Gradient estimates for a weighted nonlinear parabolic equation and applications

This paper derives elliptic gradient estimates for positive solutions to a nonlinear parabolic equation defined on a complete weighted Riemannian manifold. Applications of these estimates yield Liouville-type theorem, parabolic Harnack inequalities and bounds on weighted heat kernel on the lower boundedness assumption for Bakry-Émery curvature tensor.

Mathematical Basis of G Spaces

This paper represents some basic mathematic theories for G[Formula: see text] spaces of functions that can be used for weakened weak (W2) formulations, upon which the smoothed finite element methods (S-FEMs) and the smoothed point interpolation methods (S-PIMs) are based for solving mechanics problems. We first introduce and prove properties of G[Formula: see text] spaces, such as the lower boundedness and convergence of the norms, which are in contrast with H1spaces. We then prove the equivalence of the Gsnorms and its corresponding semi-norms. These mathematic theories are important and essential for the establishment of theoretical frame and the development of relevant numerical approaches. Finally, numerical examples are presented by using typical S-FEM models known as the NS-FEM and [Formula: see text]S-FEM to examine the properties of a smoothed method based on Gsspaces, in comparison with the standard FEM with weak formulation.

Reverse Hypercontractivity for Subharmonic Functions

Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, A, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, e−tA, can be bounded below from Lp to Lq when p, q and t are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions.

Duality and quasi-normability for complexity spaces

<p>The complexity (quasi-metric) space was introduced in [23] to study complexity analysis of programs. Recently, it was introduced in [22] the dual complexity (quasi-metric) space, as a subspace of the function space [0,) ^ω. Several quasi-metric properties of the complexity space were obtained via the analysis of its dual.</p> <p>We here show that the structure of a quasi-normed semilinear space provides a suitable setting to carry out an analysis of the dual complexity space. We show that if (E,) is a biBanach space (i.e., a quasi-normed space whose induced quasi-metric is bicomplete), then the function space (B*_E, _B* ) is biBanach, where B*_E = {f : E Σ^∞_n=0 2^-n( V ) } and _B* = Σ^∞_n=0 2^-n We deduce that the dual complexity space admits a structure of quasinormed semlinear space such that the induced quasi-metric space is order-convex, upper weightable and Smyth complete, not only in the case that this dual is a subspace of [0,)^ω but also in the general case that it is a subspace of F^ω where F is any biBanach normweightable space. We also prove that for a large class of dual complexity (sub)spaces, lower boundedness implies total boundedness. Finally, we investigate completeness of the quasi-metric of uniform convergence and of the Hausdorff quasi-pseudo-metric for the dual complexity space, in the context of function spaces and hyperspaces, respectively.</p>

Lower bounds for normal structure coefficients

SynopsisUsing some new expressions for the weakly convergent sequences coefficient WCS(X) the lower boundednessis proved, where δ(-) is the (Clarkson) modulus of convexity. We also define a modulus of noncompact convexity concerning nearly uniformly convex spaces which is used to obtain another lower bound for WCS(X). The computation of this modulus in Ip-spaces shows that our second lower bound is the best possible in these spaces.