We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.
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Estimates for Dirichlet-to-Neumann Maps as Integro-differential Operators
Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a nonlinear Dirichlet-to-Neumann mapping that is constructed using a solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the Lévy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the Lévy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings.
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- PAR ID:
- 10097955
- Date Published:
- Journal Name:
- Potential analysis
- ISSN:
- 1572-929X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Grid-free Monte Carlo methods based on the
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We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L‐T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.
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Nonlocal gradient operators are prototypical nonlocal differential operators that are very important in the studies of nonlocal models. One of the simplest variational settings for such studies is the nonlocal Dirichlet energies wherein the energy densities are quadratic in the nonlocal gradients. There have been earlier studies to illuminate the link between the coercivity of the Dirichlet energies and the interaction strengths of radially symmetric kernels that constitute nonlocal gradient operators in the form of integral operators. In this work we adopt a different perspective and focus on nonlocal gradient operators with a non-spherical interaction neighborhood. We show that the truncation of the spherical interaction neighborhood to a half sphere helps making nonlocal gradient operators well-defined and the associated nonlocal Dirichlet energies coercive. These become possible, unlike the case with full spherical neighborhoods, without any extra assumption on the strengths of the kernels near the origin. We then present some applications of the nonlocal gradient operators with non-spherical interaction neighborhoods. These include nonlocal linear models in mechanics such as nonlocal isotropic linear elasticity and nonlocal Stokes equations, and a nonlocal extension of the Helmholtz decomposition.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s11118-019-09776-w
