Xavier Ros-Oton

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 2s, with s ∈ (0,1). We consider the class of nonlocal operators L∗ ⊂L 0, which consists of all the infinitesimal generators of stable Levy processes belonging to the class L0 of Caffarelli-Silvestre. For fully nonlinear operators I elliptic with respec tt oL∗ ,w e prove that solutions to Iu = f in Ω, u = 0 in R n \ Ω, satisfy u/d s ∈ C s−ϵ (Ω) for all ϵ> 0, where d is the distance to ∂ Ωa ndf ∈ L ∞ . We expect the Holder exponent s − ϵ to be optimal (or almost optimal) for general right hand sides f ∈ L ∞ . Moreover, we also expect the class L∗ to be the largest scale invariant subclass of L0 for which this result is true. In this direction, we show that the class L0 is too large for all solutions to behave like d s . The constants in all the estimates in this paper remain bounded as th eo rder of the equation approaches 2.

Nonlocal elliptic equations in bounded domains : a survey

In this paper we survey some results on the Dirichlet problem ( Lu = f in u = g in R n n for nonlocal operators of the form Lu(x) = PV Z Rn u(x) u(x + y) K(y)dy: 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 L 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 they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the eld. 2010 Mathematics Subject Classication: 47G20, 60G52, 35B65.

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 Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| n+σ is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case a ij ≡ 0 and for σ = 2 in case that (a ij ) is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for L in star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation to n and σ). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian (− Δ) s (here s > 1) or the fractional p-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin.

Stable inversion of Abel equations

Stable inversion plays a key role in the solution of the exact tracking control problem in nonminimum phase systems. However, the general methods developed so far for the computation of stable inverses require backwards time numeric integration of the internal dynamics equation, which yields high sensitivity to external disturbances and/or structured uncertainties. This article introduces an iterative technique that provides periodic, closed-form analytic expressions uniformly convergent to the exact periodic solution of a certain class of Abel ODE written in the normal form. The method is then applied to the output voltage tracking of periodic references in DC-DC boost power converters through a state feedback indirect control scheme. The procedure lies on a number of assumptions for which sufficient conditions involving system parameters and reference candidates are derived. It also allows one to attenuate the effect of bounded, piecewise constant load disturbances using dynamic compensation. Simulation results validate the proposed algorithm.

Sharp isoperimetric inequalities via the ABP method

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of

REGULARITY OF STABLE SOLUTIONS UP TO DIMENSION 7 IN DOMAINS OF DOUBLE REVOLUTION

\mathbb{R}^n

Pohozaev identities for anisotropic integrodifferential operators

. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. 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 Pacella.

Periodic solutions with nonconstant sign in Abel equations of the second kind

We consider the class of semi-stable positive solutions to semilinear equations − Δu = f(u) in a bounded domain Ω ⊂ ℝ n of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n − m variables. We assume 2 ≤ m ≤ n − 2. When the domain is convex, we establish a priori L p and bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead to the boundedness of the extremal solution of − Δu = λf(u) in every convex domain of double revolution when n ≤ 7. The boundedness of extremal solutions is known when n ≤ 4 for any domain Ω, and in dimensions 5 ≤ n ≤ 9 in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities f in dimensions 5 ≤ n ≤ 9.

Higher-order boundary regularity estimates for nonlocal parabolic equations

ABSTRACT 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 plays the role that ∂u∕∂ν plays in the second-order case. Here, u is any solution to Lu = f(x,u) in Ω, with u = 0 in ℝn∖Ω, and d is the distance to ∂Ω.

Obstacle problems and free boundaries: an overview

Abstract The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation x ↦ x − 1 , and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.