Indiana Univ Math Research Articles

On a question of Bhatia, Friedland and Jain II

Let p 1 < p 2 < ⋯ < p n be positive numbers and r a non-negative real number. The Loewner matrix associated with the function x r + 1 given by L r + 1 = [ p i r + 1 − p j r + 1 p i − p j ] and matrix P r = [ ( p i + p j ) r ] (the Hadamard inverse of rth Hadamard power of well-known Cauchy matrix) have same inertia. A question was left open in Inertia of Loewner matrices. Indiana Univ Math J. 2016;65(4):1251–1261 by Bhatia, Friedland and Jain to find a connection between these two matrix families. We aim to answer this question firmly in terms of a congruence relation between L r + 1 and P r . Indeed, a non-singular matrix X over R is explicitly obtained such that X ′ P r X = L r + 1 in this paper.

Alexandrov’s Theorem for Anisotropic Capillary Hypersurfaces in the Half-Space

In this paper, we show that any embedded capillary hypersurface in the half-space with anisotropic constant mean curvature is a truncated Wulff shape. This extends Wente’s result (Pac J Math 88:387–397, 1980. https://doi.org/10.2140/pjm.1980.88.387) to the anisotropic case and He–Li–Ma–Ge’s result (Indiana Univ Math J 58(2):853–868, 2009. https://doi.org/10.1512/iumj.2009.58.3515) to the capillary boundary case. The main ingredients in the proof are a new Heintze-Karcher inequality and a new Minkowski formula, which have their own interest.

John—Nirenberg-Q Spaces via Congruent Cubes

To shed some light on the John—Nirenberg space, the authors of this article introduce the John—Nirenberg-Q space via congruent cubes, \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\), which, when p = ∞ and q = 2, coincides with the space Qα (ℝn) introduced by Essén, Janson, Peng and Xiao in [Indiana Univ Math J, 2000, 49(2): 575-615]. Moreover, the authors show that, for some particular indices, \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\) coincides with the congruent John—Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\). Furthermore, the authors characterize \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\) via mean oscillations, and then use this characterization to study the dyadic counterparts. Also, the authors obtain some properties of composition operators on such spaces. The main novelties of this article are twofold: establishing a general equivalence principle for a kind of ‘almost increasing’ set function that is here introduced, and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.

$$L^{p}$$ Estimates and Weighted Estimates of Fractional Maximal Rough Singular Integrals on Homogeneous Groups

In this paper, we study the \(L^{p}\) boundedness and \(L^{p}(w)\) boundedness (\(1<p<\infty \) and w a Muckenhoupt \(A_{p}\) weight) of fractional maximal singular integral operators \(T_{\Omega ,\alpha }^{\#}\) with homogeneous convolution kernel \(\Omega (x)\) on an arbitrary homogeneous group \({\mathbb {H}}\) of dimension \({\mathbb {Q}}\). We show that if \(0<\alpha <{\mathbb {Q}}\), \(\Omega \in L^{1}(\Sigma )\) and satisfies the cancellation condition of order \([\alpha ]\), then for any \(1<p<\infty \), $$\begin{aligned} \Vert T_{\Omega ,\alpha }^{\#}f\Vert _{L^{p}({\mathbb {H}})}\lesssim \Vert \Omega \Vert _{L^{1}(\Sigma )}\Vert f\Vert _{L_{\alpha }^{p}({\mathbb {H}})}. \end{aligned}$$where for the case \(\alpha =0\), the \(L^p\) boundedness of rough singular integral operator and its maximal operator were studied by Tao (Indiana Univ Math J 48:1547–1584, 1999) and Sato (J Math Anal Appl 400:311–330, 2013), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if \(0\le \alpha <{\mathbb {Q}}\) and \(\Omega \) satisfies the same cancellation condition but a stronger condition that \(\Omega \in L^{q}(\Sigma )\) for some \(q>{\mathbb {Q}}/\alpha \), then for any \(1<p<\infty \) and \(w\in A_{p}\), $$\begin{aligned} \Vert T_{\Omega ,\alpha }^{\#}f\Vert _{L^{p}(w)}\lesssim \Vert \Omega \Vert _{L^{q}(\Sigma )}\{w\}_{A_p}(w)_{A_p}\Vert f\Vert _{L_{\alpha }^{p}(w)},\ \ 1<p<\infty . \end{aligned}$$

Exterior Dirichlet and Neumann Problems and the Linked Ergodic Inverse Problems in the Entire Space

The paper is mainly concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the unit disk in {mathbb {R}}^2 and the unit ball in {mathbb {R}}^3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. The basis is the theory of semigroups of linear operators mapping a Banach space X into itself. The classical one-parameter theory for semigroups applies in the present particular applications, actually for X= L^2_{2pi } in case of the unit disk, and X=L^2(S) in the three dimensional setting, S being the unit sphere in {mathbb {R}}^3. Another tool is a Drazin-like inverse operator B for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator B is a closed, not necessarily bounded, operator. It was introduced in a paper with Butzer and Westphal (Indiana Univ Math J 20:1163–1174, 1970/1971) and extended to a generalized setting with Butzer and Koliha (J Oper Theory 62:297–326, 2009).

