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We show that all nonnegative solutions of the critical semilinear elliptic equation involving the regional fractional Laplacian are locally universally bounded. This strongly contrasts with the standard fractional Laplacian case. Secondly, we consider the fractional critical elliptic equations with nonnegative potentials. We prove compactness of solutions provided the potentials only have non-degenerate zeros. Corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow-up points of solutions. [ABSTRACT FROM AUTHOR]

We prove generalized Aleksandrov–Bakelman–Pucci maximum principles for elliptic and parabolic integro-PDEs of Hamilton–Jacobi–Bellman–Isaacs types, whose PDE parts are either uniformly elliptic or uniformly parabolic. The proofs of these results are based on the classical Aleksandrov–Bakelman–Pucci maximum principles for the elliptic and parabolic PDEs and an iteration procedure using solutions of Pucci extremal equations. We also provide proofs of nonlocal versions of the classical Aleksandrov–Bakelman–Pucci maximum principles for elliptic and parabolic integro-PDEs. [ABSTRACT FROM AUTHOR]

We consider the nonlocal Cahn–Hilliard equation with singular potential and constant mobility. Well-posedness and regularity of weak solutions are studied. Then we establish the validity of the strict separation property in dimension two. Further regularity results as well as the existence of regular finite dimensional attractors and the convergence of a weak solution to a single equilibrium are also provided. Finally, regularity results and the strict separation property are also proven for the two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes system with singular potential. [ABSTRACT FROM AUTHOR]

Entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics. Through a modified Glimm scheme, an approximate solution sequence is constructed, and then a priori estimates are provided with the methods of approximate characteristics and approximate conservation laws, which gives not only the existence and uniqueness but also the uniform total variation bounds for the entropy solutions. [ABSTRACT FROM AUTHOR]

We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p -Laplace, ∞-Laplace, mean curvature of graph, and even strongly degenerate operators, in addition to some nonlocal quasilinear operators appearing in the existing literature. Our main results are comparison, uniqueness, and existence results for viscosity solutions of linear and fully nonlinear equations involving these operators. Because of the structure of our operators, especially the existence proof is highly non-trivial and non-standard. We also identify the conditions under which the nonlocal operators converge to local quasilinear operators, and show that the solutions of the corresponding nonlocal equations converge to the solutions of the local limit equations. Finally, we give a (formal) stochastic representation formula for the solutions and provide many examples. [ABSTRACT FROM AUTHOR]

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation ∂ t u − L μ [ φ ( u ) ] = 0 . Here L μ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function φ : R → R is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain stability, L 1 -contraction, and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations. [ABSTRACT FROM AUTHOR]

We prove that for p ≥ 2 , solutions of equations modeled by the fractional p -Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in W loc 1 , p and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation s reaches 1. [ABSTRACT FROM AUTHOR]

In this paper a novel variational method is developed to construct four-end solutions in R 2 for the Allen–Cahn equation. Four-end solutions have been constructed by Del Pino, Kowalczyk, Pacard and Wei when the angle θ of the ends is close to π / 2 or 0 using the Lyapunov–Schmid reduction method, and later by Kowalczyk, Liu and Pacard using a continuation method for general θ ∈ ( 0 , π / 2 ) . By a special mountain pass argument in a restricted space, namely a set of monotone paths of monotone functions, a family of solutions in bounded domain is constructed with very good control of their nodal sets, which then leads to a four-end solution after sending the domains to R 2 . In this way, not only a family of four end solutions is constructed for any angle θ ∈ ( 0 , π / 2 ) . The Morse index of such a four-end solution is also shown to be one. [ABSTRACT FROM AUTHOR]

This paper is concerned with interior regularity of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations with Dini continuous terms. We obtain C σ regularity estimates for the nonlocal equations by perturbative methods and a version of a recursive Evans–Krylov theorem. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems { ( − Δ ) s u − λ u | x | 2 s = f ( x , u ) in Ω , u = 0 in R N ∖ Ω , u > 0 in Ω , where ( − Δ ) s , s ∈ ( 0 , 1 ) , is the fractional Laplacian operator, Ω ⊂ R N is a bounded domain with Lipschitz boundary such that 0 ∈ Ω and N > 2 s . We will mainly consider the solvability in two cases: (1) The linear problem, that is, f ( x , t ) = f ( x ) , where according to the summability of the datum f and the parameter λ we give the summability of the solution u . (2) The problem with a nonlinear term f ( x , t ) = h ( x ) t σ for t > 0 . In this case, existence and regularity will depend on the value of σ and on the summability of h . Looking for optimal results we will need a weak Harnack inequality for elliptic operators with singular coefficients that seems to be new. [ABSTRACT FROM AUTHOR]

We study 2D Euler equations on a rotating surface, subject to the effect of the Coriolis force, with an emphasis on surfaces of revolution. We bring in conservation laws that yield long time estimates on solutions to the Euler equation, and examine ways in which the solutions behave like zonal fields, building on previous works that have examined how such 2D Euler equations can account for the observed band structure of rapidly rotating planets. Specific results include both an analysis of time averages of solutions and a study of stability of stationary zonal fields. The latter study includes both analytical and numerical work. [ABSTRACT FROM AUTHOR]

In this paper we study the effect of rare mutations, driven by a marked point process, on the evolutionary behavior of a population. We derive a Kolmogorov equation describing the expected values of the different frequencies and prove some rigorous analytical results about their behavior. Finally, in a simple case of two different quasispecies, we are able to prove that the rarity of mutations increases the survival opportunity of the low fitness species. [ABSTRACT FROM AUTHOR]