Eduardo V. Teixeira

On the behavior of weak convergence under nonlinearities and applications

This paper provides a sufficient condition to guarantee the stability of weak limits under nonlinear operators acting on vector-valued Lebesgue spaces. This nonlinear framework places the weak convergence in perspective. Such an approach allows short and insightful proofs of important results in Functional Analysis such as: weak convergence in L∞ implies strong convergence in L p for all 1 ≤ p < oo, weak convergence in L 1 vs. strong convergence in L 1 and the Brezis-Lieb theorem. The final goal is to use this framework as a strategy to grapple with a nonlinear weak spectral problem on W 1,p .

Asymptotic behavior of solutions to the σk-Yamabe equation near isolated singularities

Abstractσk-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In (J. Funct. Anal. 233: 380–425, 2006) YanYan Li proved that an admissible solution with an isolated singularity at 0∈ℝn to the σk-Yamabe equation is asymptotically radially symmetric. In this work we prove that such a solution is asymptotic to a radial solution to the same equation on ℝn∖{0}. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al., we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and σk curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.

Towards sharp Bohnenblust–Hille constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust--Hille type inequalities for

Weak convergence under nonlinearities

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Geometric Approach to Nonvariational Singular Elliptic Equations

--linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust--Hille inequality are universally bounded, irrespectively of the value of

A limiting free boundary problem ruled by Aronsson’s equation

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Sharp regularity estimates for second order fully nonlinear parabolic equations

; hereafter referred as the \textit{Universality Conjecture}. In our approach, we introduce the {notions of entropy and complexity}, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to

Regularity and geometric estimates for minima of discontinuous functionals

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Free Transmission Problems

, then the optimal constants of the

Regularity estimates for fully non linear elliptic equations which are asymptotically convex

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