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| | | #redirect [[comparison principle]] |
| == Existence and uniqueness results ==
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| For a variety of nonlinear elliptic and parabolic equations, the existence of [[viscosity solutions]] can be obtained using [[Perron's method]]. The uniqueness of solutions is a consequence of the [[comparison principle]].
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| There are some equations for which this general framework does not work, for example the [[surface quasi-geostrophic equation]]. One could say that the underlying reason is that the equation is not ''purely'' parabolic, but it has one hyperbolic term.
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| == Regularity results ==
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| The regularity tools used for nonlocal equations vary depending on the type of equation.
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| === Nonlinear equations ===
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| The starting point to study the regularity of solutions to a nonlinear elliptic or parabolic equation are the [[Holder estimates]] which hold under very weak assumptions and rough coefficients. They are related to the [[Harnack inequality]].
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| For some [[fully nonlinear integro-differential equation]] with continuous coefficients, we can prove [[differentiability estimates|$C^{1,\alpha}$ estimates]].
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| Under certain hypothesis, the nonlocal [[Bellman equation]] from optimal stochastic control has classical solutions due to the [[nonlocal Evans-Krylov theorem|nonlocal version of Evans-Krylov theorem]].
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| === Semilinear equations ===
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| There are several interesting models that are [[semilinear equations]]. Those equations consists of either the [[fractional Laplacian]] or [[fractional heat equation]] plus a nonlinear term.
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| There are challenging regularity questions especially when the Laplacian interacts with gradient terms in [[Drift-diffusion equations]]. A simple method that has been successful in proving the well posedness of some semilinear equations with drift terms in the critical case (when both terms have the same scaling properties) is the [[conserved modulus of continuity approach]], often called "nonlocal maximum principle method".
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Latest revision as of 20:40, 24 April 2012