arXiv subjects

Augusto C. Ponce

Publications and source records attributed to Augusto C. Ponce.

33 records · Page 2Linked to original sources

arXiv2017

Density of bounded maps in Sobolev spaces into complete manifolds

Given a complete noncompact Riemannian manifold $N^n$, we investigate whether the set of bounded Sobolev maps $(W^{1, p} \cap L^\infty) (Q^m; N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1, p} (Q^m; N^n)$ for $1 \le p \le m$. The density always holds when $p$ is not an integer. When $p$ is an integer, the density can fail, and we prove that a quantitative trimming property is equivalent with the density. This new condition is ensured for example by a uniform Lipschitz geometry of $N^n$. As a byproduct, we give necessary and sufficient conditions for the strong density of the set of smooth maps $C^\infty (\overline{Q^m}; N^n)$ in $W^{1, p} (Q^m; N^n)$.

math.FA↗

arXiv2017

Flat solutions of the 1-Laplacian equation

For every $f \in L^N(Ω)$ defined in an open bounded subset $Ω$ of $\mathbb{R}^N$, we prove that a solution $u \in W_0^{1, 1}(Ω)$ of the $1$-Laplacian equation ${-}\mathrm{div}{(\frac{\nabla u}{|\nabla u|})} = f$ in $Ω$ satisfies $\nabla u = 0$ on a set of positive Lebesgue measure. The same property holds if $f \not\in L^N(Ω)$ has small norm in the Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.

math.AP↗

arXiv2017

Gauge-measurable functions

In 1973, E.J. McShane proposed an alternative definition of the Lebesgue integral based on Riemann sums, where gauges are used decide what tagged partitions are allowed. Such an approach does not require any preliminary knowledge of Measure theory. We investigate in this paper a definition of measurable functions also based on gauges. Its relation to the gauge-integrable functions that satisfy McShane's definition is obtained using elementary tools from Real Analysis. We show in particular a dominated integration property of gauge-measurable functions.

math.CA↗

arXiv2017

Weak approximation by bounded Sobolev maps with values into complete manifolds

We have recently introduced the trimming property for a complete Riemannian manifold $N^{n}$ as a necessary and sufficient condition for bounded maps to be strongly dense in $W^{1, p}(B^m; N^{n})$ when $p \in \{1, \dotsc, m\}$. We prove in this note that even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.

math.FA↗

arXiv2016

The role of interplay between coefficients in the $G$-convergence of some elliptic equations

We study the behavior of the solutions $u$ of the linear Dirichlet problems $- \mathrm{div} (M(x) \nabla u) + a(x) u = f(x)$ with respect to perturbations of the matrix $M(x)$ (with respect to the $G$-convergence) and with respect to perturbations of the nonnegative coefficient $a(x)$ and of the right hand side $f(x)$ satisfying the condition $|f (x)| \leq Q \, a (x)$.

math.AP↗

arXiv2016

On formulae decoupling the total variation of BV functions

In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non-local functionals. One special case shows that rescalings of the fractional Laplacian of a function $u\in SBV$ converge strictly to the singular portion of $Du$.

math.FA↗

arXiv2014

Strong maximum principle for Schrödinger operators with singular potential

We prove that for every $p > 1$ and for every potential $V \in L^p$, any nonnegative function satisfying $-Δu + V u \ge 0$ in an open connected set of $\mathbb{R}^N$ is either identically zero or its level set $\{u = 0\}$ has zero $W^{2, p}$ capacity. This gives an affirmative answer to an open problem of Bénilan and Brezis concerning a bridge between Serrin-Stampacchia's strong maximum principle for $p > \frac{N}{2}$ and Ancona's 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 Schrödinger operator with diffuse measure data.

math.AP↗

arXiv2014

Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell$ simply connected manifolds when $s \ge 1$

Given a compact manifold $N^n \subset \mathbb{R}^ν$, $s \ge 1$ and $1 \le p < \infty$, we prove that the class of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when $N^n$ is $\lfloor sp \rfloor$ simply connected. For $sp$ integer, we prove weak density of smooth maps with values into $N^n$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of $\mathbb{R}^ν$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, sp}$.

