arXiv subjects
Publications and source records attributed to Francesco Esposito.
We provide a classification result of positive solutions with a non-removable singularity at the origin to the subcritical Lane-Emden equation $-\Delta_p u = u^q$ in $\mathbb{R}^N \setminus \{0\}$, where $p\in(1,N)$, $q\in (p-1,p^*-1)$ and $u^{q-p+1} \in L^{\frac{N}{p}}(\mathbb{R}^N \setminus B_R)$.
In this paper we consider a quasilinear singular anisotropic elliptic problem with a p-sublinear perturbation. We prove uniqueness of weak solutions and we provide necessary and sufficient conditions for the existence of weak solutions in the same spirit of the celebrated paper by Lazer and McKenna.
Kato developed an exotic Deligne-Langlands correspondence using a geometric model for the multiparameter affine Hecke algebra of type C, based on his exotic nilpotent cone. Achar-Henderson and Springer showed that this exotic nilpotent is intimately related to another, apparently simpler variety called the enhanced nilpotent cone. Each of these is defined as the Hilbert nullcone of a polar module, the exotic Sp(2n)-module and the enhanced GL(n)-module, respectively. In this paper we conduct a detailed study of the geometry of these two modules, by introducing the Jordan stratification, simultaneously generalising classical results on the adjoint representation as well as the symmetric space associated to (gl(2n), sp(2n)). One of the key tools we develop is the theory of induced orbits in the enhanced and exotic nilpotent cones, following the work of Lusztig-Spaltenstein. Our main application is a classification of sheets in these modules, inspired by a theorem of Borho.
The aim of this paper is to prove radial symmetry results for positive weak solutions with finite energy to the following quasilinear doubly critical system \begin{equation} \begin{cases} -Δ_p u\,=γ\frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ ναu^{α-1} v^β& \text{in}\quad \mathbb{R}^n \\ -Δ_p v\,=γ\frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ νβu^αv^{β-1} & \text{in}\quad\mathbb{R}^n, \end{cases} \end{equation} where $1 1$ such that $α+ β= p^*=np/(n-p)$ and $ν>0$.
In this work, we investigate the approach via flipclasses to the Combinatorial Invariance Conjecture for Kazhdan--Lusztig polynomials of all Coxeter groups. We prove the combinatorial invariance of Kazhdan--Lusztig $\widetilde{R}$-polynomials of Weyl groups modulo $q^7$ and of Kazhdan--Lusztig $\widetilde{R}$-polynomials of type $A$ Weyl groups modulo $q^8$. As a consequence, the Combinatorial Invariance Conjecture holds for all intervals up to length 8 in Weyl groups and up to length 10 in type $A$ Weyl groups.
In this paper we prove rigidity results for classical solutions to the stationary 2D Euler equations in $\mathbb{R}^2$. Assuming that the velocity field has finite energy and that the stagnation set is connected, we prove that the corresponding stream function solves an autonomous semilinear elliptic equation. Under some extra conditions on the vorticity near infinity we can also prove that the streamlines are concentric circles. The proofs include several energy estimates on the behavior of the stream function at infinity, as well as an adaptation of the continuous Steiner symmetrization to our setting.
Kapranov and Schechtman defined the category FP of factorized perverse sheaves on Sym(C) smooth along the stratification given by multiplicities and with values in a braided monoidal category V. We define for each d in N the category FP^{\leq d} of factorized perverse sheaves on U_{n\leq d}Sym^{n}(C) and the category FP_{\leq d} of factorized perverse sheaves on the open subset of Sym(C) consisting of multi-sets with multiplicities bounded by d. We prove that the natural restriction functor from FP_{\leq d} to \FP^{\leq d} is an equivalence for any d in N, and that FP^{\leq 1} and \FP_{\leq 1} are equivalent to V. We show that the full direct image *, the extension by zero ! and the intermediate extension !* induce functors from FP_{\leq d} to \FP. In addition, we show that the families (FP^{\leq d})_{d in N} and (\FP_{\leq d})_{d in N} fit into systems of categories, compatible with restrictions and extensions, whose inverse limit is FP.
Motivated by an equivalence of categories established by Kapranov and Schechtman, we introduce, for each non-negative integer d, the category of connected bialgebras modulo d+1. We show that these categories fit into an inverse system of categories whose inverse limit category is equivalent to the category of connected bialgebras. In addition, we extend the notion of approximation of connected bialgebras to those that are not necessarily generated in degree 1 and show that, for connected bialgebras in the category of Yetter-Drinfeld modules over a Hopf algebra, approximation is compatible with cocycle twisting.
