Abstract In this paper, we investigate the stability of a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. We employ variational methods to prove the stability in the functional inequality setting. Moreover, overcoming the absence of an explicit extremal function, we employ the asymptotic behavior of extremal functions to calculate some crucial estimates. By utilizing the finite‐dimensional reduction method, we establish a sharp stability result in the critical points setting.

A simple proof of reverse Sobolev inequalities on the sphere and Sobolev trace inequalities on the unit ball

Abstract This work studies time averages of an observable $h(t,X_t)$ , where $X_t$ is the solution to a time-inhomogeneous stochastic differential equation (SDE) driven by drift, b ( t , x ), and diffusion, $\sigma(t{,}{\kern.5pt}x)$ , that change sufficiently slowly in time. In this quasistatic regime we derive an approximation to the time average that is computable from properties of the time-homogeneous SDEs driven by $b(t,\cdot)$ and $\sigma(t,\cdot)$ with fixed t ; specifically, we utilize $\log$ -Sobolev inequalities for the instantaneous invariant distribution and generator for each t . We obtain explicit non-asymptotic error bounds on this quasistatic approximation, both in the form of concentration inequalities and bounds on the expected value. The error bounds demonstrate a competition between the speed of convergence to the instantaneous invariant distributions and their rate of change, matching the intuition that underlies the quasistatic approximation.

Sharp radial Sobolev inequalities with inverse square potentials and applications to nonlinear Schrödinger equations

Abstract Quasi-isometries are a versatile type of maps that preserve the large-scale geometry of spaces, while introducing significant local distortions. Following Kanai’s work, which established the invariance of various analytic and geometric properties under quasi-isometries, this paper generalizes isoperimetric and Sobolev inequalities for exponents less than the manifold’s dimension, proving both that they are equivalent and preserved by quasi-isometries.

This paper investigates a non-local system involving the fractional p-Laplacian with a critical Sobolev-Hardy exponent. The system is characterized by coupled nonlinear equations with fractional diffusion terms, Hardy-type singular potentials, and critical nonlinearities. Specifically, we consider 0\\ {\\rm in }\\ \\Omega, \\quad u = v = 0 \\ {\\rm in } \\ \\mathbb{R}^n \\setminus\\Omega. \\end{array} \\right. \\] ]]> ( S ) { ( − Δ p ) s u − μ | u | p − 2 u | x | sp = λa ( x ) | u | q − 2 u | x | θ + c ( x ) pα α + β | u | α − 2 u | v | β | x | t in Ω , ( − Δ p ) s v − μ | v | p − 2 v | x | sp = γb ( x ) | v | q − 2 v | x | θ + c ( x ) pβ α + β | u | α | v | β − 2 v | x | t in Ω , u , v > 0 in Ω , u = v = 0 in R n ∖ Ω . where Ω ⊂ R N is a smooth bounded domain containing the origin ( 0 ∈ Ω ), N>sp, s ∈ ( 0 , 1 ) , and θ , t ∈ [ 0 , sp ) . The parameters satisfy 1<q<p, 1 $ ]]> α , β > 1 with α + β = p s ∗ ( t ) , where p s ∗ ( t ) = p ( N − t ) N − sp is the critical Sobolev–Hardy exponent. The parameters λ and γ are positive, and 0 ≤ μ < μ H , where μ H is the optimal constant in the fractional Hardy–Sobolev inequality. The functions a ( x ) , b ( x ) , and c ( x ) are non-negative, continuous, and have compact support in Ω, satisfying additional technical conditions. The fractional p-Laplacian ( − Δ p ) s is defined for smooth functions as: ( − Δ p ) s u ( x ) = 2 lim ϵ ↘ 0 ⁡ ∫ R N ∖ B ϵ | u ( x ) − u ( y ) | p − 2 ( u ( x ) − u ( y ) ) | x − y | N + sp d y , x ∈ R N . where B ϵ is a ball of radius ϵ centered at x. We use the Nehari manifold method combined with the fibering maps, in order to show the existence of at least two positive solutions for the system under suitable conditions on the parameters ( λ , γ ) . The analysis involves variational techniques, careful estimates to handle the critical exponent and singular terms, and compactness arguments to overcome the lack of compactness in the problem. This work extends previous results on fractional p-Laplacian systems by incorporating Hardy-type singularities and critical nonlinearities, providing new insights into the multiplicity of solutions in non-local settings.

