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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 prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in L q with q > n. Our approach is based on the scaling and comparison arguments of [13] , and we show that all our assumptions are sharp. As a consequence of our results, we deduce the C 1 , α regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem. [ABSTRACT FROM AUTHOR]

In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form Lu = 0 in Ω, u = g in RN \ Ω, in non-smooth domains Ω. When g is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right-hand side for which the boundary regularity is well understood. Here, we study the case in which g ∈ C0,α, and establish the optimal Hölder regularity of u up to the boundary. Our results extend previous results of Grubb for C∞ domains Ω. [ABSTRACT FROM AUTHOR]

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, min{(-Δ)su, u - φ} = 0 in Rn, for general obstacles φ. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in those of Garofalo-Petrosyan to all s ∈ (0, 1). [ABSTRACT FROM AUTHOR]

We study the regularity of the free boundary in the fully nonlinear thin obstacle problem. Our main result establishes that the free boundary is C 1 near any regular point. This extends to the fully nonlinear setting the celebrated result of Athanasopoulos–Caffarelli–Salsa [1] . The proofs we present here are completely independent from those in [1] , and do not rely on any monotonicity formula. Furthermore, an interesting and novel feature of our proofs is that we establish the regularity of the free boundary without classifying blow-ups, a priori they could be non-homogeneous and/or non-unique. We do not classify blow-ups but only prove that they are 1D on { x n = 0 } . [ABSTRACT FROM AUTHOR]

We find and prove new Pohozaev identities and integration by parts type formulas for anisotropic integrodifferential operators of order 2s, withs∈(0,1). These identities involve local boundary terms, in which the quantityplays the role that∂u∕∂νplays in the second-order case. Here,uis any solution toLu = f(x,u) in Ω, withu = 0 in ℝn∖Ω, anddis the distance to∂Ω. [ABSTRACT FROM AUTHOR]

We study the obstacle problem for integro-differential operators of order 2 s, with $$s\in (0,1)$$ . Our main result establish that the free boundary is $$C^{1,\gamma }$$ and $$u\in C^{1,s}$$ near all regular points. Namely, we prove the following dichotomy at all free boundary points $$x_0\in \partial \{u=\varphi \}$$ : where d is the distance to the contact set $$\{u=\varphi \}$$ . Moreover, we show that the set of free boundary points $$x_0$$ satisfying (i) is open, and that the free boundary is $$C^{1,\gamma }$$ and $$u\in C^{1,s}$$ near those points. These results were only known for the fractional Laplacian [2], and are completely new for more general integro-differential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicity-type formula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integro-differential operators: we establish similar regularity results for obstacle problems with convex operators. [ABSTRACT FROM AUTHOR]

We establish sharp interior and boundary regularity estimates for solutions to ∂ t u − L u = f ( t , x ) in I × Ω , with I ⊂ R and Ω ⊂ R n . The operators L we consider are infinitesimal generators of stable Lévy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that u is C 2 s + α in x and C 1 + α 2 s in t , whenever f is C α in x and C α 2 s in t . In the case f ∈ L ∞ , we prove that u is C 2 s − ϵ in x and C 1 − ϵ in t , for any ϵ > 0 . On the other hand, we study the boundary regularity of solutions in C 1 , 1 domains. We prove that for solutions u to the Dirichlet problem the quotient u / d s is Hölder continuous in space and time up to the boundary ∂Ω, where d is the distance to ∂Ω. This is new even when L is the fractional Laplacian. [ABSTRACT FROM AUTHOR]

We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside Ω, decreasing energy, and convergence to a constant as t→∞. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition. For concreteness, one may think that our nonlocal analogue of the classical Neumann condition ∂νu = 0 on ∂Ω consists in the nonlocal prescription ∫Ωu(x) − u(y)/∣x − y∣n+2s dy = 0 for x ∈ Rn \ Ω. We made an effort to keep all the arguments at the simplest possible technical level, in order to clarify the connections between the different scientific fields that are naturally involved in the problem, and make the paper accessible also to a wide, non-specialistic public (for this scope, we also tried to use and compare different concepts and notations in a somehow more unified way). [ABSTRACT FROM AUTHOR]

