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      Local and nonlocal boundary conditions for \(\mu\)-transmission and fractional elliptic pseudodifferential operators

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          Abstract

          A classical pseudodifferential operator \(P\) on \(R^n\) satisfies the \(\mu\)-transmission condition relative to a smooth open subset \(\Omega \), when the symbol terms have a certain twisted parity on the normal to \(\partial\Omega \). As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic \(P\) in full scales of Sobolev spaces with a singularity \(d^{\mu -k}\), \(d(x)=\operatorname{dist}(x,\partial\Omega)\). Examples include fractional Laplacians \((-\Delta)^a\) and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on \(R^n\setminus \Omega \) reduce to problems supported on \(\bar\Omega\), and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces \(F^s_{p,q}\) and \(B^s_{p,q}\), including H\"older-Zygmund spaces \(B^s_{\infty ,\infty}\). This leads to optimal H\"older estimates, e.g. for Dirichlet solutions of \((-\Delta)^au=f\in L_\infty (\Omega)\), \(u\in d^aC^a(\bar\Omega)\) when \(0<a<1\), \(a\ne 1/2\) (in \(d^aC^{a-\epsilon}(\bar\Omega)\) when \(a=1/2\)).

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          The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

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            Regularity theory for fully nonlinear integro-differential equations

            We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump L\`evy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior \(C^{1,\alpha}\) regularity for general fully nonlinear integro-differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.
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              Singular Green operators and their spectral asymptotics

              Gerd Grubb (1984)
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                Author and article information

                Journal
                2014-03-27
                2014-12-20
                Article
                10.2140/apde.2014.7.1649
                1403.7140
                dccf616a-efa8-4c2c-ac07-bd94af717abd

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                35J75, 35S15, 45E99, 46E35, 58J40
                Anal. PDE 7 (2014) 1649-1682
                Title slightly changed, 34 pages
                math.AP math.FA

                Analysis,Functional analysis
                Analysis, Functional analysis

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