Geometric Locations of Points Equally Distance from Two Given Geometric Figures. Part 4: Geometric Locations of Points Equally Remote from Two Spheres

The article deals with the geometric locations of points equidistant from two spheres. In all variants of the mutual position of the spheres, the geometric places of the points are two surfaces. When the centers of the spheres coincide with the locus of points equidistant from the spheres, there will be spheres equal to the half-sum and half-difference of the diameters of the original spheres. In three variants of the relative position of the initial spheres, one of the two surfaces of the geometric places of the points is a two-sheet hyperboloid of revolution. It is obtained when: 1) the spheres intersect, 2) the spheres touch, 3) the outer surfaces of the spheres are removed from each other. In the case of equal spheres, a two-sheeted hyperboloid of revolution degenerates into a two-sheeted plane, more precisely, it is a second-order degenerate surface with a second infinitely distant branch. The spheres intersect - the second locus of the points will be the ellipsoid of revolution. Spheres touch - the second locus of points - an ellipsoid of revolution, degenerated into a straight line, more precisely into a zero-quadric of the second order - a cylindrical surface with zero radius. The outer surfaces of the spheres are distant from each other - the second locus of points will be a two-sheet hyperboloid of revolution. The small sphere is located inside the large one - two coaxial confocal ellipsoids of revolution. In all variants of the mutual position of spheres of the same diameters, the common geometrical place of equidistant points is a plane (degenerate surface of the second order) passing through the middle of the segment perpendicular to it, connecting the centers of the original spheres. The second locus of points equidistant from two spheres of the same diameter can be either an ellipsoid of revolution (if the original spheres intersect), or a straight (cylindrical surface with zero radius) connecting the centers of the original spheres when the original spheres touch each other, or a two-sheet hyperboloid of revolution (if continue to increase the distance between the centers of the original spheres).

Weighted W 1, p (·)-Regularity for Degenerate Elliptic Equations in Reifenberg Domains

Let w be a Muckenhoupt A 2(ℝ n ) weight and Ω a bounded Reifenberg flat domain in ℝ n . Assume that p (·):Ω → (1, ∞) is a variable exponent satisfying the log-Hölder continuous condition. In this article, the authors investigate the weighted W 1, p (·)(Ω, w)-regularity of the weak solutions of second order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coefficients belong to weighted BMO spaces with small norms.

Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

We study the following parabolic nonlocal 4-th order degenerate equation: u t = - [ 2 ⁢ π ⁢ H ⁢ ( u x ) + ln ⁡ ( u x ⁢ x + a ) + 3 2 ⁢ ( u x ⁢ x + a ) 2 ] x ⁢ x , u_{t}=-\Bigl{[}2\pi H(u_{x})+\ln(u_{xx}+a)+\frac{3}{2}(u_{xx}+a)^{2}\Bigr{]}_{% xx}, arising from the epitaxial growth on crystalline materials. Here H denotes the Hilbert transform, and a > 0 {a>0} is a given parameter. By relying on the theory of gradient flows, we first prove the global existence of a variational inequality solution with a general initial datum. Furthermore, to obtain a global strong solution, the main difficulty is the singularity of the logarithmic term when u x ⁢ x + a {u_{xx}+a} approaches zero. Thus we show that, if the initial datum u 0 {u_{0}} is such that ( u 0 ) x ⁢ x + a {(u_{0})_{xx}+a} is uniformly bounded away from zero, then such property is preserved for all positive times. Finally, we will prove several higher regularity results for this global strong solution. These finer properties provide a rigorous justification for the global-in-time monotone solution to the epitaxial growth model with nonlocal elastic effects on vicinal surface.

Representations of degenerate poly-Bernoulli polynomials

As is well known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some of their properties were investigated. The aim of this paper is to further study some properties of the degenerate poly-Bernoulli polynomials by using three formulas coming from the recently developed ‘λ-umbral calculus’. In more detail, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement polynomials.

Liouville-type theorem for high order degenerate Lane-Emden system

<p style='text-indent:20px;'>In this paper, we are concerned with the following high order degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE111000">\begin{document}$\left\{ \begin{align} &amp; {{(-A)}^{m}}u={{v}^{p}} \\ &amp; {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y&gt;0 \right\}, \\ &amp; u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>where the operator <inline-formula><tex-math id="M1">\begin{document}$ A: = y\partial_{y}^2+a\partial_{y}+\Delta_{x}, \;a\geq 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ n+2a&gt;2m, m\in \mathbb{Z}^+,\;p,\,q\geq 1 $\end{document}</tex-math></inline-formula>. We prove the non-existence of positive smooth solutions for <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p,\, q&lt;\frac{n+2a+2m}{n+2a-2m} $\end{document}</tex-math></inline-formula>, and classify positive solutions for <inline-formula><tex-math id="M4">\begin{document}$ p = q = \frac{n+2a+2m}{n+2a-2m} $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M5">\begin{document}$ \frac{1}{p+1}+\frac{1}{q+1}&gt;\frac{n+2a-2m}{n+2a} $\end{document}</tex-math></inline-formula>, we show the non-existence of positive, ellipse-radial, smooth solutions. Moreover we prove the non-existence of positive smooth solutions for the high order degenerate elliptic system of inequalities <inline-formula><tex-math id="M6">\begin{document}$ (-A)^{m}u\geq v^p, (-A)^{m}v\geq u^q, u\geq 0, v\geq 0, $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{R}_+^{n+1} $\end{document}</tex-math></inline-formula> for either <inline-formula><tex-math id="M8">\begin{document}$ (n+2a-2m)q&lt;\frac{n+2a}{p}+2m $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M9">\begin{document}$ (n+2a-2m)p&lt;\frac{n+2a}{q}+2m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M10">\begin{document}$ p,q&gt;1 $\end{document}</tex-math></inline-formula>.</p>

A class of inverse problems for fractional order degenerate evolution equations

The criteria of the well-posedness is obtained for an inverse problem to a class of fractional order in the sense of Caputo degenerate evolution equations with a relatively bounded pair of operators and with the generalized Showalter–Sidorov initial conditions. It is formulated in terms of the relative spectrum of the pair and of the characteristic function of the problem. Sufficient conditions of the unique solvability are obtained for a similar problem with the Cauchy initial condition. For these purposes the unique solvability of the same inverse problem was studied for the equation with a bounded operator near an unknown function, which is solved with respect to the fractional derivative. General results are applied to the inverse problem research for the time fractional system of equations describing the dynamics of a viscoelastic fluid in the weakly degenerate and the strongly degenerate cases.