Realization of the fractional Laplacian with nonlocal exterior conditions via forms method

被引:19
作者
Claus, Burkhard [1 ]
Warma, Mahamadi [2 ]
机构
[1] Tech Univ Dresden, Inst Anal, D-01062 Dresden, Germany
[2] George Mason Univ, Dept Math Sci, Fairfax, VA 22030 USA
关键词
Fractional Laplacian; Forms method; Dirichlet; Neumann; and Robin exterior conditions; Submarkovian semigroup; Ultracontractivity; Domination of semigroups; BOUNDARY-CONDITIONS; EXTENSION PROBLEM; MU-TRANSMISSION; DIRICHLET; REGULARITY;
D O I
10.1007/s00028-020-00567-0
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Let omega subset of R-n (n >= 1) be a bounded open set with a Lipschitz continuous boundary. In the first part of the paper, using the method of bilinear forms we give a characterization of the realization in L2(omega) of the fractional Laplace operator (-Delta)s (0<s<1) with the nonlocal Neumann and Robin exterior conditions. Contrarily to the classical local case s=1, it turns out that the nonlocal (Robin and Neumann) exterior conditions can be incorporated in the form domain. We show that each of the above operators generates a strongly continuous submarkovian semigroup which is also ultracontractive. In the second part, we prove that the semigroup corresponding to the nonlocal Robin exterior condition is always sandwiched between the fractional Dirichlet semigroup and the fractional Neumann semigroup.
引用
收藏
页码:1597 / 1631
页数:35
相关论文
共 40 条
  • [1] [Anonymous], 2011, De Gruyter Stud. Math.
  • [2] [Anonymous], 2002, THESIS U ULM
  • [3] External optimal control of nonlocal PDEs
    Antil, Harbir
    Khatri, Ratna
    Warma, Mahamadi
    [J]. INVERSE PROBLEMS, 2019, 35 (08)
  • [4] Dirichlet and Neumann boundary conditions: What is in between?
    Arendt, W
    Warma, M
    [J]. JOURNAL OF EVOLUTION EQUATIONS, 2003, 3 (01) : 119 - 135
  • [5] The Laplacian with Robin boundary conditions on arbitrary domains
    Arendt, W
    Warma, M
    [J]. POTENTIAL ANALYSIS, 2003, 19 (04) : 341 - 363
  • [6] Arendt W., 2012, Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, Operator Theory: Advances and Applications, P47, DOI DOI 10.1007/978-3-0348-0297-04
  • [7] Local Elliptic Regularity for the Dirichlet Fractional Laplacian
    Biccari, Umberto
    Warma, Mahamadi
    Zuazua, Enrique
    [J]. ADVANCED NONLINEAR STUDIES, 2017, 17 (02) : 387 - 409
  • [8] Nonlocal Tug-of-War and the Infinity Fractional Laplacian
    Bjorland, C.
    Caffarelli, L.
    Figalli, A.
    [J]. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 2012, 65 (03) : 337 - 380
  • [9] Blumenthal Robert M, 1960, Trans. Am. Math. Soc, V95, P263, DOI [DOI 10.1090/S0002-9947-1960-0119247-6, 10.1090/S0002-9947-1960-0119247-6]
  • [10] Censored stable processes
    Bogdan, K
    Burdzy, K
    Chen, ZQ
    [J]. PROBABILITY THEORY AND RELATED FIELDS, 2003, 127 (01) : 89 - 152