In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.
Louis Nirenberg was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.
Luis Angel Caffarelli is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic. Informally, and perhaps less directly, since Hilbert's concept of a "regular variational problem" identifies precisely a variational problem whose Euler–Lagrange equation is an elliptic partial differential equation with analytic coefficients, Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of partial differential equations, any solution function inherits the relatively simple and well understood structure from the solved equation. Hilbert's nineteenth problem was solved independently in the late 1950s by Ennio De Giorgi and John Forbes Nash, Jr.
Richard Melvin Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions.
Leon Melvyn Simon, born in 1945, is a Leroy P. Steele Prize and Bôcher Prize-winning mathematician, known for deep contributions to the fields of geometric analysis, geometric measure theory, and partial differential equations. He is currently Professor Emeritus in the Mathematics Department at Stanford University.
Victor Bangert is Professor of Mathematics at the Mathematisches Institut in Freiburg, Germany. His main interests are differential geometry and dynamical systems theory. He specialises in the theory of closed geodesics, wherein one of his significant results, combined with another one due to John Franks, implies that every Riemannian 2-sphere possesses infinitely many closed geodesics. He also made important contributions to Aubry–Mather theory.
Luigi Ambrosio is a professor at Scuola Normale Superiore in Pisa, Italy. His main fields of research are the calculus of variations and geometric measure theory.
In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. The Perron method works by finding the largest subharmonic function with boundary values below the desired values; the "Perron solution" coincides with the actual solution of the Dirichlet problem if the problem is soluble.
In differential geometry, Bernstein's problem is as follows: if the graph of a function on Rn−1 is a minimal surface in Rn, does this imply that the function is linear? This is true in dimensions n at most 8, but false in dimensions n at least 9. The problem is named for Sergei Natanovich Bernstein who solved the case n = 3 in 1914.
In the theory of partial differential equations, an a priori estimate is an estimate for the size of a solution or its derivatives of a partial differential equation. A priori is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. One reason for their importance is that if one can prove an a priori estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem.
Fritz Gassmann (1899–1990) was a Swiss mathematician and geophysicist.
Lee Albert Rubel was a mathematician known for his contributions to analog computing.
Giuseppe Mingione is an Italian mathematician who is active in the fields of partial differential equations and calculus of variations.
Henry Christian Wente was an American mathematician, known for his 1986 discovery of the Wente torus, an immersed constant-mean-curvature surface whose existence disproved a conjecture of Heinz Hopf.
Donatella Danielli is a professor of mathematics at Arizona State University and is known for her contributions to partial differential equations, calculus of variations and geometric measure theory, with specific emphasis on free boundary problems.
Erich Hans Rothe was a German-born American mathematician, who did research in mathematical analysis, differential equations, integral equations, and mathematical physics. He is known for the Rothe method used for solving evolution equations.
Ulisse Stefanelli is an Italian mathematician. He is currently professor at the Faculty of Mathematics of the University of Vienna. His research focuses on calculus of variations, partial differential equations, and materials science.
Klaus Schmitt is an American mathematician doing research in nonlinear differential equations, and nonlinear analysis.
