This is a continuation of our earlier work [14] on the Monge-Ampère obstacle problem D^2 v = v^q χ_{v>0}, v ≥ 0 convex with q ∈ [0,n), where we studied the regularity of the strictly convex part of the free boundary. In this work, we examine the non-strictly convex part of the free boundary and establish optimal dimension bounds for its flat portion. Additionally, we investigate the strong maximum principle and a stability property for this Monge-Ampère obstacle problem.

We establish Schauder estimates for linearized very fast diffusion equations with Dirichlet boundary conditions in bounded smooth domains. These estimates provide an important ingredient for deriving optimal boundary regularity and long-time dynamics of solutions to very fast diffusion equations.

We prove the optimal global regularity of admissible solutions to a transformed very fast diffusion equation in the range -1<p<0, posed on smooth bounded domains with zero Dirichlet boundary data and initial data comparable to the distance function. More precisely, we establish existence and uniqueness and show that solutions belong to C^1,p+1() in space for every positive time and are C^∞ in time uniformly up to the boundary. Moreover, all their time derivatives belong to C^1,p+1(), and the exponent p+1 is optimal. These regularity estimates further yield fine long-time asymptotics toward the friendly giant solution, including a first-order expansion in the C^1,p+1() topology and an improved convergence rate for the relative error in C^p+1().

This paper continues our work Jin et al. (Extremal Alexandrov estimates: singularities, obstacles, and stability. arXiv:2602.06468 . 2026) on sharp Alexandrov estimates. We obtain a sharp global uniform distance estimate from a convex function to the class of unimodular convex quadratic polynomials in terms of the total variation of its Monge–Ampère defect measure relative to Lebesgue measure. The estimate has an explicit optimal constant, and the inequality is strict in the regime of positive finite defect mass. In this regime we further prove asymptotic rigidity at infinity: every such convex function admits a unique quadratic asymptote with an explicit convergence rate, and satisfies a sharp affine invariant global Alexandrov estimate with equality if and only if the function solves the isolated singularity problem or the hyperplane obstacle problem. Standard subsolution methods are not well suited to this measure-theoretic setting and typically do not yield sharp constants, while the sharp Alexandrov estimates developed in our earlier work (Jin et al. 2026) play a central role here. As an application, for entire solutions of Monge–Ampère equations with multiple (possibly infinitely many) isolated singularities, we give an explicit quantitative mass-separation condition ensuring strict convexity and hence smoothness away from the set of the isolated singularities.

The classical Alexandrov estimate controls the oscillation of a convex function by the mass of its associated Monge-Ampère measure and yields, for two convex functions of n variables with the same boundary values, a sup-norm bound with exponent 1/n in the measure discrepancy. We show that this exponent is not optimal in the small-discrepancy regime once one of the functions is non-degenerate in the sense of having Monge-Ampère density bounded above and below by two positive constants. We prove sharp quantitative estimates comparing two convex functions by the total variation of the difference of their Monge-Ampère measures: in dimensions n≥ 3 the optimal dependence is quadratic in the natural mass scale, while in dimension n=2 the optimal dependence contains a logarithmic correction. These rates are shown to be optimal for all small discrepancies. A key structural ingredient is a characterization of extremizers. We identify the pointwise minimizers and maximizers in the admissible class and prove that they are realized, respectively, by solutions to Monge-Ampère equations with an isolated singularity and by solutions to Monge-Ampère equations with a linear obstacle. This extremal description reduces the sharp estimates to a precise asymptotic analysis of these two model configurations. Assuming further that the domain and the non-degenerate reference function are C^2,α and uniformly convex, we obtain sharp pointwise two-sided asymptotics at interior points with explicit leading constants. Finally, in dimensions n≥ 3 we establish a stability phenomenon: if the pointwise estimate is nearly saturated, then the measure discrepancy must concentrate near the point at the natural scale, quantifying rigidity of almost-extremal configurations.

In this paper, we investigate the extinction behavior of nonnegative solutions to the Sobolev critical fast diffusion equation in bounded smooth domains with the Dirichlet zero boundary condition. Under the two-bubble energy threshold assumption on the initial data, we prove the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady state or a blowing-up bubble.