We consider entire solutions to L u = f ( u ) in R 2 , where L is a nonlocal operator with translation invariant, even and compactly supported kernel K . Under different assumptions on the operator L , we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions. [ABSTRACT FROM AUTHOR]

Given an arbitrary convex cone of ℝn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in ℝn satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativity of a Bakry--Émery Ricci tensor. Even though our weights are nonradial, balls are still minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. [ABSTRACT FROM AUTHOR]

We study the regularity of minimizers of the functional E (u) : = [ u ] H s (Ω) 2 + ∫ Ω f u. This corresponds to understanding solutions for the regional fractional Laplacian in Ω ⊂ R N. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in H s (Ω) (i.e., Neumann problem), or in the case of Dirichlet condition u ∈ H 0 s (Ω) when s > 1 2. Our main result establishes the existence of a constant α s ∈ (0 , 1 − s) satisfying 2 s + α s > 1 such that for all α ∈ (0 , α s) the solution u ∈ C 2 s + α (Ω ‾) in the Neumann case, and u / δ 2 s − 1 ∈ C 1 + α (Ω ‾) in the Dirichlet case. Here, δ is the distance to ∂Ω. We also show the optimality of our result: these estimates fail for α > α s , even when f and ∂Ω are C ∞. [ABSTRACT FROM AUTHOR]

We establish sharp regularity estimates for solutions to L u = f in Ω ⊂ R n , L being the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on S n − 1 , called the spectral measure. First, we study the interior regularity of solutions to L u = f in B 1 . We prove that if f is C α then u belong to C α + 2 s whenever α + 2 s is not an integer. In case f ∈ L ∞ , we show that the solution u is C 2 s when s ≠ 1 / 2 , and C 2 s − ϵ for all ϵ > 0 when s = 1 / 2 . Then, we study the boundary regularity of solutions to L u = f in Ω, u = 0 in R n ∖ Ω , in C 1 , 1 domains Ω. We show that solutions u satisfy u / d s ∈ C s − ϵ ( Ω ‾ ) for all ϵ > 0 , where d is the distance to ∂Ω. Finally, we show that our results are sharp by constructing two counterexamples. [ABSTRACT FROM AUTHOR]

We study the interior regularity of solutions to the Dirichlet problem L u = g in Ω, u = 0 in R n ∖ Ω , for anisotropic operators of fractional type L u ( x ) = ∫ 0 + ∞ d ρ ∫ S n − 1 d a ( ω ) 2 u ( x ) − u ( x + ρ ω ) − u ( x − ρ ω ) ρ 1 + 2 s . Here, a is any measure on S n − 1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a ∈ C ∞ ( S n − 1 ) and g is C ∞ ( Ω ) , solutions are known to be C ∞ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is L ∞ ( S n − 1 ) , solutions are only known to be C 3 s inside Ω. We prove here that, for general measures a , solutions are C 1 + 3 s − ϵ inside Ω for all ϵ > 0 whenever Ω is convex. When a ∈ L ∞ ( S n − 1 ) , we show that the same holds in all C 1 , 1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C 3 s + ϵ for any ϵ > 0 – even if g and Ω are C ∞ . [ABSTRACT FROM AUTHOR]

We study solutions to L u = f in Ω ⊂ R n , being L the generator of any, possibly non-symmetric , stable Lévy process. On the one hand, we study the regularity of solutions to L u = f in Ω, u = 0 in Ω c , in C 1 , α domains Ω. We show that solutions u satisfy u / d γ ∈ C ε ∘ (Ω ‾) , where d is the distance to ∂Ω, and γ = γ (L , ν) is an explicit exponent that depends on the Fourier symbol of operator L and on the unit normal ν to the boundary ∂Ω. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded C 1 , α domains. We do it via a new efficient approximation argument, which exploits the Hölder regularity of u / d γ. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian. [ABSTRACT FROM AUTHOR]

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernelsKsatisfying thatK(y)|y|n+σis nondecreasing along rays from the origin, for some σ ∈ (0, 2) in caseaij ≡ 0 and for σ = 2 in case that (aij) is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems forLin star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation tonand σ). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian (− Δ)s(heres > 1) or the fractionalp-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin. [ABSTRACT FROM AUTHOR]

