arXiv subjects
Publications and source records attributed to Augusto C. Ponce.
We investigate a class of nonlocal gradients featuring distinct homogeneities at zero and infinity. We establish a representation formula for such doubly homogeneous operators and derive associated Sobolev-type inequalities. We also propose open questions linked to our results, suggesting directions for future research inspired by the work of Haim Brezis.
Given a parabolic cylinder $Q =(0,T)\timesΩ$, where $Ω\subset \mathbb{R}^{N}$ is a bounded domain, we prove new properties of solutions of \[ u_t-Δ_p u = μ\quad \text{in $Q$} \] with Dirichlet boundary conditions, where $μ$ is a finite Radon measure in $Q$. We first prove a priori estimates on the $p$-parabolic capacity of level sets of $u$. We then show that diffuse measures (i.e.\@ measures which do not charge sets of zero parabolic $p$-capacity) can be strongly approximated by the measures $μ_k = (T_k(u))_t-Δ_p(T_k(u))$, and we introduce a new notion of renormalized solution based on this property. We finally apply our new approach to prove the existence of solutions of $$ u_t-Δ_{p} u + h(u)=μ\quad \text{in $Q$,} $$ for any function $h$ such that $h(s)s\geq 0$ and for any diffuse measure $μ$; when $h$ is nondecreasing we also prove uniqueness in the renormalized formulation. Extensions are given to the case of more general nonlinear operators in divergence form.
We prove an integral representation formula for the distributional normal derivative of solutions of $$ \left\{ \begin{aligned} - Δu + V u &= μ&& \text{in $Ω$,}\\ u &= 0 && \text{on $\partialΩ$,} \end{aligned} \right. $$ where $V \in L_{\mathrm{loc}}^1(Ω)$ is a nonnegative function and $μ$ is a finite Borel measure on $Ω$. As an application, we show that the Hopf lemma holds almost everywhere on $\partialΩ$ when $V$ is a nonnegative Hopf potential.
This manuscript develops a framework for the strong approximation of Sobolev maps with values in compact manifolds, emphasizing the interplay between local and global topological properties. Building on topological concepts adapted to VMO maps, such as homotopy and the degree of continuous maps, it introduces and analyzes extendability properties, focusing on the notions of $\ell$-extendability and its generalization, $(\ell, e)$-extendability. We rely on Fuglede maps, providing a robust setting for handling compositions with Sobolev maps. Several constructions -- including opening, thickening, adaptive smoothing, and shrinking -- are carefully integrated into a unified approach that combines homotopical techniques with precise quantitative estimates. Our main results establish that a Sobolev map $u \in W^{k, p}$ defined on a compact manifold of dimension $m > kp$ can be approximated by smooth maps if and only if $u$ is $(\lfloor kp \rfloor, e)$-extendable with $e = m$. When $e < m$, the approximation can still be carried out using maps that are smooth except on structured singular sets of rank $m - e - 1$.
We present results for Choquet integrals with minimal assumptions on the monotone set function through which they are defined. They include the equivalence of sublinearity and strong subadditivity independent of regularity assumptions on the capacity, as well as various forms of standard measure theoretic convergence theorems for these non-additive integrals, e.g. Fatou's lemma and Lebesgue's dominated convergence theorem.
Given a finite nonnegative Borel measure $m$ in $\mathbb{R}^{d}$, we identify the Lebesgue set $\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}$ of the vector-valued function $$V_{s}(x) = \int_{\mathbb{R}^{d}}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \in \mathcal{L}(V_{s})$ if and only if the integral above has a principal value at $a$ and $$\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the precise representative of $V_{s}$ at $a$ coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.
We prove that each Borel function $V : Ω\to [-\infty, +\infty]$ defined on an open subset $Ω\subset \mathbb{R}^{N}$ induces a decomposition $Ω= S \cup \bigcup_{i} D_{i}$ such that every function in $W^{1,2}_{0}(Ω) \cap L^{2}(Ω; V^{+} dx)$ is zero almost everywhere on $S$ and existence of nonnegative supersolutions of $-Δ+ V$ on each component $D_{i}$ yields nonnegativity of the associated quadratic form $\int_{D_{i}} (|\nabla ξ|^2+Vξ^2)$.
We prove that every finite Borel measure $μ$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $μ\lfloor_{E_k}$ that are bounded from above by the Hausdorff content $\mathcal{H}_\infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to show the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.
We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator $- Δ+ V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(Ω)$. More precisely, if $u \in W_0^{1, 2}(Ω) \cap L^2(Ω; V \mathrm{d}x)$ satisfies $- Δu + V u = f$ on $Ω$ for some nonnegative datum $f \in L^\infty(Ω)$, $f \not\equiv 0$, then we show that at every point $a \in \partialΩ$ where the classical normal derivative $\partial u(a) / \partial n$ exists and satisfies the Poisson representation formula, one has $\partial u(a) / \partial n > 0$ if and only if the boundary value problem $$ \begin{cases} \begin{aligned} - Δv + V v &= 0 && \text{in $Ω$,} \\ v &= ν&& \text{on $\partialΩ$,} \end{aligned} \end{cases} $$ involving the Dirac measure $ν= δ_a$ has a solution. More generally, we characterize the nonnegative finite Borel measures $ν$ on $\partialΩ$ for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.
