Hölder Regularity of Solutions of Pde's: A Geometrical View

Forzani, Liliana; Aimar, Hugo; Toledano, R.

doi:10.1081/pde-100106130

Cited by 13 publications

(18 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This implies that e j − e i < c log 2 1 and the proof of the lemma is complete. We are now in a position to prove the following selection theorem of importance in the following sections.…”

Section: Where C Depends Only On K and The Constants α And β In Definmentioning

confidence: 74%

“…1 We first note that if x 0 ∈ B t (z), then by quasi triangle inequality B t (z) ⊆ B 2K t (x 0 ) and therefore, by the doubling property we get…”

Section: Another Approach To Prove Harnack Inequalitymentioning

confidence: 99%

“…We mention the recent paper [1] by Aimar, Forzani and Toledano extending the method from [8] to quasi metric spaces. These authors assume conditions involving metric rings and the boundary of metric balls and use a Vitali-type covering lemma by Aimar [2].…”

Section: Introductionmentioning

confidence: 99%

“…These authors assume conditions involving metric rings and the boundary of metric balls and use a Vitali-type covering lemma by Aimar [2]. In addition, since their Calderón-Zygmund type decomposition, [1,Theorem 4.1], holds only for open sets, they can only handle semicontinuous functions and obtain Harnack inequality for continuous functions. In the applications, the semicontinuity of solutions is in general not known and it is more natural to make assumptions on the geometry of the underlying space rather than on the class of functions considered.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Covering theorems, inequalities on metric spaces and applications to PDE’s

2008

View full text Add to dashboard Cite

We establish a covering lemma of Besicovitch type for metric balls in the setting of Hölder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions of power decay, critical density and double ball property and with the aid of the covering theorem we show how these notions are related. Next we present an axiomatic procedure to establish Harnack inequality that permits to handle both divergence and non divergence linear equations.

show abstract

Section: Where C Depends Only On K and The Constants α And β In Definmentioning

confidence: 74%

“…1 We first note that if x 0 ∈ B t (z), then by quasi triangle inequality B t (z) ⊆ B 2K t (x 0 ) and therefore, by the doubling property we get…”

Section: Another Approach To Prove Harnack Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Covering theorems, inequalities on metric spaces and applications to PDE’s

2008

View full text Add to dashboard Cite

show abstract

“…Growth Lemma is a fundamental tool to study qualitative properties and regularity of solutions in bounded and unbounded domain. Recent review on Growth Lemma and its applications was published in [11] (see also [13]).…”

Section: Introductionmentioning

confidence: 99%

On Phragmen — Lindelof Principle for Non-Divergence Type Elliptic Equations and Mixed Boundary Conditions

Ibraguimov¹,

Nazarov

2017

Vestn. Volgogr. gos. univ., Ser. 1. Math. Phy. and Comp. Model.

View full text Add to dashboard Cite

Abstract. The paper is dedicated to qualitative study of the solution of the Zaremba-type problem in Lipschitz domain with respect to the elliptic equation in non-divergent form. Main result is Landis type Growth Lemma in spherical layer for Mixed Boundary Value Problem in the class of "admissible domain". Based on the Growth Lemma Phragmén -Lindelöf theorem is proved at junction point of Dirichlet boundary and boundary over which derivative in non-tangential direction is defined.

show abstract

Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

Bonfiglioli

2019

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

We outline several results of Potential Theory for a class of linear partial differential operators L of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for L ; under different geometrical assumptions on L (mainly, under global doubling/Poincaré assumptions), it is described how to obtain an invariant, non-homogeneous Harnack inequality. When L is equipped with a global fundamental solution Γ , further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on L ensuring that such a Γ exists.

show abstract

Hölder Regularity of Solutions of Pde's: A Geometrical View

Cited by 13 publications

References 11 publications

Covering theorems, inequalities on metric spaces and applications to PDE’s

Covering theorems, inequalities on metric spaces and applications to PDE’s

On Phragmen — Lindelof Principle for Non-Divergence Type Elliptic Equations and Mixed Boundary Conditions

Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

Contact Info

Product

Resources

About