This paper continues our work [19] on sharp Alexandrov estimates. We obtain a sharp global uniform distance estimate from a convex function to the class of unimodular convex quadratic polynomials in terms of the total variation of its Monge-Ampère defect measure relative to Lebesgue measure. The estimate has an explicit optimal constant, and the inequality is strict in the regime of positive finite defect mass. In this regime we further prove asymptotic rigidity at infinity: every such convex function admits a unique quadratic asymptote with an explicit convergence rate, and satisfies a sharp affine invariant global Alexandrov estimate with equality if and only if the function solves the isolated singularity problem or the hyperplane obstacle problem. Standard subsolution methods are not well suited to this measure-theoretic setting and typically do not yield sharp constants, while the sharp Alexandrov estimates developed in our earlier work [19] play a central role here. As an application, for entire solutions of Monge-Ampère equations with multiple (possibly infinitely many) isolated singularities, we give an explicit quantitative mass-separation condition ensuring strict convexity and hence smoothness away from the set of the isolated singularities.

We establish Łojasiewicz–Simon type gradient inequalities for the Yamabe functional on bounded smooth domains near configurations consisting of finitely many concentrating bubbles, with or without a regular component. A weighted decomposition separates the finite-dimensional spectral and bubble parameters from an infinite-dimensional coercive remainder. We obtain quantitative estimates for the parameters, the remainder, and the corresponding energy gaps, in both the regular-plus-bubbling and the pure-bubbling regimes. These inequalities quantify the deviation from such surfaces and are essential for studying the dynamics of related parabolic flows.

We study convex solutions to the Monge-Ampère obstacle problem det D^2 v=g v^qχ_{v>0}, v ≥ 0, where q ∈ [0,n) is a constant and g is a bounded positive function. This problem emerges from the L_p Minkowski problem. We establish C^1, α regularity for the strictly convex part of the free boundary ∂{v=0}. Furthermore, when g ∈ C^α, we prove a Schauder-type estimate. As a consequence, when g≡ 1, we obtain a Liouville theorem for entire solutions with unbounded coincidence sets {v=0}. Combined with existing results, this provides a complete classification of entire solutions for the case q=0.

In this article,we survey the classic achievements and the latest research progress on the isolated singularity problem of the Yamabe equation.We present an alternative proof for the classical theorems obtained by Caffarelli et al.(1989),and list several unresolved problems.

Let μ≢0 be a nonnegative locally finite periodic Borel measure on ℝ^n. We show that any convex solution to the Monge-Ampère equation D^2 u = μ in ℝ^n admits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends, in full generality, the earlier works for the case μ=f(x) d x: when log f ∈ C^α, it was established by Caffarelli and Li; and when log f is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Ampère equations with nonnegative periodic measures.

In this paper, we study the sharp constants in fractional Sobolev inequalities associated with the regional fractional Laplacian in domains.

We prove global Hölder gradient estimates for bounded positive weak solutions of fast diffusion equations in smooth bounded domains with the homogeneous Dirichlet boundary condition, which then lead us to establish their optimal global regularity. This solves a problem raised by Berryman and Holland in 1980.

We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirich let boundary condition after certain waiting time T*. More precisely, we show that solutions are C-2,C-alpha((Omega) over bar) in space, with alpha = 1/m, and C-infinity in time (uniformly in x is an element of (Omega) over bar), fort > 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-gamma)), and we prove that the convergence holds in the C-1,C-alpha((Omega) over bar) topology.

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.

In this paper, we systematically study weak solutions of a linear singular or degenerate parabolic equation in a mixed divergence form and nondivergence form, which arises from the linearized fast diffusion equation and the linearized porous medium equation with the homogeneous Dirichlet boundary condition. We prove the Hölder regularity of their weak solutions.

In this paper, we study existence of solutions to a conformally invariant integral equation involving Poisson-type kernels. Such integral equation has a stronger non-local feature and is not the dual of any PDE. We obtain the existence of solutions in the antipodal symmetry class.

We derive local estimates of positive solutions to the conformal Q -curvature equation n+2m (-Delta)mu = K(x)un-2m in 12\? near their singular set ?, where 12 subset of Rn is an open set, K(x) is a positive continuous function on 12, ? is a closed subset of Rn, 2 < m < n/2 and m is an integer. Under certain flatness conditions at critical points of K on ?, we prove that u(x) < C[dist(x, ?)]-(n-2m)/2 when the upper Minkowski dimension of ? is less than (n - 2m)/2.(c) 2023 Elsevier Inc. All rights reserved.

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.

We study a Sobolev critical fast diffusion equation in bounded domains with the Brezis-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.