Math Anal Research Articles

Convergence rates for the Allen–Cahn equation with boundary contact energy: the non-perturbative regime

We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90^circ . We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order {varepsilon ^{frac{1}{2}}} for general contact angles alpha in (0,pi ). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order varepsilon ; again for general contact angles alpha in (0,pi ). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).

Set-valued equilibrium problems based on the concept of null set with applications

In this paper, by means of the null set defined by Wu (J Math Anal Appl. 2019;472:1741–1761), we introduce a set-valued equilibrium problem based on the null set, where the objective mapping takes values in a hyperspace equipped with a convex cone. Moreover, we obtain new existence results for set-valued equilibrium problems defined on compact or noncompact sets. Some applications are given to set optimization problems, to a set-valued variational inequality, to saddle point theorems for set-valued mappings, and to a generalized noncooperative game.

Ill-Posedness Issues on (abcd)-Boussinesq System

In this paper, we consider the Cauchy problem for (abcd)-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona et al. (J Nonlinear Sci 12:283–318, 2002, Nonlinearity 17:925–952, 2004), describes a small-amplitude waves on the surface of an inviscid fluid, and is derived as a first order approximation of incompressible, irrotational Euler equations. We mainly establish the ill-posedness of the system under various parameter regimes, which generalize the result of one-dimensional BBM–BBM case by Chen and Liu (Anal Math 121:299–316, 2013). Among results established here, we emphasize that the ill-posedness result for two-dimensional BBM–BBM system is optimal. The proof follows from an observation of the high to low frequency cascade present in nonlinearity, motivated by Bejenaru and Tao (J Funct Anal 233:228–259, 2006).

A generalized Wigner–Yanase skew information

We generalize a two-parameter extended Wigner–Yanase skew information given by Z. Zhang [J Math Anal Appl. 2021;497(1): 124851] to any general operator monotone function. We prove several fundamental properties including the joint concavity and an uncertainty relation for the proposed generalized Wigner–Yanase skew information.

Local Existence of Solutions to a Navier–Stokes-Nonlinear-Schrödinger Model of Superfluidity

In Pitaevskii (Sov Phys JETP 35(8):282–287, 1959), a micro-scale model of superfluidity was derived from first principles, to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model couples two of the most fundamental PDEs in mathematics: the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations (NSE). In this article, we show the local existence of solutions—strong in wavefunction and velocity, weak in density—to this system in a smooth bounded domain in 3D, by deriving the required a priori estimates. (We will also establish an energy inequality obeyed by the weak solutions constructed in Kim (SIAM J Math Anal 18(1):89–96, 1987) for the incompressible, inhomogeneous NSE.) To the best of our knowledge, this is the first rigorous mathematical analysis of a bidirectionally coupled system of the NLS and NSE.

Some A-spectral radius inequalities for A-bounded Hilbert space operators

Let $$r_A(T)$$ denote the A-spectral radius of an operator T which is bounded with respect to the seminorm induced by a positive operator A on a complex Hilbert space $${\mathcal {H}}$$ . In this paper, we aim to establish several A-spectral radius inequalities for products, sums and commutators of A-bounded operators. Some applications of our results are provided. Moreover, we give an affirmative answer to the question recently posed by Baklouti and Namouri (Banach J Math Anal 16:12, https://doi.org/10.1007/s43037-021-00167-1 , 2022) regarding the connection between the notions of A-spectral radius and A-spectrum for A-bounded operators.