We establish that for every function $u \in L^1_\mathrm{loc}(Ω)$ whose distributional Laplacian $Δu$ is a signed Borel measure in an open set $Ω$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $Ω$ with respect to the weak-$L^{\frac{N}{N-1}}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $Δu$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u = α\}$ and $\{\nabla u = e\}$, for every $α\in \mathbb{R}$ and $e \in \mathbb{R}^N$. Our proofs rely on an adaptation of Calderón and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.
Given any Borel function $V : Ω\to [0, +\infty]$ on a smooth bounded domain $Ω\subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schrödinger operator $-Δ+ V$ in $Ω$ holds in each Sobolev-connected component of $Ω\setminus Z$, where $Z \subset Ω$ is the set of points which cannot carry a Green's function for $- Δ+ V$. More generally, we show that the equation $- Δu + V u = μ$ has a distributional solution in $W_{0}^{1, 1}(Ω)$ for a nonnegative finite Borel measure $μ$ if and only if $μ(Z) = 0$.
We establish the Hopf boundary point lemma for the Schrödinger operator $-Δ+ V$ involving potentials $V$ that merely belong to the space $L^{1}_{loc}(Ω)$. More precisely, we prove that among all supersolutions $u$ of $-Δ+ V$ which vanish on the boundary $\partialΩ$ and are such that $V u \in L^{1}(Ω)$, if there exists one supersolution which satisfies $\partial u/\partial n < 0$ almost everywhere on $\partialΩ$ with respect to the outward unit vector $n$, then such a property holds for every nontrivial supersolution in the same class. We rely on the existence of nontrivial solutions of the nonhomogeneous Dirichlet problem with boundary datum in $L^{\infty}(\partialΩ)$.
We show how a strong capacitary inequality can be used to give a decomposition of any function in the Sobolev space $W^{k,1}(\mathbb{R}^d)$ as the difference of two non-negative functions in the same space with control of their norms.
We prove the Boxing inequality: $$\mathcal{H}^{d-α}_\infty(U) \leq Cα(1-α)\int_U \int_{\mathbb{R}^{d} \setminus U} \frac{\mathrm{d}y \, \mathrm{d}z}{|y-z|^{α+d}},$$ for every $α\in (0,1)$ and every bounded open subset $U \subset \mathbb{R}^d$, where $\mathcal{H}^{d-α}_\infty(U)$ is the Hausdorff content of $U$ of dimension $d -α$ and the constant $C > 0$ depends only on $d$. We then show how this estimate implies a trace inequality in the fractional Sobolev space $W^{α, 1}(\mathbb{R}^d)$ that includes Sobolev's $L^{\frac{d}{d - α}}$ embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as $α$ tends to $0$ and $1$, recovering in particular the classical inequalities of first order. Their counterparts in the full range $α\in (0, d)$ are also investigated.
We study the minimization of the cost functional \[ F(μ) = \lVert u - u_d \rVert_{L^p(Ω)} + α\lVert μ\rVert_{\mathcal{M}(Ω)}, \] where the controls $μ$ are taken in the space of finite Borel measures and $u \in W_0^{1, 1}(Ω)$ satisfies the equation $- Δu + g(u) = μ$ in the sense of distributions in $Ω$ for a given nondecreasing continuous function $g : \mathbb{R} \to \mathbb{R}$ such that $g(0) = 0$. We prove that $F$ has a minimizer for every desired state $u_d \in L^1(Ω)$ and every control parameter $α> 0$. We then show that when $u_d$ is nonnegative or bounded, every minimizer of $F$ has the same property.
We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces $W^{α, 1}$ of order $0 < α< 1$.
We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data $$ \left\{ \begin{alignedat}{2} -Δu + Vu & = μ&& \quad \text{in } Ω,\\ u & = 0 && \quad \text{on } \partial Ω. \end{alignedat} \right. $$ We characterize the finite measures $μ$ for which this problem has a solution for every nonnegative potential $V$ in the Lebesgue space $L^p(Ω)$ with $1 \le p \le N/2$. The full answer can be expressed in terms of the $W^{2,p}$ capacity for $p > 1$, and the $W^{1,2}$ (or Newtonian) capacity for $p = 1$. We then prove the existence of a solution of the problem above when $V$ belongs to the real Hardy space $H^1(Ω)$ and $μ$ is diffuse with respect to the $W^{2,1}$ capacity.
This monograph is the core of my book "Elliptic PDEs, Measures and Capacities: From the Poisson equation to Nonlinear Thomas-Fermi Problems" which has received the 2014 EMS Monograph Award and is available in the series EMS Tracts in Mathematics published by the European Mathematical Society. Many chapters have been thoroughly rewritten during the book preparation. The manuscript here has kept the original presentation and concerns linear and nonlinear Dirichlet problems involving $L^1$ data and more generally measure data, based on Stampacchia's definition of weak solution. I explain some of the main tools: linear regularity theory, maximum principles, Kato's inequality, method of sub and supersolutions, and the Perron method. The semilinear Dirichlet problem need not have a solution for every finite measure. I give characterizations of measures for which the problem has a solution with polynomial and exponential nonlinearities in connection with capacities and Hausdorff measures. Finally, the reader will find a different approach to the concept of reduced measure introduced in collaboration with H. Brezis and M. Marcus, which has not been retained in my EMS book due to personal time constraints.