An Optimal Estimate for the Anisotropic Logarithmic Potential

This paper introduces the new annulus body to establish the optimal lower bound for the anisotropic logarithmic potential as the complement to the theory of its upper bound estimate which has already been investigated. The connections with convex geometry analysis and some metric properties are also established. For the application, a polynomial dual log-mixed volume difference law is deduced from the optimal estimate.

Analyticity and Regge asymptotics in virtual Compton scattering on the nucleon

We test the consistency of the data on the nucleon structure functions with analyticity and the Regge asymptotics of the virtual Compton amplitude. By solving a functional extremal problem, we derive an optimal lower bound on the maximum difference between the exact amplitude and the dominant Reggeon contribution for energies $$\nu $$ ν above a certain high value $$\nu _h(Q^2)$$ ν h ( Q 2 ) . Considering in particular the difference of the amplitudes $$T_1^\text {inel}(\nu , Q^2)$$ T 1 inel ( ν , Q 2 ) for the proton and neutron, we find that the lower bound decreases in an impressive way when $$\nu _h(Q^2)$$ ν h ( Q 2 ) is increased, and represents a very small fraction of the magnitude of the dominant Reggeon. While the method cannot rule out the hypothesis of a fixed Regge pole, the results indicate that the data on the structure function are consistent with an asymptotic behaviour given by leading Reggeon contributions. We also show that the minimum of the lower bound as a function of the subtraction constant $$S_1^\text {inel}(Q^2)$$ S 1 inel ( Q 2 ) provides a reasonable estimate of this quantity, in a frame similar, but not identical to the Reggeon dominance hypothesis.

Minimization of eigenvalues for the Camassa–Holm equation

A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem [Formula: see text] We will study in this paper the optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation with the Neumann boundary condition when the [Formula: see text] norm of potentials is given. First, we will study the optimal lower bound for the smallest eigenvalue in the measure differential equations to make our results more applicable. Second, Based on the relationship between the minimization problem of the smallest eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation.

Optimal Estimates for Steklov Eigenvalue Gaps and Ratios on Warped Product Manifolds

We consider an $n$-dimensional smooth Riemannian manifold $M^n=[0,R)\times \mathbb{S}^{n-1}$ endowed with a warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Suppose that $M$ has strictly convex boundary. First, for the classical Steklov eigenvalue problem, we derive an optimal lower (upper, respectively) bound for its eigenvalue gaps in terms of $h^{\prime}(R)/h(R)$ when $n\geq 2$ and $Ric_g\geq 0$ ($\leq 0$, respectively). Second, in the same spirit, for two 4th-order Steklov eigenvalue problems studied by Kuttler and Sigillito in 1968, we deduce an optimal lower bound for their eigenvalue gaps in terms of either $h^{\prime}(R)/h^3(R)$ or $h^{\prime}(R)/h(R)$ when $n=2$ and the Gaussian curvature is nonnegative. We also consider optimal estimates on the eigenvalue ratios for these eigenvalue problems.