256 research outputs found
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
Automatització industrial a les indústries
L'objectiu d'aquest article és oferir una visió de l'automatització industrial a les indústries.Donem a conèixer els components més importants que l'integren, el seu comportament, tenint com a element fonamental de control l'autòmat programable, del qual coneixerem el funcionament i el comportament en un sistema industrial automatitzat.The aim of the present article is to give an overview on the industrial automation at industries. Their main components and behaviour, keeping the programmable automaton as the basic control element, which their behaviour and performance within an automated industrial system is shown
Exploring the role of materials in policy change: innovation in low energy housing in the UK
We find and prove new Pohozaev identities and integration by parts type formulas for anisotropic integrodifferential operators of order 2s, with s¿(0,1). These identities involve local boundary terms, in which the quantity (Formula presented.) plays the role that ¿u/¿¿ plays in the second-order case. Here, u is any solution to Lu = f(x,u) in O, with u = 0 in RnO, and d is the distance to ¿O.Peer ReviewedPostprint (author's final draft
Sharp isoperimetric inequalities via the ABP
Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in Rn satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativeness of a Bakry-Emery Ricci tensor. Even that our weights are nonradial, still balls are minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and PacellaPeer ReviewedPostprint (published version
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