zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains. (English) Zbl 1209.31005
Summary: The purpose of this paper is to extend some results of potential theory for elliptic operators to the fractional Laplacian (-Δ) α/2 , 0<α<2, in a bounded C 1,1 domain D in n . In particular, we introduce a new Kato class K α (D), and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian.
MSC:
31C99Generalizations in potential theory
References:
[1]Janicki, A.; Weron, A.: Simulation and chaotic behavior of α-stable processes, (1994)
[2]Klater, J.; Shlesinger, M. F.; Zumofen, G.: Beyond Brownian motion, Phys. today 49, No. 2, 33-39 (1996)
[3]Aizenman, M.; Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators, Comm. pure appl. Math. 35, 209-271 (1982) · Zbl 0459.60069 · doi:10.1002/cpa.3160350206
[4]Ben Othman, S.; Mâagli, H.; Masmoudi, S.; Zribi, M.: Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems, Nonlinear anal. 71, 4137-4150 (2009) · Zbl 1177.35091 · doi:10.1016/j.na.2009.02.073
[5]Bliedtner, J.; Hansen, W.: Potential theory. An analytic and probabilistic approach to balayage, (1986) · Zbl 0706.31001
[6]Chung, K. L.; Zhao, Z.: From Brownian motion to Schrödinger’s equation, (1995)
[7]Lazer, A. C.; Mckenna, P. J.: On a singular nonlinear elliptic boundary value problem, Proc. amer. Math. soc. 111, 721-730 (1991) · Zbl 0727.35057 · doi:10.2307/2048410
[8]Mâagli, H.: Perturbation semi-linéaire des résolvantes et des semi-groupes, Potential anal. 3, 61-87 (1994) · Zbl 0797.47038 · doi:10.1007/BF01047836
[9]Mâagli, H.: Inequalities for the Riesz potentials, Arch. inequal. Appl. 1, 285-294 (2003) · Zbl 1050.31005
[10]Mâagli, H.; Zribi, M.: Existence and estimates of solutions for singular nonlinear elliptic problems, J. math. Anal. appl. 263, 522-542 (2001) · Zbl 1030.35064 · doi:10.1006/jmaa.2001.7628
[11]Mâagli, H.; Zribi, M.: On a new Kato class and singular solutions of a nonlinear elliptic equation in bounded domain of rn, Positivity 9, 667-686 (2005) · Zbl 1131.35335 · doi:10.1007/s11117-005-2782-z
[12]Port, S.; Stone, C.: Brownian motion and classical potential theory, Probab. math. Statist. (1978) · Zbl 0413.60067
[13]Widman, K. O.: Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. scand. 21, 17-37 (1967) · Zbl 0164.13101
[14]Zhao, Z.: Green function for Schrödinger operator and conditional Feynman–Kac gauge, J. math. Anal. appl. 116, 309-334 (1986) · Zbl 0608.35012 · doi:10.1016/S0022-247X(86)80001-4
[15]Zeddini, N.: Positive solutions for a singular nonlinear problem on a bounded domain in R2, Potential anal. 18, 97-118 (2003) · Zbl 1028.34030 · doi:10.1023/A:1020559619108
[16]Chen, Z. Q.; Song, R.: Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. ann. 312, 465-501 (1998) · Zbl 0918.60068 · doi:10.1007/s002080050232
[17]Bogdan, K.; Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Studia math. 133, 53-92 (1999) · Zbl 0923.31003
[18]Chen, Z. Q.; Song, R.: General gauge and conditional gauge theorems, Ann. probab. 30, 1313-1339 (2002) · Zbl 1017.60086 · doi:10.1214/aop/1029867129
[19]Kim, P.: Relative Fatou’s theorem for (-Δ)α/2-harmonic functions in bounded κ-fat open sets, J. funct. Anal. 234, 70-105 (2006) · Zbl 1100.31008 · doi:10.1016/j.jfa.2005.12.001
[20]Chen, Z. Q.; Kim, P.; Song, R.: Heat kernel estimates for Dirichlet fractional Laplacian, J. eur. Math. soc. 12, 1307-1329 (2010) · Zbl 1203.60114 · doi:10.4171/JEMS/231 · doi:http://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=12&iss=5&rank=9
[21]Bogdan, K.: Representation of α-harmonic functions in Lipschitz domains, Hiroshima math. J. 29, 227-243 (1999) · Zbl 0936.31008
[22]Chen, Z. Q.; Song, R.: Martin boundary and integral representation for harmonic function of symmetric processes, J. funct. Anal. 159, 267-294 (1998) · Zbl 0954.60003 · doi:10.1006/jfan.1998.3304
[23]Bogdan, K.: The boundary Harnack principle for the fractional Laplacian, Studia math. 123, No. 1, 43-80 (1997) · Zbl 0870.31009
[24]Blumenthal, R. M.; Getoor, R. K.; Ray, D. B.: On the distribution of first hits for the symmetric stable processes, Trans. amer. Math. soc. 99, 540-554 (1961) · Zbl 0118.13005 · doi:10.2307/1993561
[25]Hmissi, F.: Fonctions harmoniques pour LES potentiels de Riesz sur la boule unité, Expo. math. 12, 281-288 (1994) · Zbl 0814.31004
[26]Chen, Z. Q.; Kim, P.: Stability of martin boundary under nonlocal Feynman–Kac perturbations, Probab. theory related fields 128, 525-564 (2004) · Zbl 1051.31004 · doi:10.1007/s00440-006-0317-8