Entire function and its linear C-shift operator sharing a set of small functions CM

This paper has been devoted to study the relation between an entire function of hyper order and its linear c-shift operator sharing a set of small functions. Our first theorem improves a very recent result of the present authors [Meromorphic function of restricted hyper-order sharing one or two sets with its linear C-shift operator. Anal Math. 2021;47(4):747–779.], while the corollary deducted from the same completely provides an answer to an open question posed by Liu [Meromorphic functions sharing a set with applications to difference equations. J Math Anal Appl. 2009;359:384–393]. Most importantly, our another theorem improves and extends a result due to Li [Entire functions sharing a finite set with their difference operators. Comput Methods Funct Theory. 2012;12(1):307–328] which in turn rectifies an erroneous result of Ma-Qi-Zhang [Further results about a special Fermat-type difference equation. J Funct Spaces. 2020. Article ID 3205357]. We have provided a handful number of examples in support of our certain claims. In the last section, as an application of our main result, we extend that of Liu-Ma-Zhai [The generalized Fermat type difference equations. Bull Korean Math Soc. 2018;55(6):1845–1858].

On the H\xf6lder Regularity of a Linear Stochastic Partial-Integro-Differential Equation with Memory

In light of recent work on particles fluctuating in linear viscoelastic fluids, we study a linear stochastic partial-integro-differential equation with memory that is driven by a stationary noise on a bounded, smooth domain. Using the framework of generalized stationary solutions introduced in McKinley and Nguyen (SIAM J Math Anal 50(5):5119–5160, 2018), we provide conditions on the differential operator and the noise to obtain the existence as well as Hölder regularity of the stationary solutions for the concerned equation. As an application of the regularity results, we compare to analogous classical results for the stochastic heat equation. When the 1d stochastic heat equation is driven by white noise, solutions are continuous with space and time regularity that is Hölder (1/2-epsilon ) and (1/4-epsilon ) respectively. When driven by colored-in-space noise, solutions can have a range of regularity properties depending on the structure of the noise. Here, we show that the particular form of colored-in-time memory that arises in viscoelastic diffusion applications, satisfying what is called the Fluctuation–Dissipation relationship, yields sample paths that are Hölder (1/2-epsilon ) and (1/2-epsilon ) in space and time.

Invariant vector means and complementability of Banach spaces in their second duals

Let X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of X^{**}. We then show that X is complemented in X^{**} if and only if there exists an invariant mean M:ell _infty (S,X)rightarrow X. This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).

New stability result of a partially dissipative viscoelastic Timoshenko system with a wide class of relaxation function

This paper is concerned with the following partially dissipative viscoelastic Timoshenko system with damping mechanism acting only on the shear force, and with Dirichlet boundary conditions. We consider a very general relaxation function Under appropriate conditions on ξ and H, we establish a general stability result. The result is obtained under the assumption of equal speed of wave propagation. This work extends and generalizes many results in literature such as Alves et al. [On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J Math Anal. 2019;51(6):4520–4543].

Partial regularity for the optimal p-compliance problem with length penalization

We establish a partial \(C^{1,\alpha }\) regularity result for minimizers of the optimal p-compliance problem with length penalization in any spatial dimension \(N\ge 2\), extending some of the results obtained in Chambolle et al. (SIAM J Math Anal 49(2):1166–1224, 2017) and Bulanyi and Lemenant (ESAIM Control Optim Calc Var 27:35, 2021). The key feature is that the \(C^{1,\alpha }\) regularity of minimizers for some free boundary type problem is investigated with a free boundary set of codimension \(N-1\). We prove that every optimal set cannot contain closed loops, cannot contain quadruple points, and it is \(C^{1,\alpha }\) regular at \({\mathcal {H}}^{1}\)-a.e. point for every \(p\in (N-1,+\infty )\).

Hankel determinants of linear combinations of moments of orthogonal polynomials, II

We present a formula that expresses the Hankel determinants of a linear combination of length d+1 of moments of orthogonal polynomials in terms of a dtimes d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

Zero-curvature point of minimal graphs

Motivated by a classical result of Finn and Osserman (J Anal Math 343(12):351–364, 1964), who proved that the Jenkins–Serrin graph over the square inscribed in the unit disk is extremal for the Gaussian curvature of the point O (so-called centre) of the minimal graphs above the center 0 of unit unit disk, provided the tangent plane is horizontal, we ask and answer to the question concerned the extremal of "second derivative" of the Gaussian curvature of such graphs provided that its curvature at O is zero. We prove that the extremals are certain Jenkins–Serrin graph over the regular hexagon inscribed in the unit disk, provided that the Gaussian curvature vanishes and the tangent plane is horizontal at the centre.

