A priori estimates for the difference of solutions to quasi-linear elliptic equations

Biegert, Markus

doi:10.1007/s00229-010-0367-z

Cited by 21 publications

(7 citation statements)

References 17 publications

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“…Proposition 2.4. (see [6]) Let a, b ∈ R N and r ∈ (1, ∞). Then there exists a constant c r > 0 such that 13) and also in this case there is a constant c r ∈…”

Section: Preliminaries and Intermediate Resultsmentioning

confidence: 99%

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

Lancia¹,

Vélez-Santiago²,

Vernole³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

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“…Proposition 2.4. (see [6]) Let a, b ∈ R N and r ∈ (1, ∞). Then there exists a constant c r > 0 such that 13) and also in this case there is a constant c r ∈…”

Section: Preliminaries and Intermediate Resultsmentioning

confidence: 99%

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

Lancia¹,

Vélez-Santiago²,

Vernole³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…where s, r ∈ R n and p > 1. But as noted in [13, p. 74] (see also [1,Section 4]), this expression is nonnegative. Thus, T is monotone.…”

Section: P(•)-poincaré Implies P(•)-neumannmentioning

confidence: 93%

“…Now consider the domain E − . By [13, p. 43] (see also [1,Section 4]) we have that for r, s ∈ R n and 1 < p ≤ 2,…”

Section: P(•)-poincaré Implies P(•)-neumannmentioning

confidence: 99%

Poincaré Inequalities and Neumann Problems for the Variable Exponent Setting

Cruz-Uribe¹,

Penrod²,

Rodney³

2021

Preprint

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We extend the results of [5], where we 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(•)-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.

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“…By virtue of Theorem 3.2, it suffices to show that u p(·), ≤ C u Wμ ( ,∂ ) for all u ∈ W 1,p(·) ( ), and for some constant C > 0. To show this assertion, one just follow the same argument as in [3,Theorem 4.24], with the help of Theorem 3.11.…”

Section: The Variable Exponent Maz'ya Spacementioning

confidence: 93%

“…REMARK 6.5. By following the arguments given in the beautiful result by Biegert [3,Theorem 5.10], it may be possible to show that if we assume the conditions of Corollary 6.4, the (unique) weak solution of (6.27) is bounded on . Since this is not the main purpose of this article, we do not go into further details here.…”

Section: Some Examples Of Non-smooth Domainsmentioning

confidence: 98%

Embedding and Trace Results for Variable Exponent Sobolev and Maz'ya Spaces on Non-Smooth Domains

Vélez-Santiago¹

2015

Glasgow Math. J.

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We establish interior and trace embedding results for Sobolev functions on a class of bounded non-smooth domains. Also, we define the corresponding generalized Maz'ya spaces of variable exponent, and obtain embedding results similar as in the constant case. Some relations between the variable exponent Maz'ya spaces and the variable exponent Sobolev spaces are also achieved. At the end, we give an application of the previous results for the well-posedness of a class of quasi-linear equations with variable exponent.

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A priori estimates for the difference of solutions to quasi-linear elliptic equations

Abstract: A priori estimates for finite-difference approximations for the first and second-order derivatives are obtained for solutions of parabolic equations described in the title.

Cited by 21 publications

References 17 publications

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

Poincaré Inequalities and Neumann Problems for the Variable Exponent Setting

Embedding and Trace Results for Variable Exponent Sobolev and Maz'ya Spaces on Non-Smooth Domains

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