We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the symmetric group. This conjecture has the advantage of being combinatorial in nature. The appendix by Grant T. Barkley and Christian Gaetz discusses the related notion of double hypercubes and proves an analogous conjecture for these in the case of co-elementary intervals.
In this paper we consider a quasilinear elliptic and critical problem with Dirichlet boundary conditions in presence of the anisotropic $p$-Laplacian. The critical exponent is the usual $p^{\star}$ such that the embedding $W^{1,p}_{0}(Ω) \subset L^{p^{\star}}(Ω)$ is not compact. We prove the existence of a weak positive solution in presence of both a $p$-linear and a $p$-superlinear perturbation. In doing this, we have to perform several precise estimates of the anisotropic Aubin-Talenti functions which can be of interest for further problems. The results we prove are a natural generalization to the anisotropic setting of the classical ones by Brezis-Nirenberg \cite{BN}.
In this paper we study the effect of the Hardy potential on existence, uniqueness and optimal summability of solutions of the mixed local-nonlocal elliptic problem $$-Δu + (-Δ)^s u - γ\frac{u}{|x|^2}=f \text{ in } Ω, \ u=0 \text{ in } \mathbb{R}^n \setminus Ω,$$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ containing the origin and $γ> 0$. In particular, we will discuss the existence, non-existence and uniqueness of solutions in terms of the summability of $f$ and of the value of the parameter $γ$.
We define a specialization map between cohomology algebras of quiver Grassmannians of Dynkin type and we prove that it is surjective in type A. This generalizes a result of Lanini and Strickland.
We provide a characterization of parallelizable compact complex manifolds and their quotients using holomorphic symmetric differentials. In particular we show that compact complex manifolds of Kodaira dimension 0 having strongly semiample cotangent bundle are parallelizable manifolds, while compact complex manifolds of Kodaira dimension 0 having weakly semiample cotangent bundle are quotients of parallelizable manifolds. The main constructions used involve considerations about semiampleness of vector bundles, which are themselves of interest. As a byproduct we prove that compact manifolds having Kodaira dimension 0 and weakly sermiample cotangent bundle have infinite fundamental group, and we conjecture that this should be the case for all compact complex manifolds not of general type with weakly semiample cotangent bundle.
In this work, we investigate a novel approach to the Combinatorial Invariance Conjecture of Kazhdan--Lusztig polynomials for the symmetric group. Using the new concept of flipclasses, we introduce some combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of $q^h$ of the Kazhdan--Lusztig $\widetilde{R}$-polynomials, for $h\leq 6$. This recipe depends only on the isomorphism class (as a poset) of the interval indexing the polynomial and thus provides new evidence for the Combinatorial Invariance Conjecture.
We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.
Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by $\mathbb{C}^\times$-equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First we give a complete description of the cases in which the finite $W$-algebra is a universal filtered quantization of the slice, building on the work of Lehn--Namikawa--Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply-laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type we prove that the finite $W$-algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally we apply this result to give a presentation of the subregular finite $W$-algebra in type B as a quotient of a shifted Yangian.
The aim of this paper is to deal with the anisotropic doubly critical equation $$-Δ_p^H u - \fracγ{[H^\circ(x)]^p} u^{p-1} = u^{p^*-1} \qquad \text{in } \R^N,$$ where $H$ is in some cases called Finsler norm, $H^\circ$ is the dual norm, $1<p<N$, $0 \leq γ< \left((N-p)/p\right)^p$ and $p^*=Np/(N-p)$. In particular, we provide a complete asymptotic analysis of $u \in \mathcal{D}^{1,p}(\R^N)$ near the origin and at infinity, showing that this solution has the same features of its euclidean counterpart. Some of the techniques used in the proofs are new even in the Euclidean framework.
This work deals with a family of Hardy-Sobolev doubly critical system defined in $\mathbb{R}^n$. More precisely, we provide a classification of the positive solutions, whose expressions comprise multiplies of solutions of the decoupled scalar equation. Our strategy is based on the symmetry of the solutions, deduced via a suitable version of the moving planes technique for cooperative singular systems, joint with the study of the asymptotic behavior by using the Moser's iteration scheme.