We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-...

where a > 0 is some constant and x̄ ∈ R. Hypothesis (2) was removed by Caffarelli, Gidas and Spruck in [6]; this is important for applications. Such Liouville type theorems have been extended to general conformally invariant fully nonlinear equations by Li and Li ([22]-[25]); see also related works of Viaclovsky ([35]-[36]) and Chang, Gursky and Yang ([11]-[12]). The method used in [19], as well...

US and European pedestrian crash cases were analyzed to determine frequency of injury by body region and by the vehicle component identified as the injury source. US pedestrian data was drawn from the Pedestrian Crash Data Study (PCDS). European pedestrian data was drawn from the German In-Depth Accident Study (GIDAS). Results were analyzed in terms of both serious injury (AIS 3+) and disabling...

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction...

Various Liouville type theorems for conformally invariant equations have been obtained by Obata ([9]), Gidas, Ni and Nirenberg ([4]), Caffarelli, Gidas and Spruck ([1]), Viaclovsky ([10] and [11]), Chang, Gursky and Yang ([2] and [3]), and Li and Li ([5], [6] and [7]). See e. g. theorem 1.3 and remark 1.6 in [6] where these results (except for the one in [7]) are stated more precisely. In this ...

In this paper we study a class of nonvariational elliptic systems, by using the Gidas-Spruck Blow-up method. first, we obtain a priori estimates, and then using Leray-Schauder topological degree theory, we establish the existence of positive radial solutions vanishing at infinity.

A safe introduction of automated driving systems on urban roads requires a thorough understanding the traffic conflicts and accidents. This is paramount to constructively safeguard these systems, i.e., design system that exhibits an adequate performance even in critical situations. In this work, we present approach gather knowledge by analyzing German In-Depth Accident Study (GIDAS) database, w...