math.FA↗

arXiv2013

Nonlinear elliptic equations with measures revisited

We study the existence of solutions of the nonlinear problem $$ \left\{ \begin{alignedat}{2} -Δu + g(u) & = μ& & \quad \text{in } Ω,\\ u & = 0 & & \quad \text{on } \partial Ω, \end{alignedat} \right. $$ where $μ$ is a Radon measure and $g : \mathbb{R} \to \mathbb{R}$ is a nondecreasing continuous function with $g(0) = 0$. This equation need not have a solution for every measure $μ$, and we say that $μ$ is a good measure if the Dirichlet problem above admits a solution. We show that for every $μ$ there exists a largest good measure $μ^* \leq μ$. This reduced measure has a number of remarkable properties.

math.AP↗

arXiv2013

Kato's inequality when $Δu$ is a measure

We extend the classical Kato's inequality in order to allow functions $u \in L^1_\mathrm{loc}$ such that $Δu$ is a Radon measure. This inequality has been applied by Brezis, Marcus, and Ponce to study the existence of solutions of the nonlinear equation $- Δu + g(u) = μ$, where $μ$ is a measure and $g : \mathbb{R} \to \mathbb{R}$ is an increasing continuous function.

math.AP↗

arXiv2013

Strong approximation of fractional Sobolev maps

Brezis and Mironescu have announced several years ago that for a compact manifold $N^n \subset \mathbb{R}^ν$ and for real numbers $0 < s < 1$ and $1 \le p < \infty$ 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 $W^{s, p}(Q^m; N^n)$ when the homotopy group $π_{\lfloor sp \rfloor}(N^n)$ of order $\lfloor sp \rfloor$ is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that $N^n$ is $\lfloor sp \rfloor$ simply connected, we give a shorter proof of their result. Our proof for $sp \ge 1$ is based on the existence of a retraction of $\mathbb{R}^ν$ onto $N^n$ except for a small subset in the complement of $N^n$ and on the Gagliardo-Nirenberg interpolation inequality for maps in $W^{1, q} \cap L^\infty$. In contrast, the case $sp < 1$ relies on the density of step functions on cubes in $W^{s, p}$.

math.FA↗

arXiv2012

Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite measures which do not charge sets of sigma-finite Hausdorff measure H^{N-1} can be written as an L^1 perturbation of the divergence of a continuous vector field. The main tool is a property of approximation of measures in terms of the Hausdorff content.

math.AP↗

arXiv2012

Limit solutions of the Chern-Simons equation

We investigate the scalar Chern-Simons equation $-Δu + e^u(e^u-1) = μ$ in cases where there is no solution for a given nonnegative finite measure $μ$. Approximating $μ$ by a sequence of nonnegative $L^1$ functions or finite measures for which this equation has a solution, we show that the sequence of solutions of the Dirichlet problem converges to the solution with largest possible datum $μ^# \le μ$ and we derive an explicit formula of $μ^#$ in terms of $μ$. The counterpart for the Chern-Simons system with datum $(μ, ν)$ behaves differently and the conclusion depends on how much the measures $μ$ and $ν$ charge singletons.

math.AP↗

arXiv2009

Isolated Boundary Singularities of Semilinear Elliptic Equations

Given a smooth domain $Ω\subset\RR^N$ such that $0 \in \partialΩ$ and given a nonnegative smooth function $ζ$ on $\partialΩ$, we study the behavior near 0 of positive solutions of $-Δu=u^q$ in $Ω$ such that $u = ζ$ on $\partialΩ\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x \to 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.

math.AP↗

arXiv2009

Closure of Smooth Maps in $W^{1,p}(B^3;S^2)$

For every $2 < p < 3$, we show that $u \in W^{1,p}(B^3; S^2)$ can be strongly approximated by maps in $C^\infty(\Bar{B}^3; S^2)$ if, and only if, the distributional Jacobian of $u$ vanishes identically. This result was originally proved by Bethuel-Coron-Demengel-Helein, but we present a different strategy which is motivated by the $W^{2,p}$-case.

math.FA↗