It is shown how one can get upper bounds for ju ? vj when u and v are the (viscosity) solutions of ut ? (Dxu))xu = 0 and vt ? (Dxv))xv = 0; respectively, in (0;1) with Dirichlet boundary conditions. Similar results are obtained for some other parabolic equations as well, including certain equations in divergence form. 1. Introduction In this paper we study… (More)
Sufficient conditions are given for a generalized viscosity solution of an elliptic boundary value problem to satisfy the boundary values in the strong sense.
The equation () where D t and D x are fractional derivatives of order and is studied. It is shown that if f = f(t; x), h 1 = h 1 (x), and h 2 = h 2 (t) are HH older-continuous and f(0; 0) = 0, then there is a solution such that D t u and D x u are HH older-continuous as well. This is proved by rst considering an abstract fractional evolution equation and… (More)
Subdivision cardinal interpolation schemes that preserve functions of positive type are shown to be related to orthonormal multiresolutions. The interpolating function is the solution to a certain optimimization problem, and this makes it possible to derive error estimates, in particular for Lagrange iterative interpolation schemes.
The regularity of solutions of the equation ? D t (u ? u 0) (t; x) + c(t; x)u x (t; x) = f(t; x); t; x 0; where D t denotes the fractional derivative, is studied. It is shown that if c and f are HH older-continuous in either t or x and c is strictly positive, then both u x and D t (u ? u 0) are HH older continuous as well (in either t or x).