Volume 25, Issue 4 - Special Issue: Geometric and functional inequalities and applications 2025

A Schrödinger operator that is bounded below and has a unique positive ground state can be transformed into a Dirichlet form operator by the ground state transformation. If the resulting Dirichlet form operator is hypercontractive, Davies and Simon call the Schrödinger operator “intrinsically hypercontractive”. I will show that if one adds a suitable potential onto an intrinsically hypercontractive Schrödinger operator it remains intrinsically hypercontractive. The proof uses a fortuitous relation between the WKB equation and logarithmic Sobolev inequalities. All bounds are dimension independent. The main theorem will be applied to several examples.

We consider the problem of minimizing the lowest eigenvalue of the Schrödinger operator −Δ + V in L 2 ( R d ) ${L}^{2}({\mathbb{R}}^{d})$ when the integral ∫ e − tV  d x is given for some t > 0. We show that the eigenvalue is minimal for the harmonic oscillator and derive a quantitative version of the corresponding inequality.

In this paper, we study the drifted Laplacian Δ f on a hypersurface M in a Ricci shrinker ( M ̄ , g , f ) $\left(\bar{M},g,f\right)$ . We prove that the spectrum of Δ f is discrete for immersed hypersurfaces with bounded weighted mean curvature in a Ricci shrinker with a mild condition on the potential function. Next, we give a lower bound for the first nonzero eigenvalue of Δ f when the hypersurface is an embedded f -minimal one. This estimate contains the case of compact minimal hypersurfaces in a positive Einstein manifold, in particular Choi and Wang’s estimate for minimal hypersurfaces in a round sphere. The estimate also recovers the ones of Ding-Xin and Brendle-Tsiamis on self-shrinkers.

In this paper, we first establish the Talenti comparison principle for anisotropic p -Laplacian equation with Robin boundary conditions. This achievement not only extends classical Talenti comparison result for Laplacian equation with Robin boundary conditions in (A. Alvino, C. Nitsch, and C. Trombetti, “A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions,” Commun. Pure Appl. Math. , vol. 76, no. 3, pp. 585–603, 2023, doi:10.1002/cpa.22090), but also for the anisotropic Laplacian cases in (R. Sannipoli, “Comparison results for solutions to the anisotropic Laplacian with Robin boundary conditions,” Nonlinear Anal. , vol. 214, no. 112615, 2022, doi:10.1016/j.na.2021.112615). Furthermore, we also obtain the rigidity for this kind of Talenti comparison principle. This naturally covers the rigidity for Talenti comparison principle involving anisotropic Laplacian operator which has not been proved before. As an application of rigidity result for anisotropic Talenti comparison principle, we furthermore establish the rigidity of anisotropic Faber–Krahn inequality with Robin boundary conditions and naturally solve the rigidity for classical Faber–Krahn inequality with Robin boundary condition established by Alvino-Nitsch-Trombetti in (A. Alvino, C. Nitsch, and C. Trombetti, “A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions,” Commun. Pure Appl. Math. , vol. 76, no. 3, pp. 585–603, 2023, doi:10.1002/cpa.22090). Significantly, our results also answer some open problems contained in (A. L. Masiello and G. Paoli, “A rigidity result for the Robin torsion problem,” J. Geom. Anal. , vol. 33, no. 5, p. 149, 2023, doi:10.1007/s12220-023-01202-3) and (A. L. Masiello and G. Paoli, “Rigidity results for the p-Laplacian poisson problem with robin boundary conditions,” J. Optim. Theor. Appl. , vol. 202, pp. 628–648, 2024, doi:10.1007/s10957-024-02442-1).

On general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using the Rumin complex has been considered in (M. Rumin, “Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups,” C. R. Acad. Sci., Paris Sér. I Math. , vol. 329, no. 11, pp. 985–990, 1999, M. Rumin, “Sub-Riemannian limit of the differential form spectrum of contact manifolds,” Geom. Funct. Anal. , vol. 10, no. 2, pp. 407–452, 2000), where the author introduced a 0-order pseudodifferential operator on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this 0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge-Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address in full generality, the main issue being that the Rumin exterior differential d c is not homogeneous on arbitrary Carnot groups. In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators, and we introduce three possible definitions of hypoelliptic Hodge-Laplacians. We compare how these three possible Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups.

This paper explores the existence and properties of basic eigenvalues and eigenfunctions associated with the Riemannian Laplacian on closed, connected Riemannian manifolds featuring an effective isometric action by a compact Lie group. Our primary focus is on investigating the potential existence of homeomorphic yet not diffeomorphic smooth manifolds that can accommodate invariant metrics sharing common basic spectra. We establish the occurrence of such scenarios for specific homotopy spheres and connected sums. Moreover, the developed theory demonstrates that the ring of invariant admissible scalar curvature functions fails to recover the smooth structure in many examples. We show the existence of homotopy spheres with identical rings of invariant scalar curvature functions, irrespective of the underlying smooth structure.

Using the notion of a Bessel pair, we study the Hardy type inequalities in the setting of Dunkl operator. We also establish a general symmetrization principle for weighted Hardy type inequalities with Dunkl operator in the situation that the standard Schwarz symmetrization is not applicable.

The present article investigates the existence, multiplicity and regularity of weak solutions of problem involving a combination of critical Hartree-type nonlinearity along with singular and discontinuous nonlinearities (see ( P λ ) $\left({\mathcal{P}}_{\lambda }\right)$ below). By applying variational methods and using the notion of generalized gradients for Lipschitz continuous functional, we obtain the existence and the multiplicity of weak solutions for some suitable range of λ and γ . Finally by studying the L ∞ -estimates and boundary behaviour of weak solutions, we prove their Hölder and Sobolev regularity.

We study the asymptotic behavior of positive least energy solutions to the weakly coupled nonlinear Schrödinger systems with nearly critical exponents. There are at most three kinds of asymptotic behaviors for positive solutions to two-coupled systems: (i) both two components converge; (ii) one component blows up while the other one converges; (iii) both two components blow up. Here we prove that any type of these three asymptotic behaviors will happen under different circumstances.

We consider the sharpness of functional inequality which we call Rellich–Hardy inequality with power weight for curl-free vector fields on R N ${\mathbb{R}}^{N}$ . This inequality can be considered as an intermediate between Hardy–Leray and Rellich–Leray inequalities and its best constant was originally found by Tertikas-Zographopoulos (“Best constants in the Hardy–Rellich inequalities and related improvements,” Adv. Math. , vol. 209, no. 2, pp. 407–459, 2007) for unconstrained fields. Under the curl-free condition, we compute a new best value of the constant in the same inequality and show it is unattainable. This paper is a sequel to (N. Hamamoto and F. Takahashi, “Sharp Hardy–Leray and Rellich–Leray inequalities for curl-free vector fields,” Math. Ann. , vol. 379, no. 1, pp. 719–742, 2021, N. Hamamoto and F. Takahashi, “Sharp Hardy-Leray inequality for curl-free fields with a remainder term,” J. Funct. Anal. , vol. 280, no. 1, 2021, Art. no. 108790).

This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a ( p , N )-Laplace Schrödinger equation with logarithmic nonlinearity and critical exponential nonlinearity in the sense of Trudinger-Moser in the whole Euclidean space R N ${\mathbb{R}}^{N}$ . Through the use of smooth variational methods, penalization techniques, and the application of the Lusternik–Schnirelmann category theory, we establish a connection between the number of positive solutions and the topological properties of a set in which the potential function achieves its minimum values.