Communications in Contemporary Mathematics

The paper is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second-order elliptic problems by Chipot and Rougirel in [On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, Commun. Contemp. Math. 4(1) (2002) 15–44], where the force functions are considered on the cross-section of domains, we prove the non-local counterpart of their result.

Recently in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89(1–2) (2014) 21–35] Yeressian established a weighted estimate for solutions of non-local Dirichlet problems which exhibit the asymptotic behavior. The case when $s=1/2$ was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this paper, we extend this result to each order between $0$ and $1$.

Consider in a Hilbert space $H$ the Cauchy problem $(P_0)$: $u^{\prime }(t)+Au(t)+Bu(t)\ni f(t),t\in [0,T];u(0)=u_0$, and associate with it the second-order problem $(P_{𝜀})$: $-𝜀u^{\prime \prime }(t)+u^{\prime }(t)+Au(t)+Bu(t)\ni f(t),t\in [0,T];u(0)=u_0,u^{\prime }(T)=0$, where $A:D(A)\subset H\to H$ is a (possibly set-valued) maximal monotone operator, $B:H\to H$ is a Lipschitz operator, and $𝜀$ is a positive small parameter. Note that $(P_{𝜀})$ is an elliptic-like regularization of $(P_0)$ in the sense suggested by Lions in his book on singular perturbations. We prove that the solution $u_{𝜀}$ of $(P_{𝜀})$ approximates the solution $u$ of $(P_0)$: $\parallel u_{𝜀}-u{\parallel }_{C([0,T];H)}={𝒪}({𝜀}^{1/4})$. Applications to the nonlinear heat equation as well as to the nonlinear telegraph system and the nonlinear wave equation are presented.

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality

It is well known that the category of finite sets and cospans, composed by pushout, contains the universal special commutative Frobenius algebra. In this paper, we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.

The Hirzebruch ${\chi }_y$-genus and Poincaré polynomial share some similar features. In this paper, we investigate two of their similar features simultaneously. Through this process we shall derive several new results as well as reprove and improve some known results.

Let $\alpha \in (\frac{1}{2},1)$, $\mathrm{\Omega }$ be a bounded open domain in ${ℝ}^N$ ($N\ge 2$) with a $C^2$ boundary $\partial \mathrm{\Omega }$ and $\omega$ be a Hausdorff measure on $\partial \mathrm{\Omega }$. We denote by ${\omega }_{\alpha }$

This paper deals with the study of the behavior of the wave functions of a two-component Bose–Einstein condensate near the interface, in the case of strong segregation. This yields a system of two coupled ordinary differential equations for which we want to have estimates on the asymptotic behavior, as the strength of the coupling tends to infinity. As in phase separation models, the leading order profile is a hyperbolic tangent. We construct an approximate solution and use the properties of the associated linearized operator to perturb it into a genuine solution for which we have an asymptotic expansion. We prove that the constructed heteroclinic solutions are linearly nondegenerate, in the natural sense, and that there is a spectral gap, independent of the large interaction parameter, between the zero eigenvalue (due to translations) at the bottom of the spectrum and the rest of the spectrum. Moreover, we prove a uniqueness result which implies that, in fact, the constructed heteroclinic is the unique minimizer (modulo translations) of the associated energy, for which we provide an expansion.

Sharpened forms of the concentration of measure phenomenon for classes of functions on the sphere are developed in terms of Hessians of these functions.

The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine-linear. In this paper, we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three-dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that $n+2$ fixed projective points in real $n$-dimensional projective space, through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.

For valuations on convex bodies in Euclidean spaces, there is by now a long series of characterization and classification theorems. The classical template is Hadwiger’s theorem, saying that every rigid motion invariant, continuous, real-valued valuation on convex bodies in ${ℝ}^n$ is a linear combination of the intrinsic volumes. For tensor-valued valuations, under the assumptions of isometry covariance and continuity, there is a similar classification theorem, due to Alesker. Also for the local extensions of the intrinsic volumes, the support, curvature and area measures, there are analogous characterization results, with continuity replaced by weak continuity, and involving an additional assumption of local determination. The present authors have recently obtained a corresponding characterization result for local tensor valuations, or tensor-valued support measures (generalized curvature measures), of convex bodies in ${ℝ}^n$. The covariance assumed there was with respect to the group $\mathrm{\text{O}}(n)$ of orthogonal transformations. This was suggested by Alesker’s observation, according to which in dimensions $n>2$, the weaker assumption of $\mathrm{\text{SO}}(n)$ covariance does not yield more tensor valuations. However, for tensor-valued support measures, the distinction between proper and improper rotations does make a difference. This paper considers, therefore, the local tensor valuations sharing the previously assumed properties, but with $\mathrm{\text{O}}(n)$ covariance replaced by $\mathrm{\text{SO}}(n)$ covariance, and provides a complete classification. New tensor-valued support measures appear only in dimensions 2 and 3.

We establish an integral formula on a smooth, precompact domain in a Kähler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula, we also prove some geometric inequalities when the boundary has positive Hermitian mean curvature.

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin’s overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.