Augusto C. Ponce

An estimate in the spirit of Poincaré's inequality

We show that if Omega subset of R-N, N greater than or equal to 2, is a bounded Lipschitz domain and (rho(n)) subset of L-1(R-N) is a sequence of nonnegative radial functions weakly converging to delta(0), then integral(Omega) |f - f(Omega)|(p) less than or equal to C integral(Omega)integral(Omega) |f(x)-f(y)|(p)/|x-y|(p) rho(n)(|x-y|)dx dy for all f is an element of L-P (Omega) and n greater than or equal to n(0), where f(Omega) denotes the average of f on Omega. The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu [2]. As n --> infinity we recover Poincares inequality. The case N = I requires an additional assumption on (rho(n)). We also extend a compactness result of Bourgain, Brezis and Mironescu.

The sub-supersolution method for weak solutions

We extend the method of sub and supersolutions in order to prove existence of L-1-solutions of the equation -Delta u = f(x, u) in Omega, where f is a Caratheodory function. The proof is based on Schauders fixed point theorem.

Elliptic equations with vertical asymptotes in the nonlinear term

We study the existence of solutions of the nonlinear problem {fx349-1} where μ is a bounded measure andg is a continuous nondecreasing function such thatg(0)=0. In this paper, we assume that the nonlinearityg satisfies {fx349-2} Problem (0.1) need not have a solution for every measure μ. We prove that, given μ, there exists a “closest” measure μ* for which (0.1) can be solved. We also explain how assumption (0.2) makes problem (0.1) different from the case whereg(t) is defined for everyt ∈ ℝ.

Kato's Inequality Up To the Boundary

We show that if Delta u is a finite measure in Omega then, under suitable assumptions on u near partial derivative Omega, Delta u(+) is also a finite measure in Omega. We also study properties of the normal derivatives partial derivative u/partial derivative n and partial derivative u(+)/partial derivative n on partial derivative Omega.

Variants of Kato's inequality and removable singularities

An inequality reminiscent of Kato’s inequality is presented. Motivated by this, we discuss some criteria to decide whether a singularity of the equation Δu=g in Ω/K comes from a Radon measure or not. As an application, we extend a lemma of H. Brezis and P. L. Lions on isolated singularities to the case where the singularity lies on a compact manifold.

Strong maximum principle for Schrödinger operators with singular potential

We prove that for every p>1 and for every potential V∈Lp, any nonnegative function satisfying −Δu+Vu≥0 in an open connected set of RN is either identically zero or its level set {u=0} has zero W2,p capacity. This gives an affirmative answer to an open problem of Benilan and Brezis concerning a bridge between Serrin–Stampacchias strong maximum principle for p>N2 and Anconas strong maximum principle for p=1. The proof is based on the construction of suitable test functions depending on the level set {u=0}, and on the existence of solutions of the Dirichlet problem for the Schrodinger operator with diffuse measure data.

Strong density for higher order Sobolev spaces into compact manifolds

Given a compact manifold \(N^n\), an integer \(k \in \mathbb{N}_*\) and an exponent \(1 \le p < \infty\), we prove that the class \(C^\infty(\overline{Q}^m; N^n)\) of smooth maps on the cube with values into \(N^n\) is dense with respect to the strong topology in the Sobolev space \(W^{k, p}(Q^m; N^n)\) when the homotopy group \(\pi_{\lfloor kp \rfloor}(N^n)\) of order \(\lfloor kp \rfloor\) is trivial. We also prove the density of maps that are smooth except for a set of dimension \(m - \lfloor kp \rfloor - 1\), without any restriction on the homotopy group of \(N^n\).

How to Construct Good Measures

Given any continuous nondecreasing function g : \(\mathbb{R} \to \mathbb{R}\), with g(t) = 0, ∀t ≤ 0, we show that there always exists some positive measure μ, concentrated on a set of zero Newtonian capacity, for which the problem

On formulae decoupling the total variation of BV functions

\left\{ {\begin{array}{*{20}c} { - \Delta u + g\left( u \right) = \mu } & {in\;\Omega ,} \\ {\quad \quad \quad \quad u = 0} & {\quad on\;\partial \Omega ,} \\ \end{array} } \right.