In this paper, we establish the existence of a strictly convex hypersurface function u defined in Ω that satisfies k-Hessian curvature equations subject to the second boundary condition Du(Ω )=Ω ^* , where Ω and Ω ^* are smooth strictly convex bounded domains in ℝ^n . By utilizing the orthogonal invariance of hypersurface, we employ special vector fields generated by the infinitesimal rotations in ℝ^n+1 to establish the boundary C^2 estimates. Our work extends the results of Urbas on the second boundary value problem of Weingarten curvature equations [31] and Hessian equations [30].
In this paper we consider the obstacle problem for the elastic flow and deduce an adapted version of the Energy Dissipation Inequality. The inequality benefits from the improved regularity of weak solutions, which is another main result of this paper. We also derive the uniqueness of weak solutions as an application of the improved regularity.
Abstract We prove strong hybrid subconvex bounds simultaneously in the q and t aspects for L -functions of selfdual $${{\,\textrm{GL}\,}}_3$$ GL 3 cusp forms twisted by primitive Dirichlet characters. We additionally prove analogous hybrid subconvex bounds for central values of certain $${{\,\textrm{GL}\,}}_3 \times {{\,\textrm{GL}\,}}_2$$ GL 3 × GL 2 Rankin–Selberg L -functions. The subconvex bounds that we obtain are strong in the sense that, modulo current knowledge on estimates for the second moment of $${{\,\textrm{GL}\,}}_3$$ GL 3 L -functions, they are the natural limit of the first moment method pioneered by Li and by Blomer. The method of proof relies on an explicit $${{\,\textrm{GL}\,}}_3 \times {{\,\textrm{GL}\,}}_2 \leftrightsquigarrow \leftrightsquigarrow {{\,\textrm{GL}\,}}_4 \times {{\,\textrm{GL}\,}}_1$$ GL 3 × GL 2 ↭ ↭ GL 4 × GL 1 spectral reciprocity formula, which relates a $${{\,\textrm{GL}\,}}_2$$ GL 2 moment of $${{\,\textrm{GL}\,}}_3 \times {{\,\textrm{GL}\,}}_2$$ GL 3 × GL 2 Rankin–Selberg L -functions to a $${{\,\textrm{GL}\,}}_1$$ GL 1 moment of $${{\,\textrm{GL}\,}}_4 \times {{\,\textrm{GL}\,}}_1$$ GL 4 × GL 1 Rankin–Selberg L -functions. A key additional input is a Lindelöf-on-average upper bound for the second moment of Dirichlet L -functions restricted to a coset, which is of independent interest.
We investigate boundedness and regularity properties of weak solutions to a class of generalized logarithmic double phase equations with variable exponents. The considered operators arise from Musielak–Orlicz type energies of logarithmic double phase type and exhibit nonstandard growth features. Under general structural assumptions, we derive a priori boundedness estimates in the subcritical setting and establish boundedness of weak solutions also in the presence of critical growth terms. In addition, we prove global Hölder continuity up to the boundary by means of the De Giorgi iteration scheme, localization arguments, and the frozen functional technique. The obtained results extend several existing regularity results for double phase and related nonstandard growth problems.
Let (𝒳, d, μ ) be a metric measure space satisfying the volume doubling condition. In this paper, the authors provide a new and direct method of constructing stochastically complete (signed) heat kernels by virtue of the multiresolution analysis (for short, MRA) structure. For any β∈ (0,∞ ), two kinds of applications are given: (i) via taking a non-smooth MRA generated by Haar wavelets, such construction gives rise to a heat kernel satisfying only stable-like upper estimate of index β (no continuity and near-diagonal lower estimate); (ii) via taking a smooth MRA generated by smooth wavelets/splines, such construction gives rise to a signed heat kernel (no positivity) satisfying the stable-like upper estimate of index β , as well as the almost Lipschitz regularity and the near-diagonal lower estimate.
In this paper, we consider a stochastic Fisher-KPP equation with shifting environment perturbed by white noise. Firstly, we study the existence and uniqueness of non-negative and strong solution of this stochastic partial differential equation. Next, under strong noise, moderate noise and weak noise, we explore the influences of noise and shifting environment on propagation property of the solution, and find that shifting environment has big impacts on extinction and persistence of species, and noise restrains the propagation of the solution. Finally, we give a brief discussion about our results.
