Perron's method and Myths about nonlocal equations: Difference between pages

Revision as of 18:13, 15 July 2011

The following myths are usually heard in the corridors of some math departments and conference coffee breaks.

There are no new difficulties in nonlocal equations and everything is proved analogously as in the classical case

Nonlocal equations is a much richer class than the usual PDE. Predictably, there are some intrinsic difficulties. A common difficulty comes from the fact that fractional order operators have different scaling properties and therefore interact differently with other terms. Moreover, in certain cases there are some surprising results which do not match what one would expect from local PDE intuition. We have a list of results that are fundamentally different to the local case.

Nonlocal equations is a field in which one replaces the Laplacian by the fractional Laplacian in whatever equation and writes a paper

One can certainly do this. In some cases the classical methods would work after a simple adaptation. In other cases there is a significant difference either in the methods or in the results. Naturally, the good papers are the ones that fit into the second category. This wiki should help people learn to differentiate one from the other.

Nonlocal equations are bizarre and unnatural objects

The Starting page of this wiki should clarify the importance of nonlocal equations.

Most equations in nature are local

In fact the opposite is true. In many cases local PDEs are a good simplification though.

All statements and proofs in nonlocal equations involve gigantic formulas

Nonlocal equations usually involve integral quantities that are larger to write than usual derivatives. This is a notation problem to a large extent. Many proofs in nonlocal equations deal with long integral quantities that come from the nonlocal character of the equation. These features are there, but are rarely at the essence of the arguments. Most statements and proofs are just as conceptual as in usual PDEs.

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-'''Perron's method''', also known as the '''method of subharmonic functions''', is a technique originally introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation.<ref>{{Citation | last1=Perron | first1=O. |  title=Eine neue Behandlung der ersten Randwertaufgabe für Δu=0 | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01192395 | year=1923 | month=12 | journal=Mathematische Zeitschrift| issn=0025-5874 | volume=18 | issue=1 | pages=42–54}}
+The following myths are usually heard in the corridors of some math departments and conference coffee breaks.
-</ref> 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.
-The Perron method can be used [[viscosity solutions]] whenever a [[comparison principle]] is available and appropriate barriers can be constructed to assure the boundary conditions. The Perron solution is taken to be the largest viscosity subsolution, or the least viscosity supersolution.
-==References==
+=== There are no new difficulties in nonlocal equations and everything is proved analogously as in the classical case ===
-{{reflist}}
+Nonlocal equations is a much richer class than the usual PDE. Predictably, there are some intrinsic difficulties. A common difficulty comes from the fact that fractional order operators have different scaling properties and therefore interact differently with other terms. Moreover, in certain cases there are some surprising results which do not match what one would expect from local PDE intuition. We have a [[list of results that are fundamentally different to the local case]].
+=== Nonlocal equations is a field in which one replaces the Laplacian by the fractional Laplacian in whatever equation and writes a paper ===
+One can certainly do this. In some cases the classical methods would work after a simple adaptation. In other cases there is a significant difference either in the methods or in the results. Naturally, the good papers are the ones that fit into the second category. This wiki should help people learn to differentiate one from the other.
+=== Nonlocal equations are bizarre and unnatural objects ===
+The [[Starting page]] of this wiki should clarify the importance of nonlocal equations.
+=== Most equations in nature are local ===
+In fact the opposite is true. In many cases local PDEs are a good simplification though.
+=== All statements and proofs in nonlocal equations involve gigantic formulas ===
+Nonlocal equations usually involve integral quantities that are larger to write than usual derivatives. This is a notation problem to a large extent. Many proofs in nonlocal equations deal with long integral quantities that come from the nonlocal character of the equation. These features are there, but are rarely at the essence of the arguments. Most statements and proofs are just as conceptual as in usual PDEs.