Comments on fixed point results in classes of function with values in a b\u2013metric space

We prove and discuss several fixed point results for nonlinear operators, acting on some classes of functions with values in a b-metric space. Thus we generalize and extend a recent theorem of Dung and Hang (J Math Anal Appl 462:131–147, 2018), motivated by several outcomes in Ulam type stability. As a simple consequence we obtain, in particular, that approximate (in some sense) eigenvalues of some linear operators, acting in some function spaces, must be eigenvalues while approximate eigenvectors are close to eigenvectors with the same eigenvalue. Our results also provide some natural generalizations and extensions of the classical Banach Contraction Principle.

Multiplier Algebras of Normed Spaces of Continuous Functions

In this article, we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties of the NSCF. We show that it is often possible to construct NSCF’s which only admit constant multipliers. To do that, using a method from Mashreghi and Ransford (Anal Math Phys 9(2):899–905, 2019), we prove that any separable Banach space can be realized as a NSCF over any separable metrizable space of infinite cardinality. On the other hand, we give a sufficient condition for non-separability of a multiplier algebra.

Alberti–Uhlmann Problem on Hardy–Littlewood–Pólya Majorization

We fully describe the doubly stochastic orbit of a self-adjoint element in the noncommutative \(L_1\)-space affiliated with a semifinite von Neumann algebra, which answers a problem posed by Alberti and Uhlmann (Stochasticity and partial order: doubly stochastic maps and unitary mixing. VEB Deutscher Verlag der Wissenschaften, Berlin, 1982) in the 1980s, extending several results in the literature. It follows further from our methods that, for any \(\sigma \)-finite von Neumann algebra \({{\mathcal {M}}}\) equipped with a semifinite infinite faithful normal trace \(\tau \), there exists a self-adjoint operator \(y\in L_1({{\mathcal {M}}},\tau )\) such that the doubly stochastic orbit of y does not coincide with the orbit of y in the sense of Hardy–Littlewood–Pólya, which confirms a conjecture by Hiai (J Math Anal Appl 127:18–48, 1987). However, we show that Hiai’s conjecture fails for non-\(\sigma \)-finite von Neumann algebras. The main result of the present paper also answers the (noncommutative) infinite counterparts of problems due to Luxemburg (Proc Symp Anal Queen’s univ 83–144, 1967) and Ryff (Pac J Math 13:1379–1386, 1963) in the 1960s.

Bilinear Spherical Maximal Functions of Product Type

In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón–Zygmund theory. This operator is different from the bilinear spherical maximal function considered by Geba et al. in (Math Res Lett 20(4):675–694, 2013). We deal with lacunary and full versions of this operator, and prove weighted estimates with respect to genuine bilinear weights beyond the Banach range. Our results are implied by sharp sparse domination for both the operators, following ideas by Lacey (J Anal Math 139(2):613–635, 2019). In the case of the lacunary maximal operator we also use interpolation of analytic families of operators to address the weighted boundedness for the whole range of tuples.

