scholarly journals Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean space

2021 ◽  
pp. 2150063
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio
Keyword(s):  
Ground State ◽  
Mean Curvature ◽  
Elliptic Equations ◽  
Regularity Theory ◽  
Sobolev Inequalities ◽  
Codimension One ◽  
Semilinear Elliptic ◽  
Curvature Term ◽  
Stable Solutions ◽  
The Hardy Inequality

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [Formula: see text], all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a “ground state” substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality.

The Influence of Some Factors on Spirolactone Stability in Solution

2008 ◽  
Vol 59 (7) ◽  
Author(s):  
Daniela Lucia Muntean ◽  
Silvia Imre ◽  
Cosmina Voda
Keyword(s):  
High Performance Liquid Chromatography ◽  
Liquid Chromatography ◽  
Uv Irradiation ◽  
High Performance ◽  
Degradation Products ◽  
Simultaneous Action ◽  
Ethanolic Solution ◽  
Degradation Processes ◽  
Stable Solutions ◽  
Preformulation Study

The influence of some factors on spironolactone stability in solution was studied, by applying high-performance liquid chromatography, as a part of a pharmaceutical preformulation study in order to obtain a spironolactone solution for alopecia treatment. Solutions of 1 mg/ml spironolactone in aqueous ethanolic solution 1 : 1 and in 20 mM cyclodextrines solutions (b-, hydroxi-b- and methyl-b-cyclodextrine) was used, maintained at 8 and 22 �C, protected from light and after UV irradiation at 254 nm. The main degradation products were 7a-thiospirolactone and canrenone. The most stable solutions were the alcoholic ones and with methyl-beta-cyclodextrine, but the simultaneous action of temperature and UV irradiation allowed degradation processes after one hour of exposure, more aggressive in the presence of methyl-beta-cyclodextrine. In conclusion, for alopecia treatment with spironolactone a 1 mg/mL ethanolic solution could be used and it is recommendable the protection of treated zone.

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

scholarly journals Dynamical structure of highly eccentric discs with applications to tidal disruption events

2020 ◽  
Vol 500 (3) ◽  
pp. 4110-4125
Author(s):  
Elliot M Lynch ◽  
Gordon I Ogilvie
Keyword(s):  
Thermal Instability ◽  
Vertical Structure ◽  
Scale Height ◽  
Accretion Discs ◽  
Radiative Cooling ◽  
Dynamical Structure ◽  
Tidal Disruption ◽  
The Subject ◽  
Stable Solutions ◽  
Physical Quantities

ABSTRACT Whether tidal disruption events circularize or accrete directly as highly eccentric discs is the subject of current research and appears to depend sensitively on the disc thermodynamics. One aspect of this problem that has not received much attention is that a highly eccentric disc must have a strong, non-hydrostatic variation of the disc scale height around each orbit. As a complement to numerical simulations carried out by other groups, we investigate the dynamical structure of TDE discs using the non-linear theory of eccentric accretion discs. In particular, we study the variation of physical quantities around each elliptical orbit, taking into account the dynamical vertical structure, as well as viscous dissipation and radiative cooling. The solutions include a structure similar to the nozzle-like structure seen in simulations. We find evidence for the existence of the thermal instability in highly eccentric discs dominated by radiation pressure. For thermally stable solutions many of our models indicate a failure of the α-prescription for turbulent stresses. We discuss the consequences of our results for the structure of eccentric TDE discs.

scholarly journals Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Siran Li
Keyword(s):  
Banach Space ◽  
Discrete Geometry ◽  
Thin Shell ◽  
Euclidean Geometry ◽  
Convex Geometry ◽  
Regularity Theory ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Space Theory ◽  
Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

scholarly journals The time domain numerical method of three-dimensional conductors including radiation with lumped parameter circuit

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Souma Jinno ◽  
Shuji Kitora ◽  
Hiroshi Toki ◽  
Masayuki Abe
Keyword(s):  
Numerical Method ◽  
Time Domain ◽  
Maxwell Equations ◽  
Three Dimensional ◽  
New Formalism ◽  
Vector Potentials ◽  
Lumped Parameter ◽  
Radiation Theory ◽  
Stable Solutions ◽  
The Time Domain

AbstractWe formulate a numerical method on the transmission and radiation theory of three-dimensional conductors starting from the Maxwell equations in the time domain. We include the delay effect in the integral equations for the scalar and vector potentials rigorously, which is vital to obtain numerically stable solutions for transmission and radiation phenomena in conductors. We provide a formalism to connect the conductors to any passive lumped-parameter circuits. We show one example of numerical calculations, demonstrating that the new formalism provides stable solutions to the transmission and radiation phenomena.

Compactness of 𝑀-uniform domains and optimal thermal insulation problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hengrong Du ◽  
Qinfeng Li ◽  
Changyou Wang
Keyword(s):  
Thermal Insulation ◽  
Heat Insulation ◽  
Optimization Problems ◽  
Optimal Shape ◽  
Optimal Heat ◽  
Optimal Shapes ◽  
Stable Solutions ◽  
Uniform Domains

Abstract In this paper, we will consider an optimal shape problem of heat insulation introduced by [D. Bucur, G. Buttazzo and C. Nitsch, Two optimization problems in thermal insulation, Notices Amer. Math. Soc. 64 (2017), 8, 830–835]. We will establish the existence of optimal shapes in the class of 𝑀-uniform domains. We will also show that balls are stable solutions of the optimal heat insulation problem.

scholarly journals Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

2021 ◽  
Vol 10 (1) ◽  
pp. 1316-1327
Author(s):  
Ali Hyder ◽  
Wen Yang
Keyword(s):  
Liouville Equation ◽  
Weak Solutions ◽  
Partial Regularity ◽  
Singular Set ◽  
Stable Solutions ◽  
Gelfand Problem

Abstract We analyze stable weak solutions to the fractional Geľfand problem ( − Δ ) s u = e u i n Ω ⊂ R n . $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10s.

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