Search

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]

Abstract The solution of a parabolic problem is expected to reproduce the basic qualitative properties of the original phenomenon, such as nonnegativity/nonpositivity preservation, maximum/minimum principles and maximum norm contractivity, without which the model might lead to unrealistic quantities in conflict with physical reality. This paper presents characterizations of qualitative properties for a general class of nonlinear parabolic systems. This is a continuation of the authors' research, published in a previous paper in this journal on scalar equations. Cooperativity of the systems plays an essential role in most of the results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel projected steepest descent iterative method with frozen derivative. The classical projected steepest descent iterative method involves the computation of derivative of the nonlinear operator at each iterate. The method of this paper requires the computation of derivative of the nonlinear operator only at an initial point. We exhibit the convergence analysis of our method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, by assuming the conditional stability on a nested family of convex and compact subsets, we develop a multi-level method. In order to enhance the accuracy of approximation between neighboring levels, we couple it with the growth of stability constants. This along with a suitable discrepancy criterion ensures that the algorithm proceeds from level to level and terminates within finite steps. Finally, we discuss an inverse problem on which our methods are applicable. [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]

In this paper, we study the nonexistence and existence of radial large positive solutions for a class of Schrödinger systems with a nonlinear random operator under certain growth conditions. We also establish the necessary and sufficient conditions for the existence of radial large positive solutions by a successive iteration technique. The results of the paper improve and extend many previous work in this field of research. [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]

This paper deals with improvements of the Trudinger–Moser inequality related to the operator Q V ( u ) : = − Δ n u + V ( x ) | u | n − 2 u , where n ≥ 2 and the potential V : R n → R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E : = { u ∈ W 1 , n ( R n ) : ∫ R n V ( x ) | u | n d x < ∞ } endowed with the norm ‖ u ‖ : = [ ∫ R n ( | ∇ u | n + V ( x ) | u | n ) d x ] 1 / n and we prove that if ( u k ) is a sequence in E such that ‖ u k ‖ = 1 , u k ⇀ u ≢ 0 in E and 0 < p < p n ( u ) : = β n ( 1 − ‖ u ‖ n ) − 1 / ( n − 1 ) , then ( ⁎ ) sup k ⁡ ∫ R n Ψ ( p | u k | n / ( n − 1 ) ) d x < ∞ , where Ψ ( t ) : = e t − ∑ i = 0 n − 2 t i i ! , β n : = n ω n − 1 1 / ( n − 1 ) and ω n − 1 is the measure of the unit sphere in R n . Furthermore, p n ( u ) is sharp in the sense that there exists a sequence ( u k ) ⊂ E satisfying ‖ u k ‖ = 1 and u k ⇀ u ≢ 0 in E such that the supremum (⁎) is infinite for p ≥ p n ( u ) . As an application of the previous result we prove the following sharp form of the Trudinger–Moser inequality for the subspace E . Considering ℓ ( α ) : = sup { u ∈ E : ‖ u ‖ = 1 } ⁡ ∫ R n Ψ ∘ ν α ( u ) d x , where ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) , assuming some conditions of symmetry on V it is established (1) for 0 ≤ α < λ 1 ( V ) we have ℓ ( α ) < ∞ , (2) for α ≥ λ 1 ( V ) , ℓ ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α < λ 1 ( V ) , an extremal function for ℓ ( α ) exists. Here λ 1 ( V ) denotes the first eigenvalue of Q V ( u ) . [ABSTRACT FROM AUTHOR]

Let S = U S | S | and T = U T | T | be the polar decompositions of S and T in L ( H ) , respectively. We say that S and T have similar positive parts if | S | and | T | are similar. In this paper, we investigate properties of operators that are preserved under this similarity condition. We describe the form for the polar decomposition of an operator when the operator and its adjoint have similar positive parts. For this case, we also study the existence of invariant subspaces under this assumption. [ABSTRACT FROM AUTHOR]

In this paper we consider the system involving fully nonlinear nonlocal operators: { F α ( u ( x ) ) = C n , α P V ∫ R n G ( u ( x ) − u ( y ) ) | x − y | n + α d y = f ( v ( x ) ) , F β ( v ( x ) ) = C n , β P V ∫ R n G ( v ( x ) − v ( y ) ) | x − y | n + β d y = g ( u ( x ) ) . For carrying on the method of moving planes, a narrow region principle and a decay at infinity are established. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Furthermore, non-existence of positive solutions to the nonlinear system on a half space is derived. [ABSTRACT FROM AUTHOR]

It is a natural expectation that the mathematical models of real-life phenomena have to possess some characteristic qualitative properties of the original process. For parabolic problems the main known qualitative properties are the maximum–minimum principles, nonnegativity–nonpositivity preservation and maximum norm contractivity. These properties have a fundamental relevance concerning the validity of the mathematical or numerical model: without them, the model might produce unphysical quantities that contradict reality. For linear problems with Dirichlet boundary conditions, these properties have been thoroughly investigated and their relations have been characterized. In the present paper, we extend the linear results to nonlinear problems with general boundary conditions. Firstly, we characterize various implications between the qualitative properties. Some of them are given in general, and in certain cases we restrict our study to operators with gradient-dependent principal part or to operators with heat conduction coefficient. Secondly, we give general sufficient conditions to ensure these qualitative properties, both separately and all of them together. The relations are illustrated with several examples. [ABSTRACT FROM AUTHOR]

A remarkable result by S. Artstein-Avidan and V. Milman states that, up to pre-composition with affine operators, addition of affine functionals, and multiplication by positive scalars, the only fully order preserving mapping acting on the class of lower semicontinuous proper convex functions defined on R n is the identity operator, and the only fully order reversing one acting on the same set is the Fenchel conjugation. Here fully order preserving (reversing) mappings are understood to be those which preserve (reverse) the pointwise order among convex functions, are invertible, and such that their inverses also preserve (reverse) such order. In this paper we establish a suitable extension of these results to order preserving and order reversing operators acting on the class of lower semicontinuous proper convex functions defined on arbitrary infinite dimensional Banach spaces. [ABSTRACT FROM AUTHOR]