Uniqueness for Very Weak Solution to a Class of Elliptic Equations

2012 ◽  
Vol 457-458 ◽  
pp. 863-866
Author(s):  
Xu Juan Xu ◽  
Jian Tao Gu ◽  
Xiao Li Liu
Keyword(s):  
Weak Solution ◽  
Elliptic Equations ◽  
Very Weak Solution

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 ) .

Uniqueness for Very Weak Solution to a Class of Elliptic Equations

2012 ◽  
Vol 457-458 ◽  
pp. 863-866
Author(s):  
Xu Juan Xu ◽  
Jian Tao Gu ◽  
Xiao Li Liu
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Hodge Decomposition ◽  
Very Weak Solutions ◽  
Uniqueness Property ◽  
Very Weak Solution

This paper studies on the very weak solution to a class of elliptic equations , and acquire the uniqueness property for very weak solutions by means of the Hodge decomposition and others.

ON THE ROBIN PROBLEM FOR STOKES AND NAVIER–STOKES SYSTEMS

2006 ◽  
Vol 16 (05) ◽  
pp. 701-716 ◽  
Author(s):  
REMIGIO RUSSO ◽  
ALFONSINA TARTAGLIONE
Keyword(s):  
Boundary Layer ◽  
Weak Solution ◽  
Lipschitz Domain ◽  
Stokes System ◽  
Navier Stokes ◽  
Robin Problem ◽  
Layer Potentials ◽  
Very Weak Solution ◽  
Stokes Systems ◽  
Compact Boundary

The Robin problem for Stokes and Navier–Stokes systems is considered in a Lipschitz domain with a compact boundary. By making use of the boundary layer potentials approach, it is proved that for Stokes system this problem admits a very weak solution under suitable assumptions on the boundary datum. A similar result is proved for the Navier–Stokes system, provided that the datum is "sufficiently small".

A uniqueness principle for up ≤ (−Δ)α 2u in the Euclidean space

2016 ◽  
Vol 18 (06) ◽  
pp. 1650019 ◽  
Author(s):  
Y. Wang ◽  
J. Xiao
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Fractional Order ◽  
Elliptic Equations ◽  
Differential Inequality ◽  
Nonlinear Elliptic Equations ◽  
Local Behavior ◽  
Nonlinear Elliptic ◽  
Global And Local ◽  
Appl Math

This paper establishes such a uniqueness principle that under [Formula: see text] the fractional order differential inequality [Formula: see text] has the property that if [Formula: see text] then a non-negative weak solution to [Formula: see text] is unique, and if [Formula: see text] then the uniqueness of a non-negative weak solution to [Formula: see text] occurs when and only when [Formula: see text], thereby innovatively generalizing Gidas–Spruck’s result for [Formula: see text] in [Formula: see text] discovered in [B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981) 525–598].

scholarly journals A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Rasmita Kar
Keyword(s):  
Weak Solution ◽  
Sobolev Space ◽  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Weighted Sobolev Space ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Value Problem ◽  
Nonlinear Elliptic ◽  
Bounded Nonlinearity

We prove the existence of a weak solution for the degenerate nonlinear elliptic Dirichlet boundary-value problem Lu-μug1+hu,∇ug2=f in Ω, u=0 on ∂Ω, in a suitable weighted Sobolev space, where Ω⊂ℝn is a bounded domain and h is a continuous bounded nonlinearity.

Existence and uniqueness of solutions of an A-harmonic elliptic equation

2019 ◽  
Vol 56 (1) ◽  
pp. 13-21
Author(s):  
Mouna Kratou
Keyword(s):  
Weak Solution ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions

Abstract This paper deals with the existence and uniqueness of weak solution of a problem which involves a class of A-harmonic elliptic equations of nonhomogeneous type. Under appropriate assumptions on the function f, our main results are obtained by using Browder Theorem.

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

2018 ◽  
Vol 26 (6) ◽  
pp. 835-857 ◽  
Author(s):  
Dinh Nho Hào ◽  
Le Thi Thu Giang ◽  
Sergey Kabanikhin ◽  
Maxim Shishlenin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Weak Sense ◽  
Very Weak Solution ◽  
Local Boundary ◽  
The Cauchy Problem ◽  
Non Local

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

scholarly journals Uniqueness of weak solution for nonlinear elliptic equations in divergence form

2000 ◽  
Vol 23 (5) ◽  
pp. 313-318 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Uniqueness Result ◽  
Quasilinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Quasilinear Elliptic

We study the uniqueness of weak solutions for quasilinear elliptic equations in divergence form. Some counterexamples are given to show that our uniqueness result cannot be improved in the general case.

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