Augusto C. Ponce

Bio: Augusto C. Ponce is an academic researcher from Université catholique de Louvain. The author has contributed to research in topics: Sobolev space & Measure (mathematics). The author has an hindex of 21, co-authored 89 publications receiving 1466 citations. Previous affiliations of Augusto C. Ponce include Institute for Advanced Study & Centre national de la recherche scientifique.

A new approach to Sobolev spaces and connections to Gamma-convergence

Abstract: This is a follow-up of a paper of Bourgain, Brezis and Mironescu [2]. We study how the existence of the limit integral(Omega)integral(Omega) omega((x)- f(y))/x - y) rho(epsilon)(x - y) dxdy as epsilon down arrow 0, (1) for omega: [0,infinity) --> [0, infinity) continuous and (rhoepsilon) subset of L-1(R-N) converging to delta(0), is related to the weak regularity of f is an element of L-loc(1)(Omega). This approach gives an alternative way of defining the Sobolev spaces W-1,W-p. We also briefly discuss the Gamma-convergence of (1) with respect to the L-1(Omega)-topology.

An estimate in the spirit of Poincaré's inequality

Abstract: 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 Poincare's inequality. The case N = I requires an additional assumption on (rho(n)). We also extend a compactness result of Bourgain, Brezis and Mironescu.

Nonlinear elliptic equations with measures revisited

Abstract: We study the existence of solutions of the nonlinear problem (P) ( �u + g(u) = µ in , u = 0 on @, where µ is a Radon measure and g : R ! 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 (P) admits a solution. We show that for every µ there exists a largest good measure µ� � µ. This reduced measure has a number of remarkable properties.

Diffuse measures and nonlinear parabolic equations

Abstract: Given a parabolic cylinder Q = (0,T) ! ! , where ! " R N is a bounded domain, we prove new properties of solutions of ut # " pu = µ 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., mea- sures which do not charge sets of zero parabolic p-capacity) can be strongly approximated by the measures µk = (Tk(u))t# " p(Tk(u)),andweintroduceanewnotionofrenormalizedsolutionbasedonthisproperty. We finally apply our new approach to prove the existence of solutions of ut # " pu + h(u) = µ in Q, for any function h such that h(s)s $ 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.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Abstract: Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Nonlinear partial differential equations with applications

Abstract: Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

Nonlocal Diffusion and Applications

Abstract: Introduction.- 1 A probabilistic motivation.-1.1 The random walk with arbitrarily long jumps.- 1.2 A payoff model.-2 An introduction to the fractional Laplacian.-2.1 Preliminary notions.- 2.2 Fractional Sobolev Inequality and Generalized Coarea Formula.- 2.3 Maximum Principle and Harnack Inequality.- 2.4 An s-harmonic function.- 2.5 All functions are locally s-harmonic up to a small error.- 2.6 A function with constant fractional Laplacian on the ball.- 3 Extension problems.- 3.1 Water wave model.- 3.2 Crystal dislocation.- 3.3 An approach to the extension problem via the Fourier transform.- 4 Nonlocal phase transitions.- 4.1 The fractional Allen-Cahn equation.- 4.2 A nonlocal version of a conjecture by De Giorgi.- 5 Nonlocal minimal surfaces.- 5.1 Graphs and s-minimal surfaces.- 5.2 Non-existence of singular cones in dimension 2 5.3 Boundary regularity.- 6 A nonlocal nonlinear stationary Schrodinger type equation.- 6.1 From the nonlocal Uncertainty Principle to a fractional weighted inequality.- Alternative proofs of some results.- A.1 Another proof of Theorem A.2 Another proof of Lemma 2.3.- References.

Towards a Theory of Transition Paths

Abstract: We construct a statistical theory of reactive trajectories between two pre-specified sets A and B, i.e. the portionsof the path of a Markov process during which the path makes a transition from A to B. This problem is relevant e.g. in the context of metastability, in which case the two sets A and B are metastable sets, though the formalism we propose is independent of any such assumptions on A and B. We show that various probability distributions on the reactive trajectories can be expressed in terms of the equilibrium distribution of the process and the so-called committor functions which give the probability that the process reaches first B before reaching A, either backward or forward in time. Using these objects, we obtain (i) the distribution of reactive trajectories, which gives the proportion of time reactive trajectories spend in sets outside of A and B; (ii) the hitting point distribution of the reactive trajectories on a surface, which measures where the reactive trajectories hit the surface when they cross it; (iii) the last hitting point distribution of the reactive trajectories on the surface; (iv) the probability current of reactive trajectories, the integral of which on a surface gives the net average flux of reactive trajectories across this surface; (v) the average frequency of reactive trajectories, which gives the average number of transitions between A and B per unit of time; and (vi) the traffic distribution of reactive trajectories, which gives some information about the regions the reactive trajectories visit regardless of the time they spend in these regions.