scholarly journals On the boundary Harnack principle in Hölder domains

2021 ◽  
Vol 4 (1) ◽  
pp. 1-12
Author(s):  
Daniela De Silva ◽  
◽  
Ovidiu Savin
Keyword(s):  
Boundary Harnack Principle

scholarly journals Boundary Harnack principle for subordinate Brownian motions

2009 ◽  
Vol 119 (5) ◽  
pp. 1601-1631 ◽  
Author(s):  
Panki Kim ◽  
Renming Song ◽  
Zoran Vondraček
Keyword(s):  
Boundary Harnack Principle ◽  
Brownian Motions

scholarly journals Radial symmetry for an elliptic PDE with a free boundary

2021 ◽  
Vol 8 (26) ◽  
pp. 311-319
Author(s):  
Layan El Hajj ◽  
Henrik Shahgholian
Keyword(s):  
Free Boundary ◽  
Radial Symmetry ◽  
Regularity Of Solutions ◽  
Elliptic Pde ◽  
Linear Elliptic Equation ◽  
Boundary Harnack Principle ◽  
Partial Differential ◽  
Content Type ◽  
Moving Plane ◽  
Plane Technique

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u )  in  B 1 , 0 ≤ u > M ,  in  B 1 , u = M ,  on  ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

scholarly journals Equivalence between the boundary Harnack principle and the Carleson estimate

2008 ◽  
Vol 103 (1) ◽  
pp. 61 ◽  
Author(s):  
Hiroaki Aikawa
Keyword(s):  
Harmonic Functions ◽  
Boundary Behavior ◽  
Boundary Harnack Principle ◽  
Positive Harmonic Functions

Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this paper is to show that the boundary Harnack principle and the Carleson estimate are equivalent for arbitrary domains.

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