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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]

In this paper, we study a nonlinear boundary diffusion equation of porous medium type arising from a boundary control problem. We give a complete and sharp characterization of the asymptotic behavior of its solutions, and prove the stability of its separable solutions. [ABSTRACT FROM AUTHOR]

We show a convergence result of the fractional Laplacian for sequences of nonnegative functions without uniform boundedness near infinity. As an application, we construct a sequence of solutions to the fractional Nirenberg problem that blows up in the region where the prescribed functions are negative. This is a different phenomenon from the classical Nirenberg problem. [ABSTRACT FROM AUTHOR]

We study a Sobolev critical fast diffusion equation in bounded domains with the Brézis-Nirenberg effect. We obtain extinction profiles of its positive solutions, and show that the convergence rates of the relative error in regular norms are at least polynomial. Exponential decay rates are proved for generic domains. Our proof makes use of its regularity estimates, a curvature type evolution equation, as well as blow up analysis. Results for Sobolev subcritical fast diffusion equations are also obtained. [ABSTRACT FROM AUTHOR]

Inter‐tissue communication is instrumental to coordinating the whole‐body level behaviour for complex multicellular organisms. However, little is known about the regulation of inter‐tissue information exchange. Here we carried out genetic screens for root‐to‐shoot mobile silencing in Arabidopsis plants with a compromised small RNA‐mediated gene silencing movement rate and identified radical‐induced cell death 1 (RCD1) as a critical regulator of root–shoot communication. RCD1 belongs to a family of poly (ADP‐ribose) polymerase proteins, which are highly conserved across land plants. We found that RCD1 coordinates symplastic and apoplastic movement by modulating the sterol level of lipid rafts. The higher superoxide production in rcd1‐knockout plants resulted in lower plasmodesmata (PD) frequency and altered PD structure in the symplasm of the hypocotyl cortex. Furthermore, the mutants showed increased lateral area of tracheary pits, which reduced axial movement. Our study highlights a novel mechanism through which root‐to‐shoot long‐distance signalling can be modulated both symplastically and apoplastically. [ABSTRACT FROM AUTHOR]

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ⁎ inf type Harnack inequality of Schoen for integral equations. [ABSTRACT FROM AUTHOR]

Let (M , g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂ M , admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean ball, and consequently is achieved, if either (i) 9 ≤ n ≤ 11 and ∂ M has a nonumbilic point; or (ii) 7 ≤ n ≤ 9 , ∂ M is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work [12] by the second named author and Xiong. [ABSTRACT FROM AUTHOR]

We prove interior Hölder estimates for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation ut = |∇u|κ ⁢div(|∇u|p−2∇u), where p ∈ (1, ∞) and κ ∈ (1− p, ∞). This includes the from L∞ to C1, α regularity for parabolic p-Laplacian equations in both divergence form with κ = 0, and non-divergence form with κ = 2 − p. [ABSTRACT FROM AUTHOR]

We study a family of 3D models for the incompressible axisymmetric Euler and Navier-Stokes equations. The models are derived by changing the strength of the convection terms in the equations written using a set of transformed variables. The models share several regularity results with the Euler and Navier-Stokes equations, including an energy identity, the conservation of a modified circulation quantity, the BKM criterion and the Prodi-Serrin criterion. The inviscid models with weak convection are numerically observed to develop stable self-similar singularity with the singular region traveling along the symmetric axis, and such singularity scenario does not seem to persist for strong convection. [ABSTRACT FROM AUTHOR]

Let (M, g) be a smooth compact Riemannian manifold of dimension n with smooth boundary . Suppose that (M, g) admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) and has a nonumbilic point; or (ii) is umbilic and the Weyl tensor does not vanish at some boundary point. A crucial ingredient in the proof is the expansion of solutions to the conformal Laplacian equation with blowing up Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

We derive local estimates of positive solutions to the conformal Q -curvature equation (− Δ) m u = K (x) u n + 2 m n − 2 m in Ω ﹨ Λ near their singular set Λ, where Ω ⊂ R n is an open set, K (x) is a positive continuous function on Ω, Λ is a closed subset of R n , 2 ≤ m < n / 2 and m is an integer. Under certain flatness conditions at critical points of K on Λ, we prove that u (x) ≤ C [ dist (x , Λ) ] − (n − 2 m) / 2 when the upper Minkowski dimension of Λ is less than (n − 2 m) / 2. [ABSTRACT FROM AUTHOR]

We prove interior Hölder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p -Laplacian equation u t = | ∇ u | 2 − p div ( | ∇ u | p − 2 ∇ u ) , where 1 < p < ∞ . This equation arises from tug-of-war -like stochastic games with noise. It can also be considered as the parabolic p -Laplacian equation in non-divergence form. [ABSTRACT FROM AUTHOR]

In this paper, we establish pointwise Schauder estimates for solutions of nonlocal fully nonlinear elliptic equations by perturbative arguments. A key ingredient is a recursive Evans–Krylov theorem for nonlocal fully nonlinear translation invariant equations. [ABSTRACT FROM AUTHOR]

In this paper, we study existence, regularity, classification, and asymptotic behaviors of solutions of some Monge–Ampère equations with isolated and line singularities. We classify all solutions of det ⁡ ∇ 2 u = 1 in R n with one puncture point. This can be applied to characterize ellipsoids, in the same spirit of Serrin's overdetermined problem for the Laplace operator. In the case of having k non-removable singular points for k > 1 , modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold. We also establish existence of global solutions with general singular sets, regularity properties, and optimal estimates of the second order derivatives of generalized solutions near the singularity consisting of a point or a straight line. The geometric motivation comes from singular semi-flat Calabi–Yau metrics. [ABSTRACT FROM AUTHOR]

We study extinction profiles of solutions to fast diffusion equations with some initial data in the Marcinkiewicz space. The extinction profiles will be the singular solutions of their stationary equations. [ABSTRACT FROM AUTHOR]

We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres , it converges to the standard sphere up to a Möbius diffeomorphism. These arguments can be applied to obtain extinction profiles of solutions of some fractional porous medium equations. In the end, we use this fractional fast diffusion equation, together with its extinction profile and some estimates of its extinction time, to improve a Sobolev inequality via a quantitative estimate of the remainder term. [ABSTRACT FROM AUTHOR]

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $${(-\Delta )^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}}$$ with an isolated singularity, where $${\sigma\in (0,1)}$$ . We prove that all the solutions are asymptotically radially symmetric. When σ = 1, these have been proved by Caffarelli et al. (Comm Pure Appl Math 42:271-297, ). [ABSTRACT FROM AUTHOR]

In this paper, we give a new proof of a celebrated theorem of Jörgens which states that every classical convex solution ofhas to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions of a class of degenerate Monge-Ampère equations. We prove that every convex generalized (or Alexandrov) solution ofwhere α > −1, has to befor some constantsa > 0,band a linear function ℓ(x1,x2). This work is motivated by the Weyl problem with nonnegative Gauss curvature. [ABSTRACT FROM AUTHOR]

Abstract: We prove some symmetry property for equations with Hardy terms in cones, without any assumption at infinity. We also show symmetry property and nonexistence of entire solutions of some elliptic systems with Hardy weights. [Copyright &y& Elsevier]

Abstract: We construct a function which is a solution to in the sense of distributions, where A is continuous and for . We also give a function such that for every , u satisfies with A continuous but . This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338). To cite this article: T. Jin et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]