POSITIVE POWERS OF THE LAPLACIAN IN THE HALF-SPACE UNDER DIRICHLET BOUNDARY CONDITIONS

被引：10

作者：

Abatangelo, Nicola ^{[1
]}

Dipierro, Serena ^{[2
]}

Fall, Mouhamed Moustapha ^{[3
]}

Jarohs, Sven ^{[4
]}

Saldana, Alberto ^{[5
]}

机构：

[1] Univ Libre Bruxelles, Dept Math, CP 214,Blvd Triomphe, B-1050 Ixelles, Belgium

[2] Univ Western Australia, Dept Math & Stat, 35 Stirling Hwy, Crawley, WA 6009, Australia

[3] African Inst Math Sci AIMS Senegal, Route Joal Ctr IRD Mbour, KM 2,BP 1418, Mbour, Senegal

[4] Goethe Univ Frankfurt, Inst Math, Robert Mayer Str 10, D-60054 Frankfurt, Germany

[5] Karlsruher Inst Technol, Inst Anal, Englerstr 2, D-76131 Karlsruhe, Germany

来源：

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS | 2019年 / 39卷 / 03期

关键词：

Fractional Laplacian; Green function; Poisson kernel; s-harmonicity; Kelvin transform; nonlocal operator; FRACTIONAL LAPLACIANS; HARMONIC-FUNCTIONS; GREEN-FUNCTION; REGULARITY; EQUATIONS;

D O I：

10.3934/dcds.2019052

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

引用

页码：1205 / 1235

页数：31

共 28 条

[1]

Abatangelo N., COMM CONT MATH

[2] ON THE LOSS OF MAXIMUM PRINCIPLES FOR HIGHER-ORDER FRACTIONAL LAPLACIANS [J].

Abatangelo, Nicola ;

Jarohs, Sven ;

Saldana, Alberto .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2018, 146 (11) :4823-4835

[3] POSITIVE POWERS OF THE LAPLACIAN: FROM HYPERSINGULAR INTEGRALS TO BOUNDARY VALUE PROBLEMS NICOLA [J].

Abatangelo, Nicola ;

Jarohs, Sven ;

Saldana, Alberto .

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 2018, 17 (03) :899-922

[4] Green function and Martin kernel for higher-order fractional Laplacians in balls [J].

Abatangelo, Nicola ;

Jarohs, Sven ;

Saldana, Alberto .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2018, 175 :173-190

[5] LARGE S-HARMONIC FUNCTIONS AND BOUNDARY BLOW-UP SOLUTIONS FOR THE FRACTIONAL LAPLACIAN [J].

Abatangelo, Nicola .

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 2015, 35 (12) :5555-5607

[6] ESTIMATES NEAR THE BOUNDARY FOR SOLUTIONS OF ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS SATISFYING GENERAL BOUNDARY CONDITIONS .1. [J].

AGMON, S ;

DOUGLIS, A ;

NIRENBERG, L .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1959, 12 (04) :623-727

[7]

[Anonymous], 1981, Higher Transcendental Functions

[8]

[Anonymous], 1972, COMBINATORIAL IDENTI

[9] Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space [J].

Bachar, I ;

Mâagli, H ;

Zribi, M .

MANUSCRIPTA MATHEMATICA, 2004, 113 (03) :269-291

[10] On Kelvin transformation [J].

Bogdan, K. ;

Zak, T. .

JOURNAL OF THEORETICAL PROBABILITY, 2006, 19 (01) :89-120

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