Existence and nonexistence results of polyharmonic boundary value problems with supercritical growth

Consider the following polyharmonic problems under the Dirichlet or the Navier boundary conditions where is a bounded smooth domain and p > 1 is a subcritical power. We examine the effect of the positive parameter λ to study the existence and the nonexistence of regular solutions. When λ is large enough, we establish an existence result only undersuitable growth condition on f at zero. Our approach is based on truncation argument as well as -bounds. Also, by virtue of Pucci-Serrin's variational identity [Pucci P, Serrin J. A general variational identity. Indiana Univ Math J. 1986;35:681–703.],, we provide a nonexistence result when f has a supercritical growth at infinity and λ is small enough.

The General Dual-Polar Orlicz–Minkowski Problem

This paper gives a systematic study to the general dual-polar Orlicz–Minkowski problem (e.g., Problem 4.1). This problem involves the general dual volume \({\widetilde{V}}_G(\cdot )\) recently proposed in Gardner et al. (Calc Var PDE 58:35, 2019; 59:33, 2020) in order to study the general dual Orlicz–Minkowski problem. As \({\widetilde{V}}_G(\cdot )\) extends the volume and the qth dual volume, the general dual-polar Orlicz–Minkowski problem is “polar" to the recently initiated general dual Orlicz–Minkowski problem in Gardner et al. (Calc Var PDE 58:35, 2019; 59:33, 2020) and “dual" to the newly proposed polar Orlicz–Minkowski problem in Luo et al. (Indiana Univ Math J 69:385–420, 2020). The existence, uniqueness and continuity, if applicable, for the solutions to the general dual-polar Orlicz–Minkowski problem are established. Polytopal solutions and/or counterexamples to the general dual-polar Orlicz–Minkowski problem for discrete measures are also provided. Several variations of the general dual-polar Orlicz–Minkowski problem are discussed as well, in particular the one leading to the general Orlicz–Petty bodies.

On algebraic differential equations concerning the Riemann-zeta function and the Euler-gamma function

In this paper, we prove that ζ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in over the ring of polynomials in , are nonnegative integers. We extend the result that ζ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in over the field of complex numbers, which is proved by Li and Ye [Li BQ, Ye Z. Algebraic differential equations concerning the Riemann zeta function and the Euler gamma function. Indiana Univ Math J. 2010;59:1405–1416.].

On the Hölder regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian

In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in Silvestre (Indiana Univ Math J 55:1155–1174, 2006), we adapt the De Giorgi technique to achieve the \(C^{\gamma }\)-regularity for solutions of class \(C^{2}\) and use the geometry found in Caffarelli et al. (Math Ann 360(3–4): 681–714, 2014) to get an ABP estimate, a Harnack inequality and the interior \(C^{1, \gamma }\) regularity for viscosity solutions.

On the global existence for the compressible Euler–Poisson system, and the instability of static solutions

We consider the Cauchy problem for the barotropic Euler system coupled to Poisson equation, in the whole space. Our main aim is to exhibit a simple functional framework that allows to handle solutions with density going to zero at infinity, but that need not be compactly supported. We have in mind in particular the 3D static solution, when the polytropic index $$\gamma $$ of the gas is equal to 6/5. Our first result is the local existence of classical solutions in a simple functional framework that does not require the velocity to tend to 0 at infinity and the density to be compactly supported. Next, following the work by Grassin and Serre dedicated to the compressible Euler system Grassin and Serre (C R Acad Sci Paris Ser I 325:721–726, 1997, Grassin (Indiana Univ Math J, 47:1397–1432, 1998), we show that if the initial density is small enough, and the initial velocity is close to some reference vector field $$u_0$$ such that the spectrum of $$Du_0$$ is positive and bounded away from zero, then the corresponding classical solution is global, and satisfies algebraic time decay estimates. Compared to our recent paper (Blanc et al. in The global existence issue for the compressible Euler system with Poisson or Helmholtz coupling), we are able to handle the 3D static solution that was mentioned above, and to show its instability, within our functional framework.

Genericity of Nondegenerate Free Boundary CMC Embeddings

Let $$\Sigma ^n$$ and $$M^{n+1}$$ be smooth manifolds with smooth boundary. In this paper, following the techniques developed by White (Indiana Univ Math J 40:161–200, 1991) and Biliotti–Javaloyes–Piccione (Indiana Univ Math J, 1797–1830, 2009), we prove that, given a compact manifold with boundary $$\Sigma ^n$$ and a manifold with boundary $$M^{n+1}$$ , for a generic set of Riemannian metrics on M every free boundary CMC embedding $$\phi :\Sigma \rightarrow M$$ is non-degenerate.