We study a generalized defocusing Hartree equation with a nonlocal exchange potential and a repulsive Hartree interaction. We introduce a class of principal solutions and prove their existence and uniqueness by means of an inverse optimal problem framework. We also establish the continuous dependence of the local principal branch on the spectral parameter and the reference kernel. The construction yields explicit formulas relating principal solutions to the variational solutions of the associated inverse optimal problem. In this way, the paper extends inverse-optimization methods to inverse spectral problems for nonlocal Schrödinger operators with incomplete spectral data.
Let n≥ 3 and p be an odd prime. We show that for every number field K with ζ _p∉ K , the absolute and tame Galois groups Γ _K and Γ ^ta_K of K satisfy the strong n-fold Massey vanishing property relative to p. Our work is based on an adaptation of the proof of the Scholz–Reichardt theorem.
We investigate the Cauchy problem for the Navier–Stokes equations for viscous compressible fluids with capillarity. The linear third-order capillarity term behaves like the heat diffusion of density fluctuations, which enables us to provide an equivalent characterization of Gevrey analyticity, optimal decay, and initial regularity criteria in ℝ^d ( d ≥ 2 ). Precisely, in the critical L^p framework, it is proved that the Besov space Ḃ^σ _1_2,∞ -boundedness condition (with d/2-2d/p≤σ _1
It was conjectured by Bergelson et al. (Geom Funct Anal 19(6):1539–1596, 2010) that every Host–Kra $$\mathbb {F}_p^\omega $$ F p ω -system of order k is an Abramov system of order k . This conjecture has been verified for $$k \le p+1$$ k ≤ p + 1 . In this paper we show that the conjecture fails when $$k=5, p=2$$ k = 5 , p = 2 . We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $$f: \mathbb {F}_2^n \rightarrow \mathbb {C}$$ f : F 2 n → C of large Gowers norm $$\Vert f\Vert _{U^6(\mathbb {F}_2^n)}$$ ‖ f ‖ U 6 ( F 2 n ) which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial e ( P ), but with the property that all such phase polynomials e ( P ) are “non-measurable” in the sense that they cannot be well approximated by functions of a bounded number of random translates of f . A simpler version of our construction can also be used to answer a question of Candela et al. (Ergodic Theory Dyn Syst 43(12):3971–4040, 2023).
We study the maximal cone multiplier operator associated with truncated Fourier multipliers adapted to conic regions in frequency space. In dimension three, we establish sharp L-p bounds for the associated maximal operator, thereby completely resolving the conjectured range for all 1 < p < infinity. Our approach combines multi-scale analysis with wave packet decomposition adapted to annular and conic geometry. The same method can also be applied in the two-dimensional spherical setting to establish new ep annular inequalities, leading to a new proof of the sharp L-4 boundedness of the maximal Bochner-Riesz operator in the plane. In higher dimensions n >= 4, we solve the problem completely for p < 2 and obtain sharp partial results for certain values of p > 2, away from the conjectured critical exponent p(c) = 2(n-1)/n-2 . While the full conjectured range p > p(c) remains open, our results improve upon previously known bounds and go below the general barrier p = 2(n+1/n-1 for maximal operators associated with general Fourier integral operators satisfying the cinematic curvature condition. In particular, we identify a new dimension-dependent threshold p(n) above which we establish sharp L-p estimates for the maximal cone multiplier operator. This work extends and refines earlier results on cone multiplier problems and demonstrates how geometric and frequency-localized methods can yield sharp maximal bounds across different settings.
