1,604 research outputs found
A Regularity Result for the p-Laplacian Near Uniform Ellipticity
We consider weak solutions to a class of Dirichlet boundary value problems involving the -Laplace operator, and prove that the second weak derivatives have summability as high as it is desirable, provided p is sufficiently close to 2. And as a consequence the Holder exponent of the gradients approaches 1. We show that this phenomenon is driven by the classical Calderon-Zygmund constant. We believe that this result is particularly interesting in higher dimensions, and it is related to the optimal regularity of -harmonic mappings. which is still an open question
Remarks on regularity for -Laplacian type equations in non-divergence form
We study a singular or degenerate equation in non-divergence form modeled by
the -Laplacian, We investigate local
regularity of viscosity solutions in the full range and , and
provide local estimates in the restricted cases where is close to
2 and is close to 0.Comment: 38 page
Boundary regularity for fully nonlinear integro-differential equations
We study fine boundary regularity properties of solutions to fully nonlinear
elliptic integro-differential equations of order , with .
We consider the class of nonlocal operators , which consists of infinitesimal generators of stable L\'evy processes
belonging to the class of Caffarelli-Silvestre. For fully
nonlinear operators elliptic with respect to , we prove that
solutions to in , in ,
satisfy , where is the distance to
and .
We expect the class to be the largest scale invariant subclass
of for which this result is true. In this direction, we show
that the class is too large for all solutions to behave like
.
The constants in all the estimates in this paper remain bounded as the order
of the equation approaches 2. Thus, in the limit we recover the
celebrated boundary regularity result due to Krylov for fully nonlinear
elliptic equations.Comment: To appear in Duke Math.
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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