Abstract: We study the problem in a bounded domain , where λ is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions for all whenever Ω is, for every , convex in the -direction and symmetric with respect to . The same holds if and , or if and . These results are new even in the unit ball . [Copyright &y& Elsevier]

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $${(-\Delta)^s u =f(u)}$$ in $${\Omega, u\equiv0}$$ in $${{\mathbb R}^n\backslash\Omega}$$ . Here, $${s\in(0,1)}$$ , (−Δ) is the fractional Laplacian in $${\mathbb{R}^n}$$ , and Ω is a bounded C domain. To establish the identity we use, among other things, that if u is a bounded solution then $${u/\delta^s|_{\Omega}}$$ is C up to the boundary ∂Ω, where δ( x) = dist( x, ∂Ω). In the fractional Pohozaev identity, the function $${u/\delta^s|_{\partial\Omega}}$$ plays the role that ∂u/ ∂ν plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ∂Ω) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities. [ABSTRACT FROM AUTHOR]

We study the extremal solution for the problem $$(-\Delta )^s u=\lambda f(u)$$ in $$\Omega $$ , $$u\equiv 0$$ in $$\mathbb R ^n\setminus \Omega $$ , where $$\lambda >0$$ is a parameter and $$s\in (0,1)$$ . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions $$n<4s$$ . We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever $$n<10s$$ . In the limit $$s\uparrow 1$$ , $$n<10$$ is optimal. In addition, we show that the extremal solution is $$H^s(\mathbb R ^n)$$ in any dimension whenever the domain is convex. To obtain some of these results we need $$L^q$$ estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with $$L^p$$ data. We prove optimal $$L^q$$ and $$C^\beta $$ estimates, depending on the value of $$p$$ . These estimates follow from classical embedding results for the Riesz potential in $$\mathbb R ^n$$ . Finally, to prove the $$H^s$$ regularity of the extremal solution we need an $$L^\infty $$ estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels. [ABSTRACT FROM AUTHOR]

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: We consider the monomial weight in , where is a real number for each , and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by , and they contain the best or critical exponent (which depends on ). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space (with no weight) when written for axially symmetric functions and domains in . [Copyright &y& Elsevier]

We consider the class of semi-stable positive solutions to semilinear equations − Δu = f(u) in a bounded domain Ω ⊂ ℝ n of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n − m variables. We assume 2 ≤ m ≤ n − 2. When the domain is convex, we establish a priori L p and bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead to the boundedness of the extremal solution of − Δu = λf(u) in every convex domain of double revolution when n ≤ 7. The boundedness of extremal solutions is known when n ≤ 4 for any domain Ω, and in dimensions 5 ≤ n ≤ 9 in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities f in dimensions 5 ≤ n ≤ 9. [ABSTRACT FROM PUBLISHER]

Abstract: In this Note we present the Pohozaev identity for the fractional Laplacian. As a consequence of this identity, we prove the nonexistence of nontrivial bounded solutions to semilinear problems with supercritical nonlinearities in star-shaped domains. [Copyright &y& Elsevier]

Abstract: The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation , and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied. [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]

We investigate the regularity of the free boundary for the Signorini problem in Rn+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^{n+1}$$\end{document}. It is known that regular points are (n-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-1)$$\end{document}-dimensional and C∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^\infty $$\end{document}. However, even for C∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^\infty $$\end{document} obstacles φ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi $$\end{document}, the set of non-regular (or degenerate) points could be very large—e.g. with infinite Hn-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {H}}^{n-1}$$\end{document} measure. The only two assumptions under which a nice structure result for degenerate points has been established are when φ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi $$\end{document} is analytic, and when Δφ<0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta \varphi < 0$$\end{document}. However, even in these cases, the set of degenerate points is in general (n-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-1)$$\end{document}-dimensional—as large as the set of regular points. In this work, we show for the first time that, “usually”, the set of degenerate points is small. Namely, we prove that, given any C∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^\infty $$\end{document} obstacle, for almost every solution the non-regular part of the free boundary is at most (n-2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-2)$$\end{document}-dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian (-Δ)s\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(-\Delta )^s$$\end{document}, and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is (n-1-α∘)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-1-\alpha _\circ )$$\end{document}-dimensional for almost all times t, for some α∘>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha _\circ > 0$$\end{document}. Finally, we construct some new examples of free boundaries with degenerate points. [ABSTRACT FROM AUTHOR]

