5,339 research outputs found
Local and nonlocal boundary conditions for -transmission and fractional elliptic pseudodifferential operators
A classical pseudodifferential operator on satisfies the
-transmission condition relative to a smooth open subset , when
the symbol terms have a certain twisted parity on the normal to . As shown recently by the author, the condition assures solvability of
Dirichlet-type boundary problems for elliptic in full scales of Sobolev
spaces with a singularity ,
. Examples include fractional
Laplacians 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
reduce to problems supported on , and how the so-called "large"
solutions arise. Moreover, the results are extended to general function spaces
and , including H\"older-Zygmund spaces . This leads to optimal H\"older estimates, e.g. for Dirichlet
solutions of ,
when , (in when ).Comment: Title slightly changed, 34 page
A nonhomogeneous boundary value problem in mass transfer theory
We prove a uniqueness result of solutions for a system of PDEs of
Monge-Kantorovich type arising in problems of mass transfer theory. The results
are obtained under very mild regularity assumptions both on the reference set
, and on the (possibly asymmetric) norm defined in
. In the special case when is endowed with the Euclidean
metric, our results provide a complete description of the stationary solutions
to the tray table problem in granular matter theory.Comment: 22 pages, 2 figure
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