Dirichlet energy

English Alternative forms en Dirichlet's energy Etymology en German mathematician Peter Gustav Lejeune Dirichlet 1805 1859 =, who made significant contributions to the theory of Fourier series. Noun en analysis functional analysis Fourier analysis A quadratic functional which, given a real function defined on an open subset of ℝn, yields a real number that is a measure of how variable said function is. en 2005 Roger Moser Partial Regularity for Harmonic Maps and Related Problems World Scientific https://books.google.com.au/books?id=OzyI-xo-ZOcC&pg=PA1&dq=%22Dirichlet+energy%22%7C%22Dirichlet+energies%22&hl=en&sa=X&ved=0ahUKEwjPqM322aThAhXCXisKHaNkArMQ6AEISjAG#v=onepage&q=%22Dirichlet%20energy%22%7C%22Dirichlet%20energies%22&f=false 1 Variational principles play an important role in both geometry and physics, and one of the key problems with applications in both fields is the variational problem associated to the Dirichlet energy of maps between Riemannian manifolds. en 2011 Camillo De Lellis; Emanuele Nunzio Spadaro Q-valued Functions Revisited w:American Mathematical Society https://books.google.com.au/books?id=ACDVAwAAQBAJ&pg=PA28&dq=%22Dirichlet+energy%22%7C%22Dirichlet+energies%22&hl=en&sa=X&ved=0ahUKEwjPqM322aThAhXCXisKHaNkArMQ6AEI0gEwIQ#v=onepage&q=%22Dirichlet%20energy%22%7C%22Dirichlet%20energies%22&f=false 28 The Dirichlet energy of a function can be recovered, moreover, as the energy of the composition , where is the biLipschitz embedding in Corollary 2.2 (compare with Theorem 2.4). en 2015 Sören Bartels Numerical Methods for Nonlinear Partial Differential Equations https://books.google.com.au/books?id=aEBFBgAAQBAJ&pg=PA89&dq=%22Dirichlet+energy%22%7C%22Dirichlet+energies%22&hl=en&sa=X&ved=0ahUKEwjPqM322aThAhXCXisKHaNkArMQ6AEI2QEwIg#v=onepage&q=%22Dirichlet%20energy%22%7C%22Dirichlet%20energies%22&f=false 89 Springer Since the Dirichlet energy is weakly lower semicontinuous and strongly continuous, the linear lower-order terms are weakly continuous on , and since the finite element spaces are dense in , we verify that as . Usage notes In mathematical terms, given an open set and a function , the Dirichlet energy of is , where denotes the gradient vector field of . Translations functional that maps a function to a real number representing its variability Bulgarian: bg енергия на Дирихле f Dutch: nl Dirichlet-energie f Italian: it energia di Dirichlet f Spanish: es energía de Dirichlet f Further reading Calculus of variations Dirichlet's principle Harmonic map Laplace's equation en eponyms