1,950 research outputs found
Nonlocal elliptic equations in bounded domains: a survey
In this paper we survey some results on the Dirichlet problem for nonlocal operators of the form
We
start from the very basics, proving existence of solutions, maximum principles,
and constructing some useful barriers. Then, we focus on the regularity
properties of solutions, both in the interior and on the boundary of the
domain.
In order to include some natural operators in the regularity theory, we
do not assume any regularity on the kernels. This leads to some interesting
features that are purely nonlocal, in the sense that have no analogue for local
equations.
We hope that this survey will be useful for both novel and more experienced
researchers in the field
Boundary regularity, Pohozaev identities, and nonexistence results
In this expository paper we survey some recent results on Dirichlet problems
of the form in , in . We first discuss in detail the boundary regularity of
solutions, stating the main known results of Grubb and of the author and Serra.
We also give a simplified proof of one of such results, focusing on the main
ideas and on the blow-up techniques that we developed in \cite{RS-K,RS-stable}.
After this, we present the Pohozaev identities established in
\cite{RS-Poh,RSV,Grubb-Poh} and give a sketch of their proofs, which use
strongly the fine boundary regularity results discussed previously. Finally, we
show how these Pohozaev identities can be used to deduce nonexistence of
solutions or unique continuation properties.
The operators under consideration are integro-differential operator of
order , , the model case being the fractional Laplacian
.Comment: Survey article. To appear as a chapter in "Recent Developments in the
Nonlocal Theory" by De Gruyte
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.
Regularity of stable solutions up to dimension 7 in domains of double revolution
We consider the class of semi-stable positive solutions to semilinear
equations in a bounded domain of
double revolution, that is, a domain invariant under rotations of the first
variables and of the last variables. We assume . When
the domain is convex, we establish a priori and bounds for each
dimension , with when . These estimates lead to the
boundedness of the extremal solution of in every
convex domain of double revolution when . The boundedness of extremal
solutions is known when for any domain , in dimension
when the domain is convex, and in dimensions in the radial case.
Except for the radial case, our result is the first partial answer valid for
all nonlinearities in dimensions
Entire solutions to semilinear nonlocal equations in \RR^2
We consider entire solutions to in \RR^2, where is a
general nonlocal operator with kernel . Under certain natural assumtions
on the operator , we show that any stable solution is a 1D solution. In
particular, our result applies to any solution which is monotone in one
direction. Compared to other proofs of the De Giorgi type results on nonlocal
equations, our method is the first successfull attempt to use the Liouville
theorem approach to get flatness of the level sets
Social and Political Dimensions of Identity
We study the interior regularity of solutions to the Dirichlet problem Lu = g in Omega, u = 0 in R-nOmega, for anisotropic operators of fractional type Lu(x) = integral(+infinity)(0) dp integral(Sn-1) da(w) 2u(x) - u(x + rho w) - u(x - rho w)/rho(1+2s). Here, a is any measure on Sn-1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a is an element of C-infinity(Sn-1) and g is c(infinity)(Omega), solutions are known to be C-infinity inside Omega (but not up to the boundary). However, when a is a general measure, or even when a is L-infinity(s(n-1)), solutions are only known to be C-3s inside Omega. We prove here that, for general measures a, solutions are C1+3s-epsilon inside Omega for all epsilon > 0 whenever Omega is convex. When a is an element of L-infinity(Sn-1), we show that the same holds in all C-1,C-1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C3s+epsilon for any epsilon > 0 - even if g and Omega are C-infinity
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