caa a22 4500 60551562X CHVBK 20210128100710.0 cr unu---uuuuu 210128e20150901xx s 000 0 eng 10.1007/s00205-015-0851-4 doi (NATIONALLICENCE)springer-10.1007/s00205-015-0851-4 Kriventsov Dennis Mathematics Department, University of Texas at Austin, Austin, TX, USA aut Regularity for a Local-Nonlocal Transmission Problem [Elektronische Daten] [Dennis Kriventsov] We formulate and study an elliptic transmission-like problem combining local and nonlocal elements. Let $${\mathbb{R}^n}$$ R n be separated into two components by a smooth hypersurface Γ. On one side of Γ, a function satisfies a local second-order elliptic equation. On the other, it satisfies a nonlocal one of lower order. In addition, a nonlocal transmission condition is imposed across the interface Γ, which has a natural variational interpretation. We deduce the existence of solutions to the corresponding Dirichlet problem, and show that under mild assumptions they are Hölder continuous, following the method of De Giorgi. The principal difficulty stems from the lack of scale invariance near Γ, which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter. We then turn to the question of optimal regularity and qualitative properties of solutions, and show that (in the case of constant coefficients and flat Γ) they satisfy a kind of transmission condition, where the ratio of "fractional conormal derivatives” on the two sides of the interface is fixed. A perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface. Throughout, we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution, which has an interpretation in the context of quasigeostrophic dynamics. Springer-Verlag Berlin Heidelberg, 2015 Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 217/3(2015-09-01), 1103-1195 0003-9527 217:3<1103 2015 217 205 https://doi.org/10.1007/s00205-015-0851-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-015-0851-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Kriventsov Dennis Mathematics Department, University of Texas at Austin, Austin, TX, USA aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 217/3(2015-09-01), 1103-1195 0003-9527 217:3<1103 2015 217 205