scholarly journals Poincaré Inequalities and Neumann Problems for the p-Laplacian

2018 ◽  
Vol 61 (4) ◽  
pp. 738-753 ◽  
Author(s):  
David Cruz-Uribe ◽  
Scott Rodney ◽  
Emily Rosta
Keyword(s):  
Quadratic Form ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Poincaré Inequalities ◽  
Existence Of Weak Solutions ◽  
Neumann Problems ◽  
Poincare Inequalities ◽  
Associated Matrix ◽  
Weighted Poincaré Inequalities

AbstractWe prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.

scholarly journals Poincaré inequalities and Neumann problems for the variable exponent setting

2021 ◽  
Vol 4 (5) ◽  
pp. 1-22
Author(s):  
David Cruz-Uribe ◽  
◽  
Michael Penrod ◽  
Scott Rodney ◽  
Keyword(s):  
Variable Exponent ◽  
Growth Conditions ◽  
Linear Elliptic Equation ◽  
Poincaré Inequalities ◽  
Neumann Problems ◽  
Variable Exponent Spaces ◽  
Nonstandard Growth ◽  
Poincare Inequalities ◽  
Nonstandard Growth Conditions ◽  
Weighted Poincaré Inequalities

<abstract><p>In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.</p></abstract>

Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces

2018 ◽  
Vol 68 (4) ◽  
pp. 867-880
Author(s):  
Saeid Shokooh ◽  
Ghasem A. Afrouzi ◽  
John R. Graef
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Neumann Problems

Abstract By using variational methods and critical point theory in an appropriate Orlicz-Sobolev setting, the authors establish the existence of infinitely many non-negative weak solutions to a non-homogeneous Neumann problem. They also provide some particular cases and an example to illustrate the main results in this paper.

scholarly journals Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems with Rough Coefficients

2012 ◽  
Vol 64 (6) ◽  
pp. 1395-1414 ◽  
Author(s):  
Scott Rodney
Keyword(s):  
Functional Analysis ◽  
Quadratic Form ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Principal Part ◽  
Regularity Theory ◽  
Existence Theory ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Rough Coefficients

Abstract This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the formThe principal part ξ'P(x)ξ of the above equation is assumed to be comparable to a quadratic form Q(x,ξ)=ξ'Q(x)ξ that may vanish for non-zero ξ ∊ ℝn. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces QH1 (Θ)=W1,2(Θ,Q) and QH10(Θ)= W1,20 (Θ,Q)as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

Applications of local linking to nonlocal Neumann problems

2014 ◽  
Vol 17 (01) ◽  
pp. 1450001 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Differential Operator ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Linking Theorem ◽  
Multiplicity Result ◽  
Elliptic Differential Operator ◽  
Abstract Result ◽  
Existence Of Weak Solutions ◽  
Neumann Problems ◽  
Superlinear Growth

We study a nonlocal Neumann problem driven by a nonhomogeneous elliptic differential operator. The reaction term is a nonlinearity function that exhibits p-superlinear growth but need not satisfy the Ambrosetti–Rabinowitz condition. By using an abstract linking theorem for smooth functionals, we prove a multiplicity result on the existence of weak solutions for such problems. An explicit example illustrates the main abstract result of this paper.

scholarly journals Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Toni Heikkinen
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Space Setting ◽  
Poincare Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

Some weighted Poincare inequalities

2009 ◽  
Vol 58 (4) ◽  
pp. 1619-1638 ◽  
Author(s):  
Fausto Ferrari ◽  
Enrico Valdinoci
Keyword(s):  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Weighted Poincaré Inequalities
Sign in / Sign up

Export Citation Format

Share Document