@prefix psr: . @prefix skos: . @prefix dc: . @prefix xsd: . psr:-H07K1XD3-T skos:prefLabel "théorème de l'indice d'Atiyah-Singer"@fr, "Atiyah-Singer index theorem"@en ; a skos:Concept ; skos:broader psr:-F0LR2RP4-6 . psr:-F0LR2RP4-6 skos:narrower psr:-G310LVH4-J, psr:-XJ77SGF5-S, psr:-H07K1XD3-T ; skos:definition """Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form

{\\\\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu+G=0,\\\\,}

where

A

B

C

D

E

F

, and

G

are functions of

x

and

y

and where

{\\\\displaystyle u_{x}={\\ rac {\\\\partial u}{\\\\partial x}}}

{\\\\displaystyle u_{xy}={\\ rac {\\\\partial ^{2}u}{\\\\partial x\\\\partial y}}}

and similarly for

{\\\\displaystyle u_{xx},u_{y},u_{yy}}

. A PDE written in this form is elliptic if

{\\\\displaystyle B^{2}-AC<0,}

with this naming convention inspired by the equation for a planar ellipse. The simplest examples of elliptic PDE's are the Laplace equation,

{\\\\displaystyle \\\\Delta u=u_{xx}+u_{yy}=0}

, and the Poisson equation,

{\\\\displaystyle \\\\Delta u=u_{xx}+u_{yy}=f(x,y).}

In a sense, any other elliptic PDE in two variables can be considered to be a generalization of one of these equations, as it can always be put into the canonical form

{\\\\displaystyle u_{xx}+u_{yy}+{\\ ext{ (lower-order terms)}}=0}

through a change of variables.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation)"""@en, """En mathématiques, une équation aux dérivées partielles linéaire du second ordre, dont la forme générale est donnée par :

{\\\\displaystyle \\\\sum _{i,j=1}^{n}{a_{ij}(\\\\mathbf {x} ){\\\\dfrac {\\\\partial ^{2}f}{\\\\partial x_{i}\\\\partial x_{j}}}}+\\\\sum _{i=1}^{n}{b_{i}(\\\\mathbf {x} ){\\\\dfrac {\\\\partial f}{\\\\partial x_{i}}}}+c(\\\\mathbf {x} )f=h(\\\\mathbf {x} ),\\\\ \\\\ \\\\ \\\\mathbf {x} \\\\in U\\\\subset \\\\mathbb {R} ^{n}}

est dite elliptique en un point donné x de l'ouvert U si la matrice carrée symétrique

{\\\\displaystyle A(\\\\mathbf {x} )=\\\\left(a_{ij}\\ ight)_{1\\\\leq i,j\\\\leq n}}

des coefficients du second ordre admet des valeurs propres non nulles et de même signe.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/%C3%89quation_aux_d%C3%A9riv%C3%A9es_partielles_elliptique)"""@fr ; skos:prefLabel "elliptic partial differential equation"@en, "équation aux dérivées partielles elliptique"@fr ; a skos:Concept ; skos:exactMatch , ; skos:broader psr:-LM732D0H-P ; dc:modified "2024-10-18"^^xsd:date ; skos:inScheme psr: . psr:-XJ77SGF5-S skos:prefLabel "Laplace's equation"@en, "équation de Laplace"@fr ; a skos:Concept ; skos:broader psr:-F0LR2RP4-6 . psr: a skos:ConceptScheme . psr:-LM732D0H-P skos:prefLabel "équation aux dérivées partielles"@fr, "partial differential equation"@en ; a skos:Concept ; skos:narrower psr:-F0LR2RP4-6 . psr:-G310LVH4-J skos:prefLabel "opérateur elliptique"@fr, "elliptic operator"@en ; a skos:Concept ; skos:broader psr:-F0LR2RP4-6 .