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On Critical Points of p Harmonic Functions in the Plane
, 1994
"... We show that if u is a p harmonic function, 1 ! p ! 1; in the unit disk and equal to a polynomial P of positive degree on the boundary of this disk, then ru has at most deg P \Gamma 1 zeros in the unit disk. In this note we prove the following theorem. Theorem 1 Given p; 1 ! p ! 1; let u be a re ..."
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We show that if u is a p harmonic function, 1 ! p ! 1; in the unit disk and equal to a polynomial P of positive degree on the boundary of this disk, then ru has at most deg P \Gamma 1 zeros in the unit disk. In this note we prove the following theorem. Theorem 1 Given p; 1 ! p ! 1; let u be a
A RADO TYPE THEOREM FOR pHARMONIC FUNCTIONS IN THE PLANE
"... Abstract. We show that if u 2 C1(Ω) satises the pLaplace equation div(jrujp−2ru) = 0 in Ω n fx: u(x) = 0g, then u is a solution to the pLaplacian in the whole Ω R2. Throughout this paper we let Ω be an open set in Rn, n 2 and 1 < p < 1 a xed number. The divergence form dierential operato ..."
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operator pu = div(jruj p−2ru) is called the pLaplacian, and a continuous function u 2 W 1;ploc (Ω) is termed to be pharmonic in Ω if pu = 0 in Ω in the sense of distributions. In this note we prove the following theorem: 1. Theorem. Let n = 2 and u 2 C1(Ω). If u is pharmonic in Ω n fx: u(x) = 0g
Boundary Behaviour for p Harmonic Functions in Lipschitz and Starlike Lipschitz Ring Domains
, 2007
"... In this paper we prove new results for p harmonic functions, p 6 = 2, 1 < p < ∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of a Lipschitz domain, ..."
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In this paper we prove new results for p harmonic functions, p 6 = 2, 1 < p < ∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of a Lipschitz domain
A Rado Type Theorem for pharmonic Functions in the Plane
, 1994
"... . We show that if u 2 C 1(\Omega\Gamma satisfies the pLaplace equation div(jruj p\Gamma2 ru) = 0 in\Omega n fx : u(x) = 0g, then u is a solution to the pLaplacian in the whole\Omega ae R 2 . Throughout this paper we let\Omega be an open set in R n , n 2 and 1 ! p ! 1 a fixed number. The ..."
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. The divergence form differential operator \Delta p u = div(jruj p\Gamma2 ru) is called the pLaplacian, and a continuous function u 2 W 1;p loc(\Omega\Gamma is termed to be pharmonic in\Omega if \Delta p u = 0 in\Omega in the sense of distributions. In this note we prove the following theorem: 1. Theorem
THE FIRST L pCOHOMOLOGY OF SOME FINITELY GENERATED GROUPS AND pHARMONIC FUNCTIONS
, 2005
"... Abstract. Let G be a finitely generated infinite group and let p> 1. In this paper we make a connection between the first L pcohomology space of G and pharmonic functions on G. We also describe the elements in the first L pcohomology space of groups with polynomial growth, and we give an inclu ..."
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Abstract. Let G be a finitely generated infinite group and let p> 1. In this paper we make a connection between the first L pcohomology space of G and pharmonic functions on G. We also describe the elements in the first L pcohomology space of groups with polynomial growth, and we give
Harnack’s inequality for pharmonic functions via stochastic games
 Comm. Partial Differential Equations
, 1985
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Asymptotic statistical characterizations of pharmonic functions of two variables
 Rocky Mountain J. Math
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On the characterization of pharmonic functions on the Heisenberg group by mean value properties, preprint
, 2012
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Nonlinear elliptic partial differential equations and pharmonic functions on graphs
, 2013
"... In this article we study the wellposedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit ..."
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of the theory of viscosity solutions for partial differential equations. The equations include the graph Laplacian, the pLaplacian, the Infinity Laplacian, and the Eikonal operator on the graph.
Results 11  20
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837