For a fixed constant λ > 0 and a bounded Lipschitz domain Ω⊂ℝ^n with n ≥ 2 , we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional 𝒥_G(v;Ω ) := ∫ _Ω( ∑ _i=1^mG (|∇ v_i(x)| ) + λχ _{|v|>0}(x)) dx , where v = (v_1, … , v_m) and m ∈ℕ , exhibit optimal Lipschitz continuity on compact subsets of Ω , where G is an 𝒩 -function satisfying specific growth conditions. Furthermore, we obtain universal gradient estimates for non-negative almost-minimizers in the interior of non-coincidence sets. Our work extends the recent regularity results for weakly coupled vectorial almost-minimizers for the p-Laplacian addressed in Bayrami et al. (2024), and even the scalar case treated in da Silva et al. (2024); Dipierro et al. (2024) and Pellegrino and Teixeira (2024), thereby providing new insights and approaches applicable to a variety of non-linear one or two-phase free boundary problems with non-standard growth.
Abstract We prove that, except for two explicit families, every discrete subgroup $$G \subset \textrm{PSL}(3,\mathbb {C})$$ G ⊂ PSL ( 3 , C ) that acts properly discontinuously on a non-empty open subset of $$\mathbb{C}\mathbb{P}^2$$ C P 2 admits a unique largest $$G$$ G -invariant open set $$\Omega _G \subset \mathbb{C}\mathbb{P}^2$$ Ω G ⊂ C P 2 on which the action is properly discontinuous. We refer to $$\Omega _G$$ Ω G as the regular set of $$G$$ G . We also establish several geometric and analytic properties of $$\Omega _G$$ Ω G and compare its complement with other natural limit sets associated to the action.
We show that every sufficiently large odd integer n can be represented as n = p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^r + p_6^s + p_7^t, where each p_i is prime, and r, s, t are fixed integers satisfying either (r,s,t)=(3,4,t) with t ≥ 5 , or 3 ≤ r ≤ s ≤ 6 with t ≥ 4 and (r,s) (3,3),(3,4) .
We prove that, except for two explicit families, every discrete subgroup G ⊂PSL(3,ℂ) that acts properly discontinuously on a non-empty open subset of ℂℙ^2 admits a unique largest G -invariant open set Ω _G ⊂ℂℙ^2 on which the action is properly discontinuous. We refer to Ω _G as the regular set of G . We also establish several geometric and analytic properties of Ω _G and compare its complement with other natural limit sets associated to the action.
This paper classifies invariant surfaces in the 3-dimensional solvable Lie group Sol_3 that act as solitons under the Gauss curvature flow. Specifically, we consider solitons associated with the canonical basis of Killing vector fields {F_1, F_2, F_3} , where F_1 and F_2 generate horizontal translations and F_3 generates a scaling isometry. Rigidity results are established for F_3 -invariant surfaces, proving that certain totally geodesic vertical planes are the only F_1 - and F_2 -solitons. Finally, for F_1 -invariant surfaces, we describe the main geometric properties of F_2 - and F_3 -solitons, considering both extrinsic and intrinsic Gauss curvatures.
The aim of this paper is to construct a chain complex for a singular 2-manifold X with n-sheet cone singularities, where n ∈ℕ with n ≥ 2 , equipped with a Gutierrez-Sotomayor flow φ , that computes the intersection homology of X. By combining Morse theory and de Rham cohomology, we introduce a chain complex (C_*(X,φ ; ℝ), ∂ _*) whose generators consist of the regular singularities of φ and the representatives of the de Rham cohomology of the links of the n-sheet cone singularities. The differential ∂ _* carries a dynamical aspect, tracking the flow trajectories between consecutive singularities. The relevance of this chain complex becomes clear when we prove that its homology is isomorphic to the intersection homology of the underlying pseudomanifold.
