Tuomo Kuusi

Local behavior of fractional p-minimizers

We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate integro-differential operators.

Nonlocal Equations with Measure Data

We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo and Gallouët (J Funct Anal 87:149–169, 1989, Partial Differ Equ 17:641–655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591–613, 1992, Acta Math 172:137–161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón–Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321–381, 1965). Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.

Guide to nonlinear potential estimates

One of the basic achievements in nonlinear potential theory is that the typical linear pointwise estimates via fundamental solutions find a precise analog in the case of nonlinear equations. We give a comprehensive account of this fact and prove new unifying families of potential estimates. We also describe new fine properties of solutions to measure data problems.

The Wolff gradient bound for degenerate parabolic equations

The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of p-Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Riesz Potentials and Nonlinear Parabolic Equations

The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case.

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincaré or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher (Ck, k ≥ 1) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities), which yields, by a new “multiscale” Poincaré inequality, quantitative estimates on the sublinearity of the corrector.

Potential estimates and gradient boundedness for nonlinear parabolic systems

We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary p-Laplacean system ut div(jDuj p 2 Du) = V (x;t); and provide L 1 -bounds for the spatial gradient of solutions Du via non- linear potentials of the right hand side datum V. Such estimates are related to those obtained by Kilpelainen & Mal y (22) in the elliptic case.

The additive structure of elliptic homogenization

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in Armstrong et al. (Commun. Math. Phys. 347(2):315–361, 2016): using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

A fractional Gehring lemma, with applications to nonlocal equations

We describe a fractional version of the classical Gehring lemma. As a consequence, new self-improving regularity properties of solutions to integrodifferential equations emerge. 1. The classical Gehring lemma The Gehring lemma [7, 9] is a fundamental tool in modern nonlinear analysis, with crucial implications in several different fields, ranging from nonlinear elliptic and parabolic equations to the calculus of variations, from quasiconformal geometry to stability issues [2, 6, 11]. Its ultimate essence relies on a basic, self-improving property of certain kind of inequalities, called reverse Holder type inequalities. This can be described as follows: if one can control the L-means of a given function f ∈ L, at all scales, with similar L-means, and p > q, then the function f is necessarily better than just being in L. Starting from the original work of Gehring, there have been several different versions of this result; see [9] for a panorama. The following one, involving reverse inequalities with increasing support, can be for instance found in [8]. Theorem 1.1. Let f ∈ Lploc(Ω), p > 1 be a non-negative function such that the following reverse Holder type inequality holds whenever B is a ball in the open subset Ω ⊂ R: (∫

Bounded Correctors in Almost Periodic Homogenization

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality.