Krylov-Safonov and Talk:Introduction to nonlocal equations: Difference between pages
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Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, ) | |||
Revision as of 22:01, 13 March 2012
Broadly speaking, nonlocal equations are those in which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, $u$, against a kernel which is singular at the origin and the equation requires regularity of $u(x+y)-u(x)$ to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, )