In 1953 LeVeque proved the existence of U_m-numbers by showing that for some specially defined Liouville number λ, the mth root λ^1/m is in U_m. In this article we study the following question: let,u be an algebraic function of degree m and λ a Liouville number; under which conditions is u(λ) a U_m-number? We consider a more refined notion of ℒ-numbers, and show that, under very general assumptions, an algebraic function of degree m takes U_m-values at all ℒ-numbers.
We study a point-separation property for constructing large algebraic and topological structures in topological vector spaces. Based on linear separation, we introduce the notion of transversal lineability and provide general criteria, establish connections with classical notions, and present applications.
This paper investigates the well-posedness of an initial boundary value problem for the time-fractional wave equation involving acoustic boundary conditions, a setting that, to the best of our knowledge, is studied here for the first time. The domain ⊂ℝ^n ( n ≥ 2 ) is bounded, connected, and admits a boundary composed of two disjoint parts: homogeneous Dirichlet conditions are imposed on _0 , while reactive acoustic-type conditions are enforced on _1 . The time-fractional derivative is taken in the Caputo sense, introducing several analytical challenges due to its nonlocal nature. To address this, we develop a constructive Faedo–Galerkin approach tailored to the fractional framework and solve a general system of time-fractional ordinary differential equations that arises from the approximation process. As a key theoretical contribution, we also establish an extended version of the classical Picard–Lindelöf theorem, which is essential for handling the approximated systems. These results lay a foundational framework for analyzing more general nonlinear problems with fractional dynamics and nonstandard boundary conditions.
In this paper, we prove an abstract version of the splitting property for summing operators, extending the classical result due to A. Pietsch and B. Maurey. Our results provide a technique that may be useful in different contexts, as illustrated in the applications.
We study the existence of two positive normalized solutions for the following fractional critical Schrödinger equation: { (-Δ )^s u = λ u + |u|^2_s^*-2u + a |u|^p-2 u, in Ω , u > 0 in Ω , u = 0 on ℝ^N ∖Ω , ∫ _Ω|u|^2 d x = c, . where Ω is a bounded, smooth and star-shaped domain in ℝ^N , N > 2s , s ∈( 0, 1 ) , c > 0 and 2_ s^*:= 2N/N-2s . λ∈ℝ appears as an unknown Lagrange multiplier, and a, p are in one of the following three cases: (1) a=0, (2) a > 0 , 2 + 4 s/N< p< 2_s^*, (3) a< 0 , 2< p < 2_s^*. We firstly establish the existence of a normalized ground state solution. Then, using some novel ideas, we obtain the second positive normalized solution, which is of mountain pass type.
In this manuscript, we establish local Schauder estimates for flat viscosity solutions, that is, solutions with sufficiently small norms, to a class of fully nonlinear elliptic partial differential equations of the form F(D^2 u, x) + ⟨𝔅(x), D u ⟩ = f(x) in B_1 ⊂ℝ^n, where the operator F is differentiable, though not necessarily convex or concave. In addition, we impose suitable Dini-type continuity assumptions on the data. Our methodology is based on geometric tangential techniques, combined with compactness and perturbative arguments. This approach is strongly motivated by recent advances in the theory of nonlinear elliptic equations and free boundary problems. As a byproduct of our analysis, we also obtain an Evans–Krylov type estimate. Our results can be viewed as an extension of the work by dos Prazeres and Teixeira (Ann Sc Norm Super Pisa Cl Sci (5) 15:485–500, 2016, Theorem 2.2), now within the framework of linear drift terms and Dini continuity assumptions. Finally, we apply our results to characterize the nodal sets of flat viscosity solutions of non-convex, fully nonlinear, uniformly elliptic PDEs.
In this paper, we investigate the existence and concentration of solutions to a (p,N)-Laplace equation in ℝ^N involving a discontinuous nonlinearity and critical exponential growth. To establish the existence of solutions, we employ a penalization technique in the sense of Del Pino and Felmer adapted to a locally Lipschitz functional. Furthermore, by combining variational methods with Moser-type iteration techniques, we obtain the concentration behavior of the solutions. Our results contribute to the study of nonlinear elliptic problems with irregular nonlinearities and critical growth phenomena.