Approximation in weighted Bergman spaces and Hankel operators on strongly pseudoconvex domains

Suppose D is a bounded strongly pseudoconvex domain in $${{\mathbb {C}}}^n$$ with smooth boundary, and let $$\rho $$ be its defining function. For $$1< p<\infty $$ and $$\alpha >-1$$ , we show that the weighted Bergman projection $$P_\alpha $$ is bounded on $$L^p(D, |\rho |^\alpha dV)$$ . With non-isotropic estimates for $$\overline{\partial }$$ and Stein’s theorem on non-tangential maximal operators, we prove that bounded holomorphic functions are dense in the weighted Bergman space $$A^p(D, |\rho |^\alpha dV)$$ , and hence Hankel operators can be well defined on these spaces. For all $$1<p, q<\infty $$ we characterize bounded (resp. compact) Hankel operators from p-th weighted Bergman space to q-th weighted Lebesgue space with possibly different weights. As a consequence, we generalize the main results in Pau et al. (Indiana Univ Math J 65:1639–1673, 2016) and resolve a question posed in Lv and Zhu (Integr Equ Oper Theory, 2019).

Generic uniqueness of expanders with vanishing relative entropy

We define a relative entropy for two self-similarly expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White (Indiana Univ Math J 36(3):567–602, 1987) and using recent results of Bernstein (Asymptotic structure of almost eigenfunctions of drift laplacians on conical ends) and Bernstein-Wang (The space of asymptotically conical self-expanders of mean curvature flow), we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique.

On the Wandering Property in Dirichlet spaces

We show that in a scale of weighted Dirichlet spaces $$D_{\alpha }$$, including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in $$D_{\alpha }$$ such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance, when $$B(z)=z^k$$ and $$|\alpha | \le \frac{\log (2)}{\log (k+1)}$$, the chosen norm is the usual one in $$D_\alpha $$.

Nearly optimal stability for Serrin’s problem and the Soap Bubble theorem

We present new quantitative estimates for the radially symmetric configuration concerning Serrin’s overdetermined problem for the torsional rigidity, Alexandrov’s Soap Bubble theorem, and other related problems. The new estimates improve on those obtained in Magnanini and Poggesi (J Anal Math, 139(1), 179–205, 2019), Magnanini and Poggesi (Indiana Univ Math J, arXiv:1708.07392, 2017) and are in some cases optimal.

On a question of Bhatia, Friedland and Jain

ABSTRACT Let be positive numbers, then for any integer m the Loewner matrix associated with the function is given by A question was left open in a paper [Bhatia R, Friedland S, Jain T. Inertia of loewner matrices. Indiana Univ Math J. 2016;65:1251–1261] by Bhatia, Friedland and Jain to find formulas for the determinants of the matrices in the case We aim to answer this question firmly in terms of the Schur polynomials in this paper.

The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

We characterize the weights for the Stieltjes transform and the Calderon operator to be bounded on the weighted variable Lebesgue spaces $$L_w^{p(\cdot )}(0,\infty )$$ , assuming that the exponent function $${p(\cdot )}$$ is log-Holder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals $$\{ (0,b) : b>0\}$$ on $$(0,\infty )$$ . Our results extend those in Duoandikoetxea et al. (Indiana Univ Math J 62(3):891–910, 2013) for the constant exponent $$L^p$$ spaces with weights. We also give two applications: the first is a weighted version of Hilbert’s inequality on variable Lebesgue spaces, and the second generalizes the results in Soria and Weiss (Indiana Univ Math J 43(1):187–204, 1994) for integral operators to the variable exponent setting.

Hankel operators induced by radial Bekoll\xe9\u2013Bonami weights on Bergman spaces

We study big Hankel operators H_f^nu :A^p_omega rightarrow L^q_nu generated by radial Bekollé–Bonami weights nu , when 1<ple q<infty . Here the radial weight omega is assumed to satisfy a two-sided doubling condition, and A^p_omega denotes the corresponding weighted Bergman space. A characterization for simultaneous boundedness of H_f^nu and H_{{overline{f}}}^nu is provided in terms of a general weighted mean oscillation. Compared to the case of standard weights that was recently obtained by Pau et al. (Indiana Univ Math J 65(5):1639–1673, 2016), the respective spaces depend on the weights omega and nu in an essentially stronger sense. This makes our analysis deviate from the blueprint of this more classical setting. As a consequence of our main result, we also study the case of anti-analytic symbols.

The Second Inner Variation of Energy and the Morse Index of Limit Interfaces

In this article, we study the second variation of the energy functional associated to the Allen–Cahn equation on closed manifolds. Extending well-known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of Le (Indiana Univ Math J 60:1843–1856, 2011; J Math Pures Appl 103:1317–1345, 2015)) and Hiesmayr (arXiv:1704.07738 preprint [math.DG], 2017).

On a Subclass of Solutions of the 2D Navier–Stokes Equations with Constant Energy and Enstrophy

In this paper, we are interested in studying the existence of the nonstationary solutions in the global attractor of the 2D Navier–Stokes equations with constant energy and enstrophy. We particularly focus on a subclass of these solutions that the geometric structures have a supplementary stability property which were called “chained ghost solutions” in Tian and Zhang (Indiana Univ Math J 64:1925–1958, 2015). By solving a Galerkin system of a particular form, we show the nonexistence of the chained ghost solutions for certain cases.