An Alexandroff–Bakelman–Pucci estimate on Riemannian manifolds

In Cabré (1997) [2], Cabré established an Alexandroff–Bakelman–Pucci (ABP) estimate on Riemannian manifolds with non-negative sectional curvatures and applied it to establish the Krylov–Safonov Harnack inequality on manifolds with non-negative sectional curvatures. In the present paper, we generalize the results of [2]. We obtain an ABP estimate on manifolds with Ricci curvatures bounded from below and apply this estimate to prove the Krylov–Safonov Harnack inequality on manifolds with sectional curvatures bounded from below. We also use this ABP estimate to study Minkowski-type inequalities.     

Aleksandrov–Bakelman–Pucci maximum principles for a class of uniformly elliptic and parabolic integro-PDE

We prove generalized Aleksandrov–Bakelman–Pucci maximum principles for elliptic and parabolic integro-PDEs of Hamilton–Jacobi–Bellman–Isaacs types, whose PDE parts are either uniformly elliptic or uniformly parabolic. The proofs of these results are based on the classical Aleksandrov–Bakelman–Pucci maximum principles for the elliptic and parabolic PDEs and an iteration procedure using solutions of Pucci extremal equations. We also provide proofs of nonlocal versions of the classical Aleksandrov–Bakelman–Pucci maximum principles for elliptic and parabolic integro-PDEs.     

A stability estimate for the Aleksandrov–Fenchel inequality under regularity assumptions

We give, under appropriate regularity assumptions, a strengthening of the Aleksandrov–Fenchel inequality in the form of a stability estimate.     

The quantitative Faber–Krahn inequality for the Robin Laplacian

We prove a quantitative form of the Faber–Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.     

Harnack estimate for the H~k-flow

We obtain the Harnack estimate of the solution to H~k-flow in Euclidean space R~(n 1),for k>0.By using this estimate,we get some corollaries about the translation soliton.     

On the controllability for the Khokhlov–Zabolotskaya–Kuznetsov-like equation

Recalling the proprieties of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, we prove the controllability of moments result for the linear part of the KZK equation and its non-linear perturbation.     

On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons

We prove a lower bound estimate for the first non-zero eigenvalue of the Witten–Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons. Our results improve some previous estimates which were obtained by the first author and Sano (Asian J Math, to appear), and by Andrews and Ni (Comm Partial Differential Equ, to appear). Moreover, we extend the diameter estimate to compact self-similar shrinkers of mean curvature flow.     

Bifurcation near infinity for the Neumann problem with concave–convex nonlinearities

In this Note, we study a class of Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing terms (concave–convex nonlinearities). Using the Ambrosetti–Rabinowitz condition and related topological and variational arguments, we prove a bifurcation result for large values of the parameter.     

Spectral optimization for the Stekloff–Laplacian: The stability issue

We consider the problem of minimizing the first non-trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock–Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (1999) [2]: we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.     

On Wiener’s Theorem for functions periodic at infinity

We consider the functions periodic at infinity with values in a complex Banach space. The notions are introduced of the canonical and generalized Fourier series of a function periodic at infinity. We prove an analog of Wiener’s Theorem on absolutely convergent Fourier series for functions periodic at infinity whose Fourier series are summable with weight. The two criteria are given: for the function periodic at infinity to be the sum of a purely periodic function and a function vanishing at infinity and for a function to be periodic at infinity. The results of the article base on substantially use on spectral theory of isometric representations.     

Sharp Moser–Trudinger inequalities for the Laplacian without boundary conditions

We derive a sharp Moser–Trudinger inequality for the borderline Sobolev imbedding of $W^{2,n/2}(B_n)$ into the exponential class, where $B_n$ is the unit ball of ${\mathbb{R}}^n$. The corresponding sharp results for the spaces $W_0^{d,n/d}(\Omega )$ are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for $d=1$ and are due to Chang–Yang, Cianchi and Leckband. The proof of our result is based on a new integral representation formula for the “canonical” solution of the Poisson equation on the ball, that is, the unique solution of the equation $\mathrm{\Delta }u=f$ which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function.     

On well-posedness for the Benjamin–Ono equation

We prove existence and uniqueness of solutions for the Benjamin–Ono equation with data in \(H^{s}({\mathbb{R}})\) , s > 1/4. Moreover, the flow is hölder continuous in weaker topologies.     

Higher derivatives estimate for the 3D Navier–Stokes equation

In this article, a nonlinear family of spaces, based on the energy dissipation, is introduced. This family bridges an energy space (containing weak solutions to Navier–Stokes equation) to a critical space (invariant through the canonical scaling of the Navier–Stokes equation). This family is used to get uniform estimates on higher derivatives to solutions to the 3D Navier–Stokes equations. Those estimates are uniform, up to the possible blowing-up time. The proof uses blow-up techniques. Estimates can be obtained by this means thanks to the galilean invariance of the transport part of the equation.     

On the uniform estimate in the Calabi-Yau theorem,Ⅱ

We show that a pluripotential proof of the uniform estimate in the Calabi-Yau theorem works also in the Hermitian case.     

On Boundary-Value Problems for the Laplacian in Bounded Domains with Micro Inhomogeneous Structure of the Boundaries

We consider boundary-value problems with rapidly alternating types of boundary conditions. We present the classification of homogenized （limit） problems depending on the ratio of small parameters, which characterize the diameter of parts of the boundary with different types of boundary conditions. Also we study the respective spectral problem of this type.     

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω, we consider the problem of providing sharp lower bounds for λ ₂(Ω), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality, asserting that the disjoint unions of two equal balls minimize λ ₂ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.     

On the nonstationary linearized Navier–Stokes problem in domains with cylindrical outlets to infinity

The nonstationary linearized Navier–Stokes system is studied in the domain with cylindrical outlets to infinity in weighted function spaces with an exponential weight function. It is proved that under natural compatibility conditions there exists a unique solution with prescribed fluxes over the sections of outlets to infinity that exponentially tends in each outlet to the corresponding nonstationary Poiseuille flow.Mathematics Subject classification (1991): 35Q30