Non linear problems

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Non linear problems with Fractional Diffusions
Luis A. Caffarelli The University of Texas at
Austin
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• Non linear problems involving fractional
  diffusions
• appear in several areas of applied mathematics
• Boundary diffusion (see for instance Duvaut and
  Lions)
• or more generally calculus of variations when
  the energy
• integrals involved correspond to fractional
  derivatives.
• Fluid dynamics like in the quasi-geostrophic
  equation
• modeling ocean atmospheric interaction, or in
  the case of
• turbulent transport
• Stochastic processes of discontinuous nature
  (Levy processes) in
• applications for which random walks have jumps
  at many

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Remark the work just described is strongly based
in an extension theorem It identifies the
fractional Laplacian of a given function u(x) in
Rn with the normal derivative of an extension
v(x,y) of u(x) into the upper half space, (ygt0),
of Rn1.
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The classical example is the ½ Laplacian If v
is the harmonic extension of u ( convolution with
the Poison kernel), then the normal derivative
of v at y0 is exactly the half Laplacian of u.
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In particular, u being half harmonic simply
means that v is harmonic across y0, reducing
regularity properties of u to those of the
harmonic function v
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In fact, any other fractional power of the
Laplacian of a given function u(x) can be
realized as the normal derivative of an
appropriate extension v(x,y).
This can be interpreted as an extension into a
space of fractional dimension and suggest the
correct form of homogeneous solutions,
monotonicity formulas, truncated test functions,
etc
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• This harmonic extension has the virtue of
• reducing many global issues and arguments
• to local, more familiar methods of the
• calculus of variations.
• The global properties of the solutions are
• somehow encoded in the restriction of the
• extension v(x,y) to unit ball in one more
• dimension. (L.C and L.Silvestre, arXiv.org,
  07)

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i) The quasi-geostrophic equation
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See alsoKiselev, Nasarov, Volberg, arXiv.org06
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ii) Problems with constrains or Free
Boundary problems
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iv) Random Homogenization
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See the work of D. Cioranescu and F. Murat (1982)
where the Homogenized equation was derived for
periodic media.
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Fully non-linear equations with fractional
diffusion
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. . .
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Formally, the solution u0 to a fully non-linear
equation, its first derivatives and its second
derivatives all satisfy equations or
inequalities like (1) above. This implies that
u0 is classical (Evans Krylov)
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Thank you for your attention