Hölder condition

Abstract: We prove the boundedness of the Hardy–Littlewood maximal function on the generalized Lebesgue space Lp(·)(Rd) under a continuity assumption on p that is weaker than uniform Holder continuity. We deduce continuity of mollifying sequences and density of C∞(Ω) in W1,p(·)(Ω) . Mathematics subject classification (2000): 42B25, 46E30.

Abstract: where u is essentially the density of the gas and A is the Laplace operator. Note that, apart from constants, grad um 1 is the velocity vector and u grad um1 is the flux vector. Thus, in particular, um1 is essentially the pressure which, by Darcy's law, is also the velocity potential. For m > 1, (1) is a nonlinear equation which is parabolic for u > 0, but which degenerates when u = 0. The most striking manifestation of the degeneracy of this equation is the finite speed of propagation of disturbances. Thus, if at some instant of time a solution u of (1) has compact support, then it will continue to have compact support for all later times. In general, the transition from a region where u > 0 to one where u = 0 is not smooth and it is therefore necessary to interpret the term "solution of (1)" in some generalized sense. Our object in this paper is to study the regularity properties of a class of generalized solutions of the Cauchy problem for (1). We show that, with respect to the space variables, the velocity potential is Lipschitz continuous, the flux is continuous, and the density is Holder continuous.

Abstract: In order that isothermal parameters exist it is necessary to impose on the metric some regularity assumptions. In fact, it was shown recently by Hartman and Wintnerl that it is not sufficient to assume the functions E, F, G to be continuous. So far the weakest conditions under which the isothermal parameters are known to exist were found by Korn and Lichtenstein.2 To formulate their theorem we recall that a function f(x, y) in a domain D of the (x, y)-plane is said to satisfy a Holder condition of order X, 0