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Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations
In this article, we provide existence results for a general class of nonlocal
and nonlinear second-order parabolic equations. The main motivation comes from
front propagation theory in the cases when the normal velocity depends on the
moving front in a nonlocal way. Among applications, we present level-set
equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo
systems
Realizability algebras: a program to well order R
The theory of classical realizability is a framework in which we can develop
the proof-program correspondence. Using this framework, we show how to
transform into programs the proofs in classical analysis with dependent choice
and the existence of a well ordering of the real line. The principal tools are:
The notion of realizability algebra, which is a three-sorted variant of the
well known combinatory algebra of Curry. An adaptation of the method of forcing
used in set theory to prove consistency results. Here, it is used in another
way, to obtain programs associated with a well ordering of R and the existence
of a non trivial ultrafilter on N
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