In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDE are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which generally model phenomena that change in time. They are also important in pure mathematics, where they are fundamental to various fields of research such as differential geometry and optimal transport.
Pierre-Louis Lions is a French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Medal and the 1991 Prize of the Philip Morris tobacco and cigarette company.
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
Jürgen Kurt Moser was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations.
Louis Nirenberg was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.
Luis Ángel Caffarelli is an Argentine-American mathematician. He studies partial differential equations and their applications. Caffarelli is a professor of mathematics at the University of Texas at Austin, and the winner of the 2023 Abel Prize.
In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming, differential games or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games.
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck, Clifford Taubes, Shing-Tung Yau, Richard Schoen, and Richard Hamilton launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to the Poincaré conjecture by Grigori Perelman, completing a program initiated and largely carried out by Richard Hamilton.
Vladimir Gilelevich Maz'ya is a Russian-born Swedish mathematician, hailed as "one of the most distinguished analysts of our time" and as "an outstanding mathematician of worldwide reputation", who strongly influenced the development of mathematical analysis and the theory of partial differential equations.
Avner Friedman is Distinguished Professor of Mathematics and Physical Sciences at Ohio State University. His primary field of research is partial differential equations, with interests in stochastic processes, mathematical modeling, free boundary problems, and control theory.
Xu-Jia Wang is a Chinese-Australian mathematician. He is a professor of mathematics at the Australian National University and a fellow of the Australian Academy of Science.
James Burton Serrin was an American mathematician, and a professor at University of Minnesota.
In mathematics, k-Hessian equations are partial differential equations (PDEs) based on the Hessian matrix. More specifically, a Hessian equation is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equation. It can be written as , where , , and , are the eigenvalues of the Hessian matrix and , is a th elementary symmetric polynomial.
Giuseppe Mingione is an Italian mathematician who is active in the fields of partial differential equations and calculus of variations.
Gregory Eskin is a Russian-Israeli-American mathematician, specializing in partial differential equations.
Robert Ronald Jensen is an American mathematician, specializing in nonlinear partial differential equations with applications to physics, engineering, game theory, and finance.
Christoph Schwab is a German applied mathematician, specializing in numerical analysis of partial differential equations and boundary integral equations.
Xavier Ros Oton is a Spanish mathematician who works on partial differential equations (PDEs). He is an ICREA Research Professor and a Full Professor at the University of Barcelona.
Joel Spruck is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University with the supervision of Robert S. Finn in 1971.
Klaus Schmitt is an American mathematician doing research in nonlinear differential equations, and nonlinear analysis.
