Obstacle Problems for Integro-Differential Operators with Partially Vanishing Kernels
Tóm tắt
In this paper, we study the obstacle problem of some convex operators, which is related to normalized p-Laplacian
$$\Delta _p^s$$
. We prove that the graph of the regular free boundary is
$$C^{1,\alpha }$$
and the solution is
$$C^{1,s}$$
near these points. Moreover, we show that the set of regular free boundary points is relatively open.
Tài liệu tham khảo
Abbas, M.I., Ragusa, M.A.: On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry 13(2), 264 (2021)
Abbas, M.I., Ragusa, M.A.: Nonlinear fractional differential inclusions with non-singular Mittag–Leffler kernel. AIMS Math. 7, 20328–20340 (2022)
Attia, N., Akgül, A., Atangana, A.: Reproducing kernel method with global derivative. J. Funct. Spaces 2726735, 11 (2023)
Bjorland, C., Caffarelli, L., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230, 1859–1894 (2012)
Bjorland, C., Caffarelli, L., Figalli, A.: Non-local tug-of-war and the infinity fractional Laplacian. Commun. Pure Appl. Math. 65, 337–380 (2012)
Caffarelli, L.A., Ros-Oton, X., Serra, J.: Obstacle problems for integro-differential operators: regularity of solutions and free boundaries. Invent. Math. 208, 1155–1211 (2017)
Caffarelli, L.A., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62, 597–638 (2009)
Caffarelli, L.A., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171, 425–461 (2008)
Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Differ. Geom. 33, 635–681 (1991)
Fernandez-Real, X., Ros-Oton, X.: The obstacle problem for the fractional Laplacian with critical drift. Math. Ann. 371, 1683–1735 (2018)
Guariglia, E.: Riemann zeta fractional derivative-functional equation and link with primes. Adv. Differ. Equ. 261, 15 (2019)
Guariglia, E.: Fractional calculus of the Lerch zeta function. Mediterr. J. Math. 19 109, 11 (2022)
Kassmann, M., Rang, M., Schwab, R.: Integro-differential equations with nonlinear directional dependence. Indiana Univ. Math. J. 63, 1467–1489 (2014)
Li, C., Dao, X., Guo, P.: Fractional derivatives in complex planes. Nonlinear Anal. 71, 1857–1869 (2009)
Mohammed, P.O., Fernandez, A.: Integral inequalities in fractional calculus with general analytic kernels. Filomat 37, 3659–3669 (2023)
Peres, Y., Sheffeld, S.: Tug-of-war with noise: a game-theoretic view of the \(p\)-Laplacian. Duke Math. J. 145, 91–120 (2008)
Qi, S., Tang, L.: Boundary regularity estimates for nonlocal normalized \(p\)-Laplacian in different kind of domains. Nonlinear Anal. Theor. 199, 1–32 (2020)
Ragusa, M.A.: Commutators of fractional integral operators on vanishing-Morrey spaces. J. Glob. Optim. 40, 361–368 (2008)
Ros-Oton, X., Serra, J.: Boundary regularity for fully nonlinear integro-differential equations. Duke Math. J. 165, 2079–2154 (2016)
Ros-Oton, X., Serra, J.: Regularity theory for general stable operators. J. Differ. Equ. 260, 8675–8715 (2016)
Ros-Oton, X., Serra, J.: Boundary regularity estimates for nonlocal elliptic equations in \(C^1\) and \(C^{1,\alpha }\) domains. Ann. Mat. 196, 1637–1668 (2017)
Ros-Oton, X., Serra, J.: The structure of the free boundary in the fully nonlinear thin obstacle problem. Adv. Math. 316, 710–747 (2017)
Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)