The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

Adamowicz, Tomasz; Lundström, Niklas L. P.

doi:10.1007/s10231-015-0481-3

Cited by 11 publications

(27 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, we conclude that solutions must vanish at the same rate as |x| k(v,p) as x approaches the apex (Corollary 5.1). Similar growth estimates where proved in the setting of C 1,1 -domains in [8] and for wider classes of equations and other geometric settings in [37,38,39,45,46,5,48]. An immediate consequence of Corollary 5.1 is the boundary Harnack inequality for p-harmonic functions in planar sectors, see (5.5).…”

Section: Introductionsupporting

confidence: 58%

Estimates of $p$-harmonic functions in planar sectors

Lundström¹,

Singh²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

, where φ is the polar angle of (x, y). Let R > 0 and ω p (x) be the p-harmonic measure of ∂B(0, R) ∩ S v at x with respect to B(0, R) ∩ S v . We prove that there exists a constant C such thatwhere the exponent k(v, p) is given explicitly as a function of v and p. Using this estimate we derive local growth estimates for p-suband p-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of p-harmonic measure we also derive a sharp Phragmen-Lindelöf theorem for p-subharmonic functions in the unbounded sector S v . Moreover, if p = ∞ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in R n . Finally, when v ∈ (1/2, ∞) and p ∈ (1, ∞) we prove a uniqueness result (modulo normalization) for positive p-harmonic functions in S v vanishing on ∂S v .

show abstract

Section: Introductionsupporting

confidence: 58%

Estimates of $p$-harmonic functions in planar sectors

Lundström¹,

Singh²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…More specifically, our boundary estimate consists of a lower estimate (Theorem 5.1) and an upper estimate (Theorem 5.3) generalizing the main results of Adamowicz-Lundström [1] to viscosity solutions of more general PDEs. To prove the lower estimate we need a sligthly stronger structural assumption than above in order to build a positive barrier function (Lemma 3.2) and to this end we assume…”

Section: Introductionmentioning

confidence: 99%

Strong maximum principle and boundary estimates for nonhomogeneous elliptic equations

Lundström¹,

Olofsson²,

Toivanen³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form F (x, u, Du, D 2 u) = 0 under suitable structure conditions on the equation allowing for non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we also prove the Hopf lemma, the boundary Harnack inequality and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.

show abstract

“…For infinity-harmonic functions, see e.g. Bhattacharya [13], Lundström-Nyström [46] and for solutions to the variable exponent p-Laplace equation in smooth domains, see Adamowicz-Lundström [2]. Only few papers considered local estimates of positive p-harmonic functions vanishing near boundaries having dimension less than n − 1.…”

Section: Introductionmentioning

confidence: 99%

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

Lundström

2015

Potential Anal

Self Cite

View full text Add to dashboard Cite

We prove estimates of a p-harmonic measure, p ∈ (n − m, ∞], for sets in R n which are close to an m-dimensional hyperplane Λ ⊂ R n , m ∈ [0, n − 1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω ⊂ R n \ Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in R n , which are close to an m-dimensional hyperplane.

show abstract

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

Cited by 11 publications

References 54 publications

Estimates of $p$-harmonic functions in planar sectors

Estimates of $p$-harmonic functions in planar sectors

Strong maximum principle and boundary estimates for nonhomogeneous elliptic equations

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

Contact Info

Product

Resources

About