In this paper, we are concerned with the Hölder regularity forsolutions of the nonlocal evolutionary equation ∂tu + (−p) su = 0. Here, (−p)s is the fractional p-Laplacian, 0 <s< 1 and 1 <p< 2. We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. Our results complement the previous results for the superquadratic case when p ≥ 2.

We consider the evolutionary p-Laplace equation in R-n x (0, infinity). For p > n, we construct a solution u with a moving gradient singularity in the sense that |del u(x, t)|-> infinity for each t as x -> (t), where : [0, infinity) -> R-n is a given curve.

Morrey's classical inequality implies the H & ouml;lder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality lambda & Vert;ud Omega 1-n/p & Vert;infinity p <=integral Omega|Du|pdxfor any open set Omega & subne;Rn. This inequality is valid for functions supported in Omega and with lambda a positive constant independent of u. The crucial hypothesis is that the exponent p exceeds the dimension n. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of Omega, sharp constants, and the existence of a nontrivial u which saturates the inequality.

We study the decay (at infinity) of extremals of Morrey’s inequality in Rn. These are functions satisfying (Formula Presented) where p>n and C(p, n) is the optimal constant in Morrey’s inequality. We prove that if n≥2 then any extremal has a power decay of order β for any (Formula Presented).

We study the fractional p-Laplace equation (−p) s u = 0 for 0 < s < 1 and in the subquadratic case 1 < p < 2. We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp for a certain range of parameters. Our results complement the previous results for the superquadratic case when p ≥ 2. The arguments are based on a careful Moser-type iteration and a perturbation argument.

We propose a new finite difference scheme for the degenerate parabolic equation ∂tu-div(|∇u|p-2∇u)=f,p≥2.Under the assumption that the data is Hölder continuous, we establish the convergence of the explicit-in-time scheme for the Cauchy problem provided a suitable stability type CFL-condition. An important advantage of our approach, is that the CFL-condition makes use of the regularity provided by the scheme to reduce the computational cost. In particular, for Lipschitz data, the CFL-condition is of the same order as for the heat equation and independent of p.

We consider equations involving a combination of local and nonlocal degenerate p-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Holder continuity with an explicit Holder exponent in the general case. For certain parameters, our results also imply Holder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.

We extend some theorems for the infinity-ground state and for the infinity-potential, known for convex polygons, to other domains in the plane, by applying Alexandroff's method to the curved boundary. A recent explicit solution disproves a conjecture.

We study the sharp constant for the embedding of W01,p(Ω) into Lq(Ω), in the case 2 < p < q. We prove that for smooth connected sets, when q > p and q is sufficiently close to p, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of p−Laplace–type equations by L. Damascelli and B. Sciunzi.

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional p-Laplace type P.V. integral(Rn)vertical bar u(x) - u(y)vertical bar(p-2)(u(x) - u(y))/vertical bar x-y vertical bar(n+sp) dy = 0. Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide. (C) 2017 Elsevier Masson SAS. All rights reserved.