Abstract

In this paper we give some simple estimates of positive and negative eigenvalues for the inhomogeneous membrane boundary problem $$ -\Delta u=\lambda mu,\quad u\,|_{\partial D=0} $$ if $D\subset R^2$ is a simple connected domain and $m$ is a real bounded function (possibly changing the sign).

Keywords: Inhomogeneous membrane problem, estimation of eigenvalues.

MSC: 35P15, 35J25, 47F05

Abstract

We prove that $|f|^p$, $p>0$, behaves like a subharmonic function if $f$ is a $C^2$-function such that, for some constants $K$ and $K_0$, $$ |\Delta f(x)|\leq Kr^{-1}\sup|\nabla f|+K_0r^{-2}\sup|f|, $$ where the supremum is taken over $B_r(x)=\{\,z\,:|z-x|

Keywords: Subharmonic function, smooth function.

MSC: 31B05

Abstract

We study the existence of positive, monotonic, unbounded solutions of the equation $(r(x)y^{(n)})^{(n)}=yf(x)$. We obtain necessary and sufficient conditions for the existence of different classes of these solutions.

MSC: 34A99

Abstract

In mathematical approach to the pattern recognition there are some mathematical problems that are important for practical aims. One of them is invariance of geometrical shape with respect to the Hausdorff convergence. In this paper we prove that the limit of a convergent sequence of compact sets in $R^n$ of the same shape is again a set of the same shape or a singleton.

Keywords: Geometric shape, Hausdorff metric, general position.

MSC: 68T10, 54B20

Abstract

In this paper we consider derivatives of higher order and certain ``double'' integrals of the eigenfunctions and associated functions of the formal Sturm-Liouville operator $$ \Cal L(u)(x)=-\bigl(p(x)\,u'(x)\bigr)'+q(x)\,u(x) $$ defined on a finite or infinite interval $G\subseteq R$. We suppose that the complex-valued potential $q=q(x)$ belongs to the class $L_1^{loc}(G)$ and that piecewise continuously differentiable coefficient $p=p(x)$ has a finite number of the discontinuity points in $G$. Order-sharp upper estimates are obtained for the suprema of the moduli of the $k$-th order derivatives $(k\geq 2$) of the eigenfunctions and associated functions $\{\,\overset{i}\to{u}_{\lambda}(x)\,|\,i=0,1,\dots\,\}$ of the operator $\Cal L$ in terms of their norms in metric $L_2$ on compact subsets of $G$ (on the entire interval $G$). Also, order-sharp upper estimates are established for the integrals (over closed intervals $[y_1,y_2]\subseteq \overline G$) $$ \int_{y_1}^{y_2}\biggl(\int_a^y\overset{i}\to{u}_{\lambda}(\xi)\,d\xi\biggr)dy, \qquad \int_{y_1}^{y_2}\biggl(\int_y^b\overset{i}\to{u}_{\lambda}(\xi)\,d\xi\biggr)dy $$ in terms of $L_2$-norms of the mentioned functions when $G$ is finite. The corresponding estimates for derivatives $\overset{i}\to{u}_{\lambda}'(x)$ and integrals $\int_{y_1}^{y_2}\overset{i}\to{u}_{\lambda}(y)\,dy$ were proved in [5]--[6].

Keywords: Sturm-Liouville operator, estimation of eigenfunctions.

MSC: 34L20, 47E05

Abstract

We present the complete solution of the following problem of G. Pólya: Circular forest has a center at the origin and radius $R\geq 1$. A person is staying at the center and the trees of the radius $r$ are planted at all other lattice points of the forest. Determine the maximal value $\rho$ of the radius $r$ for which the person can see out of the forest and, in the case $r=\rho$, determine the directions in which he/she should look in order to see out of the forest.

Keywords: Lattice point, Pick's theorem.

MSC: 52C05

Abstract

The linear spaces generated by the eigenfunctions of a differential operator are well-known in applied mathematics. In this paper we examine their interpolation properties, connection with Sobolev spaces and apply these results to the solving of hyperbolic equation in Sobolev spaces of fractional order.

Keywords: Interpolation of Banach spaces, Sobolev spaces, hyperbolic equation.

MSC: 46B70, 46E35, 34G10

Abstract

A new class of generalized open sets in a topological space, called $b$-open sets, is introduced and studied. This class is contained in the class of semi-preopen sets and contains all semi-open sets and all preopen sets. It is proved that the class of $b$-open sets generates the same topology as the class of preopen sets.

Keywords: Topological space, generalized open sets, $\alpha$-set, $b$-open, semi-open, preopen, semi-preopen set.

MSC: 54A10