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- W4300660715 abstract "In this paper we consider a smooth bounded domain $Omega subset R^N$ and a parametric family of radially symmetric kernels $K_epsilon: R^N to R_+$ such that, for each $epsilon in (0,1)$, its $L^1-$norm is finite but it blows up as $epsilon to 0$. Our aim is to establish an $epsilon$ independent modulus of continuity in ${Omega}$, for the solution $u_epsilon$ of the homogeneous Dirichlet problem begin{equation*} left { begin{array}{rcll} - I_epsilon [u] &=& f & mbox{in} Omega. u &=& 0 & mbox{in} Omega^c, end{array} right . end{equation*} where $f in C(bar{Omega})$ and the operator $I_epsilon$ has the form begin{equation*} I_epsilon[u](x) = frac12int limits_{R^N} [u(x + z) + u(x - z) - 2u(x)]K_epsilon(z)dz end{equation*} and it approaches the fractional Laplacian as $epsilonto 0$. The modulus of continuity is obtained combining the comparison principle with the translation invariance of $I_epsilon$, constructing suitable barriers that allow to manage the discontinuities that the solution $u_epsilon$ may have on $partial Omega$. Extensions of this result to fully non-linear elliptic and parabolic operators are also discussed." @default.
- W4300660715 created "2022-10-03" @default.
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- W4300660715 date "2014-04-01" @default.
- W4300660715 modified "2023-10-12" @default.
- W4300660715 title "Uniform Equicontinuity for a family of Zero Order operators approaching the fractional Laplacian" @default.
- W4300660715 hasPublicationYear "2014" @default.
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