8 research outputs found
Parabolic equations associated with symmetric nonlocal operators
Felsinger M. Parabolic equations associated with symmetric nonlocal operators. Bielefeld: Bielefeld University; 2013
Local regularity for parabolic nonlocal operators
Weak solutions to parabolic integro-differential operators of order are studied. Local a priori estimates of H\"older norms and
a weak Harnack inequality are proved. These results are robust with respect to
. In this sense, the presentation is an extension of Moser's
result in 1971.Comment: 31 pages, 3 figure
Fractional-order operators: Boundary problems, heat equations
The first half of this work gives a survey of the fractional Laplacian (and
related operators), its restricted Dirichlet realization on a bounded domain,
and its nonhomogeneous local boundary conditions, as treated by
pseudodifferential methods. The second half takes up the associated heat
equation with homogeneous Dirichlet condition. Here we recall recently shown
sharp results on interior regularity and on -estimates up to the boundary,
as well as recent H\"older estimates. This is supplied with new higher
regularity estimates in -spaces using a technique of Lions and Magenes,
and higher -regularity estimates (with arbitrarily high H\"older estimates
in the time-parameter) based on a general result of Amann. Moreover, it is
shown that an improvement to spatial -regularity at the boundary is
not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in
Mathematics and Statistics: "New Perspectives in Mathematical Analysis -
Plenary Lectures, ISAAC 2017, Vaxjo Sweden
The Dirichlet problem for nonlocal operators
Felsinger M, Kaßmann M, Voigt P. The Dirichlet problem for nonlocal operators. Mathematische Zeitschrift. 2015;279(3-4):779-809.In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the "boundary data" are prescribed on the complement of a given bounded set. We formulate the problem in the classical framework of Hilbert spaces and prove unique solvability using standard techniques like the Fredholm alternative
The Dirichlet problem for nonlocal operators
Felsinger M, Kaßmann M, Voigt P. The Dirichlet problem for nonlocal operators. Mathematische Zeitschrift. 2015;279(3-4):779-809.In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the "boundary data" are prescribed on the complement of a given bounded set. We formulate the problem in the classical framework of Hilbert spaces and prove unique solvability using standard techniques like the Fredholm alternative