Abstract. This is the first of two articles dealing with the equation (−∆)
= Rn . In this first article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian -in the spirit of a result of Modica for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.
We express our sincere gratitude to the referee for a careful reading of the manuscript.International audienceWe investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge2$, $s \in (0,1)$, $\lambda \geq 0$ and $f$ is any smooth positive superlinear function. The operator $(-\Delta)^s$ stands for the fractional Laplacian, a pseudo-differential operator of order $2s$. According to the value of $\lambda$, we study the existence and regularity of weak solutions $u$
Abstract. This paper, which is the follow-up to part I, concerns the equation (−∆) s v + G ′ (v) = 0 in R n , with s ∈ (0, 1), where (−∆) s stands for the fractional Laplacian -the infinitesimal generator of a Lévy process.When n = 1, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits ±1 at ±∞) if and only if the potential G has only two absolute minima in [−1, 1], located at ±1 and satisfying G ′ (−1) = G ′ (1) = 0. Under the additional hypothesis G ′′ (−1) > 0 and G ′′ (1) > 0, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution.For n ≥ 1, we prove some results related to the one-dimensional symmetry of certain solutions -in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.
Abstract. We discuss properties (optimal regularity, nondegeneracy, smoothness of the free boundary etc.) of a variational interface problem involving the fractional Laplacian; due to the nonlocality of the Dirichlet problem, the task is nontrivial. This difficulty is bypassed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a codimension 2 (degenerate) free boundary.
We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a review of the theory of fractional Sobolev spaces and of the interpolation theory that are used in the rest of the paper.
We deal with symmetry properties for solutions of nonlocal equations of the typewhere s ∈ (0, 1) and the operator (− ) s is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation u(y, x). This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator Γ α : u| ∂R n+1More generally, we study the so-called boundary reaction equations given byunder some natural assumptions on the diffusion coefficient μ and on the nonlinearities f and g. We prove a geometric formula of Poincaré-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi.
International audienceSolutions to nonlocal equations with measurable coefficients are higher differentiable. Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by integral(Rn)integral(Rn)[u(x) - u(y)][eta(x) - eta(y)]K(x, y) dx dy = integral(Rn) f eta dx for all eta is an element of C-c(infinity) (R-n), where the kernel K( . ) is a measurable function and satisfies the bounds 1/Lambda vertical bar x - y vertical bar(n+2 alpha) <= K(x, y) <= Lambda/vertical bar x - y vertical bar(n+2 alpha) with 0 < alpha < 1, Lambda > 1, while f is an element of L-loc(q)(R-n) for some q > 2n/(n + 2 alpha). The main result states that there exists a positive, universal exponent delta equivalent to delta(n, alpha, Lambda, q) such that for every weak solution u the self-improving property u is an element of W-alpha,W-2 (R-n) double right arrow u is an element of W-loc(alpha+delta,2+delta) (R-n) holds. This differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear equations in divergence form with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable. The result is achieved by proving a new version of the Gehring lemma involving certain families of lifted reverse Holder-type inequalities in R-2n and which is implied by delicate covering and exit-time arguments. In turn, such reverse Holder inequalities are based on the concept of dual pairs, that is, pairs (mu, U) of measures and functions in R-2n which are canonically associated to solutions. We also allow for more general equations involving as a source term an integrodifferential operator whose kernel does not necessarily have to be of order alpha
Abstract. 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 & Gallouët [6,7] and Kilpeläinen & Malý [32,33]. As a consequence, we establish a number of results which 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 to prove a nonlocal analog of the classical theory due to Campanato [16]. 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.
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