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The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p -Laplace operator, namely: λ 1 (β , Ω) = min ψ ∈ W 1 , p (Ω) ∖ { 0 } ⁡ ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N − 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, anisotropic mean convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ 1 in the case β > 0 , and a upper bound in the case β < 0. As a consequence, we prove, for β > 0 , a lower bound for λ 1 (β , Ω) in terms of the anisotropic inradius of Ω and, for β < 0 , an upper bound of λ 1 (β , Ω) in terms of β. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following non-linear equations in unbounded domains Ω with exterior Dirichlet condition: { (− Δ) p s u (x) = f (u (x)) , x ∈ Ω , u (x) > 0 , x ∈ Ω , u (x) = 0 , x ∈ R n ∖ Ω , where (− Δ) p s is the fractional p -Laplacian defined as (0.1) (− Δ) p s u (x) = C n , s , p P. V. ∫ R n | u (x) − u (y) | p − 2 [ u (x) − u (y) ] | x − y | n + s p d y with 0 < s < 1 and p ≥ 2. We first establish a maximum principle in unbounded domains involving the fractional p -Laplacian by estimating the singular integral in (0.1) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p -Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the classical Laplacian [3] and for the fractional Laplacian [39] , which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p -Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on f (⋅) and on the domain Ω. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators. [ABSTRACT FROM AUTHOR]

The fractional heat operator (∂ t − Δ x) s and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to (∂ t − Δ x) s and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator. [ABSTRACT FROM AUTHOR]

In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order X D s − 1 , q (Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs. [ABSTRACT FROM AUTHOR]

In this paper we study quasilinear elliptic equations driven by the so-called double phase operator and with a nonlinear boundary condition. Due to the lack of regularity, we prove the existence of multiple solutions by applying the Nehari manifold method along with truncation and comparison techniques and critical point theory. In addition, we can also determine the sign of the solutions (one positive, one negative, one nodal). Moreover, as a result of independent interest, we prove for a general class of such problems the boundedness of weak solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis-Stokes system with nonlinear diffusion and rotation: n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ (n S (x , n , c) ⋅ ∇ c) , c t + u ⋅ ∇ c = Δ c − n c , u t + ∇ P = Δ u + n ∇ ϕ + f (x , t) and ∇ ⋅ u = 0 , in a bounded domain Ω ⊂ R 3 , where m > 0 , and ϕ : Ω ¯ → R , f : Ω ¯ × [ 0 , ∞) → R 3 and S : Ω ¯ × [ 0 , ∞) 2 → R 3 × 3 are given sufficiently smooth functions such that f is bounded in Ω × (0 , ∞) and S satisfies | S (x , n , c) | ≤ S 0 (c) (1 + n) − α for all (x , n , c) ∈ Ω ¯ × [ 0 , ∞) 2 with α > 0 and some nondecreasing function S 0 : [ 0 , ∞) → [ 0 , ∞). It is shown that if m + α > 10 9 and m + 5 4 α > 9 8 , then for any reasonably smooth initial data, the corresponding Neumann-Neumann-Dirichlet initial-boundary problem possesses a globally bounded weak solution. This extends the previous global boundedness result for m > 9 8 and α = 0 [43] , and improves that for m ≥ 1 and m + α > 7 6 [34] , or for m + α > 7 6 in the associated fluid-free system [31]. Our proof consists at its core in using, inter alia , the maximal Sobolev regularity theory to elaborately derive some spatio-temporal estimates for the signal and the fluid equations so as to decouple the system. [ABSTRACT FROM AUTHOR]

Abstract The paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in R N. Our focal point is to involve the leading, nonlinear part of the operator whose growth is described by anisotropic N -function M inhomogeneous in the space and the time variables. The main goals are proven in absence of Lavrentiev's phenomenon, to ensure which we impose a certain type of balance of interplay between the behavior of M for large | ξ | and small changes of time and space variables. Its instances are log-Hölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (possibly changing in time). New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a priori C 2 , α estimate up to the boundary for F (D 2 u) = 0 and the regularity of the free boundary of the obstacle problem for fully nonlinear operator under specific conditions for the operator and level sets of the operator. The conditions are variations of conditions for the zero set of the operator in [7]. [ABSTRACT FROM AUTHOR]