Fernando Charro

Limit Branch of Solutions as p → ∞ for a Family of Sub-Diffusive Problems Related to the p-Laplacian

We characterize the limit as p → ∞ of the branches of solutions to with λ > 0 and , where R < 1. We show that the limit set is a curve of positive viscosity solutions of the equation where and Λ > 0. The key result is a comparison principle for the limit equation from which we deduce uniqueness for the Dirichlet problem and hence the existence of the curve of solutions.

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS: COMPARISON, EXISTENCE AND UNIQUENESS

We study existence of solutions to where F is elliptic and homogeneous of degree m, and either f(λ,u) = λ uq or f(λ,u) = λ uq + ur, for 0 0. Furthermore, in the first case, we obtain that the solution is unique as a consequence of a comparison principle up to the boundary. Several examples, including uniformly elliptic operators and the infinity-laplacian, are considered.

On a Fractional Monge-Ampère Operator

In this paper we consider a fractional analogue of the Monge–Ampère operator. Our operator is a concave envelope of fractional linear operators of the form

Ireneo Peral: Forty Years as Mentor

\inf _{A\in \mathcal {A}}L_Au,

Monotonicity of solutions of Fully nonlinear uniformly elliptic equations in the half-plane☆

infA∈ALAu, where the set of operators is a degenerate class that corresponds to all affine transformations of determinant one of a given multiple of the fractional Laplacian. We set up a relatively simple framework of global solutions prescribing data at infinity and global barriers. In our key estimate, we show that the operator remains strictly elliptic, which allows to apply known regularity results for uniformly elliptic operators and deduce that solutions are classical.

On the Aleksandrov-Bakel'man-Pucci Estimate for Some Elliptic and Parabolic Nonlinear Operators

We study the following boundary value problem with a concave–convex nonlinearity: Here Ω ⊂ ℝn is a bounded domain and 1 0 such that the problem admits at least two positive solutions for 0 Λq, r. We show that where λ1(p) is the first eigenvalue of the p-Laplacian. It is worth noticing that λ1(p) is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case q = p.

Limits as p → ∞ of p-Laplacian problems with a superdiffusive power-type nonlinearity: Positive and sign-changing solutions

Abstract In this article we present a survey of the Ph.D. theses that have been completed under the advice of Ireneo Peral. Following a chronological order, we summarize the main results contained in the works of the former students of Ireneo Peral.