Disconnection and Entropic Repulsion for the Harmonic Crystal with Random Conductances

We study level-set percolation for the harmonic crystal on $${\mathbb {Z}}^d$$ , $$d \ge 3$$ , with uniformly elliptic random conductances. We prove that this model undergoes a non-trivial phase transition at a critical level that is almost surely constant under the environment measure. Moreover, we study the disconnection event that the level-set of this field below a level $$\alpha $$ disconnects the discrete blow-up of a compact set $$A \subseteq {\mathbb {R}}^d$$ from the boundary of an enclosing box. We obtain quenched asymptotic upper and lower bounds on its probability in terms of the homogenized capacity of A, utilizing results from Neukamm, Schäffner and Schlömerkemper (SIAM J Math Anal 49(3):1761–1809, 2017). Furthermore, we give upper bounds on the probability that a local average of the field deviates from some profile function depending on A, when disconnection occurs. The upper and lower bounds concerning disconnection that we derive are plausibly matching at leading order. In this case, this work shows that conditioning on disconnection leads to an entropic push-down of the field. The results in this article generalize the findings of Nitzschner (Electron J Probab 23:105, 2018) and Chiarini and Nitzschner (Probab Theory Relat Fields 177(1–2):525–575, 2020) which treat the case of constant conductances. Our proofs involve novel “solidification estimates” for random walks, which are similar in nature to the corresponding estimates for Brownian motion derived by Nitzschner and Sznitman (J Eur Math Soc. 22:2629–2672, 2020).

Structure of the Lipschitz free p-spaces $${\mathcal{F}}_p({\mathbb{Z}}^d)$$ and $${\mathcal{F}}_p({\mathbb{R}}^d)$$ for $$0<p\le 1$$

Our aim in this article is to contribute to the theory of Lipschitz free p-spaces for \(0<p\le 1\) over the Euclidean spaces \({\mathbb{R}}^d\) and \({\mathbb{Z}}^d\). To that end, we show that \({\mathcal{F}}_p({\mathbb{R}}^d)\) admits a Schauder basis for every \(p\in (0,1]\), thus generalizing the corresponding result for the case \(p=1\) by Hájek and Pernecká (J Math Anal Appl 416(2):629–646, 2014, Theorem 3.1) and answering in the positive a question that was raised by Albiac et al. in (J Funct Anal 278(4):108354, 2020). Explicit formulas for the bases of \({\mathcal{F}}_p({\mathbb{R}}^d)\) and its isomorphic space \({\mathcal{F}}_p([0,1]^d)\) are given. We also show that the well-known fact that \({\mathcal{F}}({\mathbb{Z}})\) is isomorphic to \(\ell _{1}\) does not extend to the case when \(p<1\), that is, \({\mathcal{F}}_{p}({\mathbb{Z}})\) is not isomorphic to \(\ell _{p}\) when \(0<p<1\).

AgFlow: fast model selection of penalized PCA via implicit regularization effects of gradient flow

Principal component analysis (PCA) has been widely used as an effective technique for feature extraction and dimension reduction. In the High Dimension Low Sample Size setting, one may prefer modified principal components, with penalized loadings, and automated penalty selection by implementing model selection among these different models with varying penalties. The earlier work (Zou et al. in J Comput Graph Stat 15(2):265–286, 2006; Gaynanova et al. in J Comput Graph Stat 26(2):379–387, 2017) has proposed penalized PCA, indicating the feasibility of model selection in \(\ell _2\)-penalized PCA through the solution path of Ridge regression, however, it is extremely time-consuming because of the intensive calculation of matrix inverse. In this paper, we propose a fast model selection method for penalized PCA, named approximated gradient flow (AgFlow), which lowers the computation complexity through incorporating the implicit regularization effect introduced by (stochastic) gradient flow (Ali et al. in: The 22nd international conference on artificial intelligence and statistics, pp 1370–1378, 2019; Ali et al. in: International conference on machine learning, 2020) and obtains the complete solution path of \(\ell _2\)-penalized PCA under varying \(\ell _2\)-regularization. We perform extensive experiments on real-world datasets. AgFlow outperforms existing methods (Oja and Karhunen in J Math Anal Appl 106(1):69–84, 1985; Hardt and Price in: Advances in neural information processing systems, pp 2861–2869, 2014; Shamir in: International conference on machine learning, pp 144–152, PMLR, 2015; and the vanilla Ridge estimators) in terms of computation costs.