Abstract: In this Note we present the solution of some isoperimetric problems in open convex cones of in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. When the weight is constant, the result was established by Lions and Pacella in 1990. [Copyright &y& Elsevier]

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are C 1 , α , then they are C ∞. This completes the study of regular points, initiated in [5]. In order to achieve this, we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in C k , α domains. These new estimates are the core of our paper, and extend previously known results by Grubb (for k = ∞) and by the second author and Serra (for k = 1). [ABSTRACT FROM AUTHOR]

We establish sharp higher-order Hölder regularity estimates up to the boundary for solutions to equations of the form ∂tu-Lu=f(t,x) in I×Ω where I⊂R, Ω⊂Rn and f is Hölder continuous. The nonlocal operators L that we consider are those arising in stochastic processes with jumps, such as the fractional Laplacian (-Δ)s, s∈(0,1). Our main result establishes that, if f is Cγ is space and Cγ/2s in time, and Ω is a C2,γ domain, then u/ds is Cs+γ up to the boundary in space and u is C1+γ/2s up the boundary in time, where d is the distance to ∂Ω. This is the first higher order boundary regularity estimate for nonlocal parabolic equations, and is new even for the fractional Laplacian in C∞ domains. [ABSTRACT FROM AUTHOR]

We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle φ satisfies Δφ ≤ 0 near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a (n-1)-dimensional C1,α manifold by the results obtained by Caffarelli, Salsa, and Silvestre (Invent. Math. 2008), and a set of singular points, which we prove to be contained in a union of k-dimensional C1-submanifold, k = 0, . . . , n-1. Such a complete result on the structure of the free boundary, proved by L. A. Caffarelli (Acta Math. 1977 and J. Fourier Anal. Appl. 1998), was known only in the case of the classical Laplacian, and it is new even for the Signorini problem (which corresponds to the particular case of the ½-fractional Laplacian). A key ingredient behind our results is the validity of a new non-degeneracy condition supBr(x0)(u-φ) ≥ cr2, valid at all free boundary points x0. [ABSTRACT FROM AUTHOR]

We study the regularity of the free boundary in the obstacle for the p -Laplacian, min ⁡ { − Δ p u , u − φ } = 0 in Ω ⊂ R n . Here, Δ p u = div ( | ∇ u | p − 2 ∇ u ) , and p ∈ ( 1 , 2 ) ∪ ( 2 , ∞ ) . Near those free boundary points where ∇ φ ≠ 0 , the operator Δ p is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when ∇ φ = 0 then Δ p is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where ∇ φ = 0 . On the one hand, for every p ≠ 2 we construct explicit global 2-homogeneous solutions to the p -Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not C 1 at points where ∇ φ = 0 . On the other hand, under the “concavity” assumption | ∇ φ | 2 − p Δ p φ < 0 , we show the free boundary is countably ( n − 1 ) -rectifiable and we prove a nondegeneracy property for u at all free boundary points. [ABSTRACT FROM AUTHOR]

We study the positivity and regularity of solutions to the fractional porous medium equations [ABSTRACT FROM AUTHOR]

Abstract: The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the asset prices are driven by pure‐jump Lévy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when s > 1 2, the free boundary is a C1,α graph in x and t near any regular free boundary point ( x 0 , t 0 ) ∈ ∂ { u > φ }. Furthermore, we also prove that solutions u are C1 + s in x and t near such points, with a precise expansion of the form u ( x , t ) − φ ( x ) = c 0 ( ( x − x 0 ) ⋅ e + κ ( t − t 0 ) ) + 1 + s + o ( | x − x 0 | 1 + s + α + | t − t 0 | 1 + s + α ) , with c 0 > 0 , e ∈ S n − 1, and a > 0. © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]