capillarity energy outside a semi-infinite convex obstacle modeling the nanowire. We first address the existence of global minimizers and then, in the case of rotationally symmetric nanowires, we investigate how the presence of a sharp edge affects the shape of local minimizers and the validity of Young’s law.

  PublisherDOI

• On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions

  Communications on Pure &amp Applied Analysis · 2020-01-01 · 3 citations

  preprintOpen accessSenior author

  This paper is concerned with the study of the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions. It is shown that if the free boundary of a symmetric local minimizer approaches the point where the two different conditions meet, then it must do so at an angle of $ \pi/2 $.

  PublisherDOI

Recent grants

• Variational Methods for Materials Science and Mechanics

  NSF · $308k · 2017–2022

• Modern methods in the Calculus of Variations with applications to materials science and hydrodynamics

  NSF · $132k · 2007–2011

• Singularly Perturbed and Multiscale Problems

  NSF · $120k · 2004–2008

Frequent coauthors

• Irene Fonseca

  60 shared

• Massimiliano Morini

  18 shared

• Gianni Dal Maso

  13 shared

• Giovanni Gravina

  Charles University

  7 shared

• Nicola Fusco

  University of Naples Federico II

  7 shared

• Patrizia Pucci

  University of Perugia

  6 shared

• Howard L. Levine

  Bay Medical Center

  5 shared

• Kevin McLeod

  The Open University

  5 shared

Education

• Ph.D., Mathematics

  University of Minnesota

• M.S., Mathematical Sciences

  University of Minnesota

Awards & honors

• Richard Moore Award (2019)
• Giuseppe Bartolozzi Award

Similar researchers at Carnegie Mellon University

• Alan Friezeh-74
• Michael Youngh-57
• David Kinderlehrerh-45