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The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions ≥ 3 is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of "large solutions" for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions ≤ 5 is one-dimensional, independently of the parameter s ∈ (0,1). [ABSTRACT FROM AUTHOR]

In this paper we analyze the singular set in the Stefan problem and prove the following results: [ABSTRACT FROM AUTHOR]

We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time T * . More precisely, we show that solutions are C 2 , α ⁢ (Ω ¯) in space, with α = 1 m , and C ∞ in time (uniformly in x ∈ Ω ¯ ), for t > T * . Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence O ⁢ (t - 1 - γ) , and we prove that the convergence holds in the C 1 , α ⁢ (Ω ¯) topology. [ABSTRACT FROM AUTHOR]

We study the obstacle problem for parabolic operators of the type ∂t+L$\partial _t + L$, where L is an elliptic integro‐differential operator of order 2s, such as (−Δ)s$(-\Delta)^s$, in the supercritical regime s∈(0,12)$s \in (0,\frac{1}{2})$. The best result in this context was due to Caffarelli and Figalli, who established the Cx1,s$C^{1,s}_x$ regularity of solutions for the case L=(−Δ)s$L = (-\Delta)^s$, the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually more regular than in the elliptic case. More precisely, we show that they are C1, 1 in space and time, and that this is optimal. We also deduce the C1,α$C^{1,\alpha }$ regularity of the free boundary. Moreover, at all free boundary points (x0,t0)$(x_0,t_0)$, we establish the following expansion: (u−φ)(x0+x,t0+t)=c0(t−a·x)+2+O(t2+α+|x|2+α),$$\begin{equation*} (u - \varphi)(x_0+x,t_0+t) = c_0(t - a\cdot x)_+^2\! +\! O(t^{2+\alpha }+|x|^{2+\alpha }), \end{equation*}$$with c0>0$c_0 > 0$, α>0$\alpha > 0$ and a∈Rn$a \in \mathbb {R}^n$. [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]