Some integrability estimates for solutions of the fractional \(p\)-Laplace equation
-
2085
Downloads
-
3303
Views
Authors
Shaoguang Shi
- Department of Mathematics, Linyi University, Linyi 276005, China.
Abstract
For \((\alpha,p)\in(0,1)\times (1,\infty)\), this note focuses on some integrability estimates for solutions of the following Dirichlet problem
\[
\begin{cases}
L_{\alpha,p}u(x)=g(x) \,\, \hbox{as} \,\,x\in \Omega,\\
u(x)=0 \,\, \hbox{as} \,\,x\in \mathbb{R}^{n}\backslash \Omega,
\end{cases}
\]
where \(L_{\alpha,p}\) is the fractional \(p\)-Laplace operator.
Share and Cite
ISRP Style
Shaoguang Shi, Some integrability estimates for solutions of the fractional \(p\)-Laplace equation, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 10, 5585--5592
AMA Style
Shi Shaoguang, Some integrability estimates for solutions of the fractional \(p\)-Laplace equation. J. Nonlinear Sci. Appl. (2017); 10(10):5585--5592
Chicago/Turabian Style
Shi, Shaoguang. "Some integrability estimates for solutions of the fractional \(p\)-Laplace equation." Journal of Nonlinear Sciences and Applications, 10, no. 10 (2017): 5585--5592
Keywords
- Fractional \(p\)-Laplace equation
- Dirichlet problem
- solution
MSC
References
-
[1]
B. Barrios, I. Peral, S. Vita, Some remarks about the summability of nonlocal nonlinear problems , Adv. Nonlinear Anal., 4 (2015), 91–107.
-
[2]
C. Bjorland, L. Caffarelli, A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337–380.
-
[3]
K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89–152.
-
[4]
L. A. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425–461.
-
[5]
A. Di Castro, T. Kuusi, G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807–1836.
-
[6]
E. Di Nezza, G. Palatucci, E. Valdinoci , Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573.
-
[7]
A. Garroni, R. V. Kohn, Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2613–2625.
-
[8]
Q.-Y. Guan, Z.-M. Ma, Reflected symmetric \(\alpha\)-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649–694.
-
[9]
R. Hurri-Syrjänen, A. V. Vähäkangas, Fractional Sobolev-Poincar and fractional Hardy inequalities in unbounded John domains, Mathematika, 61 (2015), 385–401.
-
[10]
A. Iannizzotto, S.-B. Liu, K. Perera, M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101–125.
-
[11]
N. S. Landkof, Foundations of modern potential theory , Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg (1972)
-
[12]
T. Leonori, I. Peral, A. Primo, F. Soria, Basic estimates for solution of elliptic and parabolic equations for a class of nonlocal operators, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 6031–6068.
-
[13]
E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–826.
-
[14]
X. Ros-Oton, J. Serra , The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275–302.
-
[15]
S.-G. Shi, J. Xiao, On fractional capacities relative to bounded open Lipschitz sets, Potential Anal., 45 (2016), 261–298.
-
[16]
S.-G. Shi, J. Xiao, Fractional capacities relative to bounded open Lipschitz sets complemented, Calc. Var. Partial Differential Equations, 56 (2017), 22 pages.
-
[17]
P. Shvartsman, On Sobolev extension domains in \(R^n\), J. Funct. Anal., 258 (2010), 2205–2245.
-
[18]
E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. (1970)
-
[19]
H. Triebel, Function spaces and wavelets on domains, EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich (2008)
-
[20]
J. Xiao, Z. Zhai, Fractional Sobolev, Moser-Trudinger Morrey-Sobolev inequalities under Lorentz norms, Problems in mathematical analysis, J. Math. Sci. (N.Y.), 166 (2010), 357–376.
-
[21]
Y. Zhou , Fractional Sobolev extension and imbedding , Trans. Amer. Math. Soc., 367 (2015), 959–979.
