Related papers: Correct Solvability, Embedding Theorems and Separa…

Elliptic and parabolic integro-differential model problems are considered in the whole space. By verifying H\"ormander condition, the existence and uniqueness is proved in L_{p}-spaces of functions whose regularity is defined by a scalable,…

In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\lambda r)y=0$, $(U-1)y^{\vee}+i(U+1)y^{\wedge}=0$, where function $p\in L_{\infty}[0,1]$ is uniformly positive, generalized function $q\in W_2^{-1}[0,1]$ is…

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and $S_{p,\tau,\theta}^{0, \overline{b}}\mathbf{B}$, $S_{p, \tau, \theta}^{0, \overline{b}}B$ -- spaces of functions with…

On the basis of the theory of Sturm--Liouville problem with distribution coefficients we get the infima and suprema of the first eigenvalue of the problem $-y" + (q-\lambda) y=0, y'(0) -k_0^2 y(0) = y'(1) + k_1^2 y(1) = 0$, where $q$…

This paper is concerned with proving some embeddings of the form \begin{equation*} F_{p_{1},q}^{s_{1}}\cdot B_{p_{2},\infty }^{s_{2}}\cdot ...\cdot B_{p_{m},\infty }^{s_{m}}\hookrightarrow F_{p,q}^{s_{1}},\quad m\geq 2. \end{equation*} The…

There is $c_{} > 0$ such that for all $f \in C[0,\pi]$ with at most $d-1$ roots inside $(0,\pi)$ $$ \sum_{1 \leq n \leq d}{ \left| \left\langle f, \sin\left( n x\right) \right\rangle \right|} \geq \kappa^{-\kappa^2 \log{\kappa}}\|f\|_{L^2}…

We characterize all the real numbers a,b,c and 1<= p,q,r<infty such that the weighted Sobolev space W_{a,b}^(q,p)(R^N\{0}) with power weights |x|^a and |x|^b is continuously embedded into L^{r}(R^N;|x|^cdx). Furthermore, we show that this…

We consider the non-linear eigenvalue equations characterizing $L^p$ into $L^q$ Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the…

The aim of this paper is twofold. On the one hand, we compute, in terms of $r$ and $s$, the indices $p$ for which $\ell_p$ isomorphically embeds into the mixed-norm separable spaces $L_s(L_r)$, $\ell_s(L_r)$, $L_s(\ell_r)$ and…

Classical Sturm-Liouville problems of $q$-difference variables are extended for symmetric discrete functions such that the corresponding solutions preserve the orthogonality property. Some illustrative examples are given in this sense.

We present a self-contained analysis of the stationary radiative transfer equation in weighted $L^p$ spaces. The use of weighted spaces allows us to derive uniform a-priori estimates for $1 \le p \le \infty$ under minimal assumptions on the…

We examine the general weighted Lane-Emden system \begin{align*} -\Delta u = \rho(x)v^p,\quad -\Delta v= \rho(x)u^\theta, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N \end{align*} where $1<p\leq\theta$ and $\rho: \mathbb{R}^N\rightarrow…

An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y={\lambda}y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations…

In this paper we study the embedding properties for the weighted Sobolev space $H^1_V(\mathbb{R}^N)$ into the Lebesgue weighted space $L^\tau_W(\mathbb{R}^N)$. Here $V$ and $W$ are diverging weight functions. The different behaviour of $V$…

In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\lambda r)y=0$, $(U-1)y^{\vee}+i(U+1)y^{\wedge}=0$, where function $p\in L_{\infty}[0,1]$ is uniformly positive, generalized functions $q,r\in W_2^{-1}[0,1]$…

We propose a new, exactly solvable Schr\"{o}dinger equation. The potential partner is given by \[{ V=}-Bp\operatorname{csch}[px]^{2}-9p(B+p)\operatorname*{sech}[3px]^{2}+(B\coth[px]-3(B+p)\tanh[3px])^{2}.\] obtained using supersymmetric…

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and spaces with mixed logarithmic smoothness. Equivalent norms of a space with mixed logarithmic smoothness are found and…

Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of…

We show the existence of a real analytic isomorphism between a space of impedance function $\rho$ of the Sturm-Liouville problem $- \rho^{-2}(\rho^2f')' + uf$ on $(0,1)$, where $u$ is a function of $\rho, \rho', \rho''$, and that of…

This article contains a characterization of when certain weighted Sobolev spaces on $\Bbb R^n$ embed compactly into $L^2(\mathbb R^n, \varphi)$. The characterization is in terms of derivatives of the weight function $\varphi$ and involves…