Non-strict singularity of optimal Sobolev embeddings

Abstract The paper is devoted to proving Allard–Michael–Simon‐type ‐Sobolev inequalities with explicit constants in the setting of Euclidean minimal submanifolds of arbitrary codimension. Our results require separate discussions for the cases and , respectively. In particular, for , we obtain an asymptotically sharp and codimension‐free Sobolev constant. Our argument is based on the optimal mass transport theory on Euclidean submanifolds and also provides an alternative, unified proof of the recent isoperimetric inequalities of Brendle (J. Amer. Math. Soc., 2021) and Brendle and Eichmair (Notices Amer. Math. Soc., 2024).

This article studies uniform measure attractors for non-autonomous stochastic reaction-diffusion equations defined on unbounded thin domains. First, the concept of uniform measure attractors is recalled, and their existence is proven. By employing uniform tail estimates, the asymptotic compactness of the processes is established, thereby overcoming the non-compactness of Sobolev embedding on unbounded thin domains. Finally, the upper semicontinuity of uniform measure attractors is demonstrated for ( n + 1 ) -dimensional unbounded thin domains collapsing into the space R n .

Abstract In this paper we prove a fractional version of a Caffarelli–Kohn–Nirenberg type interpolation inequality on hypersurfaces $$M\subset \mathbb {R}^{n+1}$$ M ⊂ R n + 1 which are boundaries of convex sets. The inequality carries a universal constant independent of M and involves the fractional mean curvature of M . In particular, it interpolates between the fractional Micheal-Simon Sobolev inequality recently obtained by Cabré, Cozzi, and the first author, and a new fractional Hardy inequality on M . Our method, when restricted to the plane case $$M=\mathbb {R}^n$$ M = R n , gives a new simple proof of the fractional Hardy inequality. To obtain the fractional Hardy inequality on a hypersurface, we establish an inequality which bounds a weighted perimeter of M by the standard perimeter of M (modulo a universal constant), and which is valid for all convex hypersurfaces M .

This paper studies the following biharmonic Choquard equation with a local nonlinear perturbation: [Formula: see text] where [Formula: see text], [Formula: see text] is the Riesz potential and [Formula: see text] has critical exponential growth in the sense of the Adams inequality. The exponent [Formula: see text] is critical with respect to the Hardy–Littlewood–Sobolev inequality. By using variational methods, we establish the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent [Formula: see text] and the critical exponential growth of [Formula: see text].

We deal with weighted Hardy–Sobolev type inequalities for functions on [Formula: see text], [Formula: see text]. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.

We study the incompressible magnetohydrodynamic system endowed with a Caputo time-fractional derivative of order [Formula: see text]. First, we reformulate the coupled momentum-induction equations by applying a double-curl projection that eliminates both the hydrodynamic pressure and the magnetic pseudo-pressure, producing a divergence-free velocity-magnetic field pair. Next, we then discretise the reformulated problem by combining a divergence-free Fourier spectral approximation in space with a variable-step L1 convolution discretisation in time. Our fully discrete method satisfies a discrete fractional kinetic-magnetic energy inequality, enforces the divergence constraints to machine precision, and reduces to the classical magnetohydrodynamic energy law as [Formula: see text], thereby ensuring asymptotic compatibility. Using discrete orthogonal-complementary convolution identities together with discrete Sobolev embeddings, we derive optimal error bounds of order [Formula: see text] on arbitrarily graded time meshes. