Let Ω ⊂ 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 L 2 (Ω) of the fractional Laplace operator (−∆) 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.2010 Mathematics Subject Classification. 35R11, 47D07, 47D06, 34B10.
In this article we study lower semicontinuous, convex functionals on real Hilbert spaces. In the first part of the article we construct a Banach space that serves as the energy space for such functionals. In the second part we study nonlinear Dirichlet forms, as defined by Cipriani and Grillo, and show, as it is well known in the bilinear case, that the energy space of such forms is a lattice. We define a capacity and introduce the notion quasicontinuity associated with these forms and prove several results, which are well known in the bilinear case.
In this article we study lower semicontinuous, convex functionals on real Hilbert spaces. In the first part of the article we construct a Banach space that serves as the energy space for such functionals. In the second part we study nonlinear Dirichlet forms, as defined by Cipriani and Grillo, and show, as it is well known in the bilinear case, that the energy space of such forms is a lattice. We define a capacity and introduce the notion quasicontinuity associated with these forms and prove several results, which are well known in the bilinear case.
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