For any subharmonic function u, we prove that |∂ _j u| ( j=1, … , n ) is upper semi-continuous, provided that the super-level sets of u can be touched from the exterior by uniform C^1,Dini domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfies this geometric condition.
The algebraic and geometric classifications of complex 3-dimensional noncommutative Jordan superalgebras are given. In particular, we obtain the algebraic and geometric classification of complex 3-dimensional Kokoris and standard superalgebras, and, due to one-to-one correspondences between suitable superalgebras, we have classifications for generic Poisson–Jordan and generic Poisson superalgebras. As a byproduct, we have the algebraic and geometric classification of the variety of complex 3-dimensional anticommutative superalgebras and their principal subvarieties: Lie, Malcev, binary Lie, Tortkara, anticommutative ℭ𝔇 -, 𝔰_4 -, anticommutative terminal superalgebras, anticommutative conservative and anticommutative quasi-conservative (rigid) superalgebras; and also prove a Grishkov–Shestakov’s conjecture for 3-dimensional binary Lie superalgebras.
In this work, we develop a periodic averaging principle for arbitrary discrete time domains, leveraging a novel definition of periodicity. This definition does not rely on the classical requirement for the time domain itself to be periodic. We implement this averaging principle across diverse discrete time domains and explore a range of periodic functions within this extended context. The paper contains several examples with numerical simulations, providing visual demonstrations of our results. This highlights the versatility of our averaging principle and its potential to understand dynamics of nonautonomous recurrences with complex temporal patterns.
Let G be a finite abelian group of order n. Let ℳ_G be the Cayley table of G and (ℳ_G) the permanent of ℳ_G . An interesting result of Hall provided a one to one correspondence between the monomials in (ℳ_G) and zero-sum sequences over G of length n. Generalizing Hall’s result, Panyushev conjectured an analogous correspondence concerning the generalized Cayley table of G. In this paper, we disprove Panyushev’s conjecture and provide a general characterization of the aforementioned correspondence. As the permanent and determinant matrix functions are special cases of immanants (which are very important objects in algebraic combinatorics), we also provide some discussions on the immanants of ℳ_G and propose some interesting conjectures.
We study foliations in ℂ^2 given by polynomial deformations of the form dH+ϵη =0 , with γ (t)⊂ H^-1(t) a family of cycles. The Poincaré first return map is of the form P(t)=t+∑ _j ϵ ^j M_j^γ (t). The functions M_j^γ are called Melnikov functions and are given by iterated integrals of orbit length at most j. We show that, for each k∈ℕ , there exists a universal Noetherianity index n_ H,γ(k) , independent of the deformation η , such that, if M_j^γ≡ 0 , for j=1,… ,n_ H,γ(k) , then M_j^γ is of orbit length j-k , for any Melnikov function M_j^γ . We call the smallest index with this property just the Noetherianity index ν _ H,γ(k) . To prove this theorem, we develop a structure theorem for Melnikov functions and use the Ritt–Raudenbush differential algebra theorem. We calculate the universal Noetherianity index n_H,γ(k) in various nontrivial examples.
Let X and Y be complex Banach spaces, B_X be the open unit ball of X and ℋL_0(B_X,Y) be the Banach space of all holomorphic Lipschitz maps f:B_X→ Y such that f(0)=0 , endowed with the Lipschitz norm. Given a Banach operator ideal 𝒜 , we use the property of 𝒜 -compactness by Carl and Stephani to introduce and study the subclass of those functions in ℋL_0(B_X,Y) for which its Lipschitz image is a relatively 𝒜 -compact subset of Y. We focus our attention on its structure as a composition Banach holomorphic Lipschitz ideal by using its connection with 𝒜 -compact linear operators through linearization/transposition techniques.
