Asymptotic of weak solutions for linear elliptic nonlocal Robin problem without Dini-continuity condition in a plane angle domain

2017 ◽  
Vol 57 (1) ◽  
Author(s):  
Krzysztof Żyjewski
Keyword(s):  
Weak Solutions ◽  
Continuity Condition ◽  
Robin Problem ◽  
Plane Angle

scholarly journals The Behaviour of Weak Solutions of Boundary Value Problems for Linear Elliptic Second Order Equations in Unbounded Cone-Like Domains

2016 ◽  
Vol 30 (1) ◽  
pp. 203-217
Author(s):  
Damian Wiśniewski
Keyword(s):  
Weak Solution ◽  
Boundary Value Problems ◽  
Weak Solutions ◽  
Boundary Value ◽  
Continuity Condition ◽  
Second Order ◽  
Second Order Equations ◽  
Elliptic Divergence

AbstractWe investigate the behaviour of weak solutions of boundary value problems (Dirichlet, Neumann, Robin and mixed) for linear elliptic divergence second order equations in domains extending to infinity along a cone. We find an exponent of the solution decreasing rate: we derive the estimate of the weak solution modulus for our problems near the infinity under assumption that leading coefficients of the equations do not satisfy the Dini-continuity condition.

Nonlocal Robin Problem for Elliptic Quasilinear Second Order Equations

2014 ◽  
Vol 14 (1) ◽  
Author(s):  
Mikhail Borsuk ◽  
Krzysztof Żyjewski
Keyword(s):  
Weak Solutions ◽  
Corner Point ◽  
Second Order ◽  
Robin Problem ◽  
Quasilinear Elliptic ◽  
Second Order Equations ◽  
Elliptic Divergence

AbstractWe investigate the behavior of weak solutions to the nonlocal Robin problem for quasilinear elliptic divergence second order equations in a neighborhood of the boundary corner point. We find an exponent of the solution decreasing rate near the boundary corner point.

scholarly journals On the regularity of very weak solutions for linear elliptic equations in divergence form

2020 ◽  
Vol 27 (5) ◽  
Author(s):  
Domenico Angelo La Manna ◽  
Chiara Leone ◽  
Roberta Schiattarella
Keyword(s):  
Weak Solution ◽  
Elliptic Equation ◽  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Continuity Condition ◽  
Very Weak Solutions ◽  
Linear Elliptic Equation ◽  
Very Weak Solution

Abstract In this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .

Dimensions of plane and solid angles and their units in the International System of Units (SI)

2020 ◽  
pp. 26-32
Author(s):  
M. I. Kalinin ◽  
L. K. Isaev ◽  
F. V. Bulygin
Keyword(s):  
Solid Angle ◽  
International System ◽  
International Committee ◽  
Plane Angle ◽  
Arc Length ◽  
Weights And Measures ◽  
International System Of Units ◽  
System Of Units ◽  
In Series ◽  
Solid Angles

The situation that has developed in the International System of Units (SI) as a result of adopting the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities, is analyzed. It is shown that the basis for such a solution was a misunderstanding of the mathematical formula relating the arc length of a circle with its radius and corresponding central angle, as well as of the expansions of trigonometric functions in series. From the analysis presented in the article, it follows that a plane angle does not depend on any of the SI quantities and should be assigned to the base quantities, and its unit, the radian, should be added to the base SI units. A solid angle, in this case, turns out to be a derived quantity of a plane angle. Its unit, the steradian, is a coherent derived unit equal to the square radian.

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