380 research outputs found
regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels
We establish interior estimates for concave nonlocal
fully nonlinear equations of order with rough kernels. Namely,
we prove that if solves in a concave
translation invariant equation with kernels in , then
belongs to , with an estimate. More
generally, our results allow the equation to depend on in a
fashion.
Our method of proof combines a Liouville theorem and a blow-up (compactness)
procedure. Due to its flexibility, the same method can be useful in different
regularity proofs for nonlocal equations
A new relation between the zero of in and the anomaly in
We present two exact relations, valid for any dilepton invariant mass region
(large and low-recoil) and independent of any effective Hamiltonian
computation, between the observables and of the angular
distribution of the 4-body decay . These relations
emerge out of the symmetries of the angular distribution. We discuss the
implications of these relations under the (testable) hypotheses of no scalar or
tensor contributions and no New Physics weak phases in the Wilson coefficients.
Under these hypotheses there is a direct relation among the observables
, and . This can be used as an independent
consistency test of the measurements of the angular observables. Alternatively,
these relations can be applied directly in the fit to data, reducing the number
of free parameters in the fit. This opens up the possibility to perform a full
angular fit of the observables with existing datasets. An important consequence
of the found relations is that a priori two different measurements, namely the
measured position of the zero () of the forward-backward asymmetry
and the value of evaluated at this same point, are
related by . Under the hypotheses of real
Wilson coefficients and being SM-like, we show that the higher the
position of the smaller should be the value of evaluated
at the same point. A precise determination of the position of the zero of
together with a measurement of (and ) at this
position can be used as an independent experimental test of the anomaly in
. We also point out the existence of upper and lower bounds for
, namely , which
constraints the physical region of the observables.Comment: 5 pages, 3 figure
Local integration by parts and Pohozaev identities for higher order fractional Laplacians
We establish an integration by parts formula in bounded domains for the
higher order fractional Laplacian with . We also obtain the
Pohozaev identity for this operator. Both identities involve local boundary
terms, and they extend the identities obtained by the authors in the case
.
As an immediate consequence of these results, we obtain a unique continuation
property for the eigenfunctions in ,
in .Comment: The sign of the boundary term in Theorem 1.5 has been correcte
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.
Nonexistence results for nonlocal equations with critical and supercritical nonlinearities
We prove nonexistence of nontrivial bounded solutions to some nonlinear
problems involving nonlocal operators of the form These operators
are infinitesimal generators of symmetric L\'evy processes. Our results apply
to even kernels satisfying that is nondecreasing along
rays from the origin, for some in case and for
in case that is a positive definite symmetric matrix.
Our nonexistence results concern Dirichlet problems for in star-shaped
domains with critical and supercritical nonlinearities (where the criticality
condition is in relation to and ).
We also establish nonexistence of bounded solutions to semilinear equations
involving other nonlocal operators such as the higher order fractional
Laplacian (here ) or the fractional -Laplacian. All these
nonexistence results follow from a general variational inequality in the spirit
of a classical identity by Pucci and Serrin
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