1,215 research outputs found
A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces
In this paper we consider scalar parabolic equations in a general non-smooth
setting with emphasis on mixed interface and boundary conditions. In
particular, we allow for dynamics and diffusion on a Lipschitz interface and on
the boundary, where diffusion coefficients are only assumed to be bounded,
measurable and positive semidefinite. In the bulk, we additionally take into
account diffusion coefficients which may degenerate towards a Lipschitz
surface. For this problem class, we introduce a unified functional analytic
framework based on sesquilinear forms and show maximal regularity for the
corresponding abstract Cauchy problem.Comment: 27 pages, 4 figure
Fractional elliptic equations, Caccioppoli estimates and regularity
Let be a uniformly elliptic operator
in divergence form in a bounded domain . We consider the fractional
nonlocal equations Here , , is the fractional power of and
is the conormal derivative of with respect to the
coefficients . We reproduce Caccioppoli type estimates that allow us to
develop the regularity theory. Indeed, we prove interior and boundary Schauder
regularity estimates depending on the smoothness of the coefficients ,
the right hand side and the boundary of the domain. Moreover, we establish
estimates for fundamental solutions in the spirit of the classical result by
Littman--Stampacchia--Weinberger and we obtain nonlocal integro-differential
formulas for . Essential tools in the analysis are the semigroup
language approach and the extension problem.Comment: 37 page
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