We investigate the long-time dynamics of a class of non-autonomous coupled pseudo-parabolic systems posed in the whole space ℝ^n . These models describe diffusive processes exhibiting both parabolic and hyperbolic features, with pseudo-parabolic perturbations governed by a small positive parameter. We prove the existence and uniqueness of weak solutions in the natural energy space and establish the existence of minimal non-autonomous pullback attractors in both the frameworks of fixed bounded sets and time-dependent tempered sets. Moreover, we establish relationships between the attractors generated in the different universes. Due to the lack of compact Sobolev embeddings in unbounded domains, the main compactness issues are addressed through a combination of uniform tail estimates and spectral decomposition techniques. Then, we study the continuity of the family of pullback attractors with respect to the perturbation parameter, proving continuity on a dense residual subset and upper semicontinuity as the perturbation parameter tends to zero.
In 1986, Andrews (Am Math Mon 93(9):708–711, 1986; Adv Math 61(2):156–164, 1986) studied the function σ (q) from Ramanujan’s “Lost” Notebook, and made several conjectures on its Fourier coefficients S(n), which count certain partition ranks. In 1988, Andrews et al. (Invent Math 91(3):391–407, 1988) famously resolved these conjectures, relating the coefficients S(n) to the arithmetic of ℚ(√(6)) ; this relationship was further expounded upon by Cohen (Invent Math 91(3):409–422, 1988) in his work on Maass waveforms, and was more recently extended by Zwegers (Q J Math 63(3):753–770, 2012) and by Li and Roehrig (Q J Math 76:367–380, 2025). A closer inspection of Andrews’ original work on σ (q) reveals additional related functions and conjectures, which we study in this paper. In particular, we study the function v_1(q) , also from Ramanujan’s “Lost” Notebook, a q-hypergeometric series with partition-theoretic Fourier coefficients V_1(n) , and prove two of Andrews’ conjectures on V_1(n) which are parallel to his original conjectures on S(n). Our methods differ from those used in Andrews et al. (Invent Math 91(3):391–407, 1988), and require a blend of novel techniques inspired by Garoufalidis’ and Zagier’s recent work on asymptotics of Nahm sums (Garoufalidis and Zagier in Ramanujan J 55:1, 2021; SIGMA Symmetry Integr Geom Methods Appl 19:Paper No. 082, 2023), with classical techniques including the Circle Method in Analytic Number Theory; our methods may also be applied to determine the asymptotic behavior of similar q-hypergeometric series of interest which are not amenable to classical techniques. We also offer explanations of additional related conjectures of Andrews, ultimately connecting the asymptotics of V_1(n) to the arithmetic of ℚ(√(-3)) .
This paper is concerned with the study of a broad class of nonsmooth dynamical systems in infinite-dimensional Hilbert spaces, each composed of a semilinear differential inclusion coupled with a nonconvex sweeping process involving a uniformly prox-regular moving set and a nonlinear perturbation. Under the appropriate conditions, we first prove the existence of mild solutions to the nonsmooth dynamical system under consideration by applying the well-known Kakutani–Ky Fan fixed point theorem for multivalued maps, the theory of measures of noncompactness and the method of variational analysis. Then, when the infinitesimal generator A is dissipative or the multi-valued perturbation F is locally Lipschitz in the Hausdorff sense, the uniqueness results are obtained. Moreover, a framework for studying the long-time dynamical behavior of the solution set to the nonsmooth systems is established, in which three theorems for determining the existence of global attractors are delivered. Finally, four concrete examples are explored and numerical experiments are carried out to demonstrate consistency with the theoretical results.
This work constructs a cluster algebra structure within the quantum cohomology ring of a flag variety. Let Fl:=Fl(N_1,… ,N_n+1) denote a partial flag variety of length n, and QH_S^*(Fl)[t]:=QH_S^*(Fl)⊗ℂ[t] be its equivariant quantum cohomology ring extended by a formal variable t, regarded as a ℚ -algebra. We establish an injective ℚ -algebra homomorphism from the A_n -type cluster algebra to the algebra QH_S^*(Fl)[t]. Furthermore, for a general quiver with potential, we propose a framework for constructing a homomorphism from the associated cluster algebra to the quantum cohomology ring of the corresponding quiver variety. The second main result addresses the conjecture of all-genus Seiberg duality for A_n -type quivers. For any quiver with potential mutation-equivalent to an A_n -type quiver, we consider the associated variety defined as the critical locus of the potential. We prove that all-genus Gromov–Witten invariants of such a variety coincide with those of the flag variety.
