In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$.

We establish a straightforward estimate for the number of open sets with fundamental group constraints needed to cover the total space of fibrations. This leads to vanishing results for simplicial volume and minimal volume entropy, e.g., for certain mapping tori.

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

In this paper, we study the principal spectral theory of age-structured models with random diffusion. First, we provide an equivalent characteristic for the principal eigenvalue, the strong maximum principle and a positive strict super-solution. Then, we use the result to investigate the effects of diffusion rate on the principal eigenvalue. Finally, we study how the principal eigenvalue affects the global dynamics of the KPP model and verify that the principal eigenvalue being zero is a critical value.

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

We consider the boundary Hardy–Hénon equation

\[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \]

$B_1(0)\subset \mathbb {R}^{N}$

$(N\geq 3)$

$1$

$0$

$p>0$

$\alpha \in \mathbb {R}$

$0< p<({N+2})/({N-2})$

$\alpha \leq -2$

$p>1$

$\alpha >-2$

$1< p<({N+2})/({N-2})$

$0< p\leq 1$

$\alpha \leq -2$

$0< p<1$

$\alpha >-2$

In this paper we are interested in comparing the spectra of two elliptic operators acting on a closed minimal submanifold of the Euclidean unit sphere. Using an approach introduced by Savo in [A Savo. Index Bounds for Minimal Hypersurfaces of the Sphere. Indiana Univ. Math. J. 59 (2010), 823-837.], we are able to compare the eigenvalues of the stability operator acting on sections of the normal bundle and the Hodge Laplacian operator acting on $1$-forms. As a byproduct of the technique and under a suitable hypothesis on the Ricci curvature of the submanifold we obtain that its first Betti's number is bounded from above by a multiple of the Morse index, which provide evidence for a well-known conjecture of Schoen and Marques & Neves in the setting of higher codimension.