Although ancient Greek and Roman sources report that Dido, the founder and first queen of Carthage was the first person who formulated a problem in Calculus of Variations, the classical existence theory is connected mainly with the names of Euler, Lagrange and Ostrogradskij. The notorious Euler-Lagrange equation is a second order Partial Differential Equation, the solvability of which ensures the existence of a minimizer of a given functional. The question of regularity for the solutions of this PDE was firstly posed by David Hilbert in his 19th and 20th problem, presented during a celebrated lecture at the International Congress of Mathematics 1900 in Paris. During the last century, these two problems gave a strong impulse to the development of the regularity theory for problems from CV and PDE. Our goal is to present some classical and new results concerning regularity properties of the solutions to the Dirichlet problem for elliptic equations and systems. We obtain essential boundedness of the solution to a class of nonlinear elliptic systems. In addition, we establish estimates in the Morrey spaces for the solutions of a kind of quasilinear and nonlinear elliptic systems.
From Dido to Morrey: Variational problems and regularity theory!
SOFTOVA PALAGACHEVA, Lyoubomira
2018-01-01
Abstract
Although ancient Greek and Roman sources report that Dido, the founder and first queen of Carthage was the first person who formulated a problem in Calculus of Variations, the classical existence theory is connected mainly with the names of Euler, Lagrange and Ostrogradskij. The notorious Euler-Lagrange equation is a second order Partial Differential Equation, the solvability of which ensures the existence of a minimizer of a given functional. The question of regularity for the solutions of this PDE was firstly posed by David Hilbert in his 19th and 20th problem, presented during a celebrated lecture at the International Congress of Mathematics 1900 in Paris. During the last century, these two problems gave a strong impulse to the development of the regularity theory for problems from CV and PDE. Our goal is to present some classical and new results concerning regularity properties of the solutions to the Dirichlet problem for elliptic equations and systems. We obtain essential boundedness of the solution to a class of nonlinear elliptic systems. In addition, we establish estimates in the Morrey spaces for the solutions of a kind of quasilinear and nonlinear elliptic systems.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
