Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional ( p , q ) {(p,q)} -Laplacian, denoted by ( - Δ ) p s 1 + ( - Δ ) q s 2 {(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}} for s 2 , s 1 ∈ ( 0 , 1 ) {s_{2},s_{1}\in(0,1)} and 1 < p , q < ∞ {1<p,q<\infty} . We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional ( p , q ) {(p,q)} -problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish a new Hopf-type maximum principle and a strong comparison principle in both situations.

Instrumental Means for Managing the Rational Behavior of Self-Organizing Autonomous Intelligent Agents

We formulate the basic principles of constructing a sign-signal control for the expedient behavior of autonomous intelligent agents in a priori undescribed conditions of a problematic environment. We clarify the concept of a self-organizing autonomous intelligent agent as a system capable of automatic goal-setting when a certain type of conditional and unconditional signal — signs appears in a problem environment. The procedures for planning the expedient behavior of autonomous intelligent agents have been developed, that imitate trial actions under uncertainty in the process of studying the regularities of transforming situations in a problem environment, which allows avoiding environmental changes in the process of self-learning that are not related to the achievement of a given goal. Boundary estimates of the proposed procedures complexity for planning expedient behavior are determined, confirming the possibility of their effective implementation on the on-board computer of the automatic control system for the expedient activity of autonomous intelligent agents. We carry out an imitation on a personal computer of the proposed procedures for planning purposeful behavior, confirming the effectiveness of their use to build intelligent problem solvers for autonomous intelligent agents in order to endow them with the ability to adapt to a priori undescribed operating conditions. The main types of connections between various conditional and unconditional signal — signs of a problem environment are structured, which allows autonomous intelligent agents to adapt to complex a priori undescribed and unstable conditions of functioning.

Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness

The primary objective of the paper is to study the existence, asymptotic boundary estimates and uniqueness of large solutions to fully nonlinear equations {H(x,u,Du,D^{2}u)=f(u)+h(x)} in bounded {C^{2}} domains {\Omega\subseteq\mathbb{R}^{n}} . Here H is a fully nonlinear uniformly elliptic differential operator, f is a non-decreasing function that satisfies appropriate growth conditions at infinity, and h is a continuous function on Ω that could be unbounded either from above or from below. The results contained herein provide substantial generalizations and improvements of results known in the literature.