In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems.
The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450. It is awarded every three years for a notable research work in analysis that has appeared during the past six years. The work must be published in a recognized, peer-reviewed venue. The current award is $5,000.
Ennio De Giorgi was an Italian mathematician who worked on partial differential equations and the foundations of mathematics.
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.
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.
Lawrence Craig Evans is an American mathematician and Professor of Mathematics at the University of California, Berkeley.
In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. J. Serrin (1955), and J. Moser generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Such results can be used to show the interior regularity of weak solutions.
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function u of two variables x,y is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of u and in the second-order partial derivatives of u. The independent variables (x,y) vary over a given domain D of R2. The term also applies to analogous equations with n independent variables. The most complete results so far have been obtained when the equation is elliptic.
The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle. It is deeply related to the study of minimal surfaces and the capacity of a set in potential theory as well. Applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics.
In mathematics, a free boundary problem is a partial differential equation to be solved for both an unknown function and an unknown domain . The segment of the boundary of which is not known at the outset of the problem is the free boundary.
Alessio Figalli is an Italian mathematician working primarily on calculus of variations and partial differential equations.
Ovidiu Vasile Savin is a Romanian-American mathematician who is active in the field of the partial differential equations.
Gheorghe Moroșanu is a Romanian mathematician known for his works in Ordinary and Partial Differential Equations, Nonlinear Analysis, Calculus of Variations, Fluid Mechanics, Asymptotic Analysis, Applied Mathematics. He earned his Ph.D. in 1981 from the Alexandru Ioan Cuza University in Iași.
Héctor Chang Lara is a Venezuelan mathematician working at CIMAT, Guanajuato unit, in Mexico. Chang received his BA in Mathematics from Simon Bolivar University in Venezuela, his MS from the University of New Mexico and his PhD in mathematics from the University of Texas at Austin, advised by Luis Caffarelli. Chang works in partial differential equations, specializing in elliptic and parabolic differential equations as well as integro-differential equations and free boundary problems.
Daniela De Silva is an Italian mathematician known for her expertise in partial differential equations. She is an associate professor of mathematics at Barnard College and Columbia University.
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.
Basilis Gidas is an applied mathematician at Brown University, interested in many applications of mathematics. Following degrees in electrical engineering, mechanical engineering, and mathematics, he obtained a combined Ph.D. in physics and nuclear engineering at the University of Michigan in 1970. He is an elected fellow of the Institute of Mathematical Statistics. He has had past appointments in various mathematics and physics departments at the Institute for Advanced Study, Rutgers University, Rockefeller University, Bielefeld University, University of Washington, and University of Michigan.
Klaus Schmitt is an American mathematician doing research in nonlinear differential equations, and nonlinear analysis.
Regularity is a property of elliptic partial differential equations such as Laplace's equation. Hilbert's nineteenth problem was concerned with this concept.
