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Abstract: We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of in Ω, in , for some and , then u is and is up to the boundary ∂Ω for some , where . For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on g we obtain higher order Hölder estimates for u and . Namely, the norms of u and in the sets are controlled by and , respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian (Ros-Oton and Serra, 2012 [19,20]). [Copyright &y& Elsevier]

Abstract: Stable inversion plays a key role in the solution of the exact tracking control problem in nonminimum phase systems. However, the general methods developed so far for the computation of stable inverses require backwards time numeric integration of the internal dynamics equation, which yields high sensitivity to external disturbances and/or structured uncertainties. This article introduces an iterative technique that provides periodic, closed-form analytic expressions uniformly convergent to the exact periodic solution of a certain class of Abel ODE written in the normal form. The method is then applied to the output voltage tracking of periodic references in DC–DC boost power converters through a state feedback indirect control scheme. The procedure lies on a number of assumptions for which sufficient conditions involving system parameters and reference candidates are derived. It also allows one to attenuate the effect of bounded, piecewise constant load disturbances using dynamic compensation. Simulation results validate the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension N ≥ 3. We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth. [ABSTRACT FROM AUTHOR]

• We develop a model on how military and trade alliances can impact defense spending. • Importance of trade and its balance with the leader influences defense spending. • The larger the leader's trade and surplus, the lower the minor partners' spending. Military spending was the main government expenditure until the 20th century, and it still represents a significant fraction of most governments' budgets. We develop a theoretical model to understand how both military and trade alliances with military leaders can impact defense spending. By increasing the costs of military aggression by a non-ally, an alliance reduces the probability of war and allows minor partners reducing their military spending in exchange for a stronger trade relationship with an alliance leader and a higher trading surplus for the latter. We test our hypotheses with data on 138 countries for 1996–2020. Our results show that the importance of the trade relationship and the trade balance with the military alliance leader is a significant driver of military spending. The greater the weight of trade with the military leader and the higher its trade surplus, the lower is the defense spending of the minor partner. [ABSTRACT FROM AUTHOR]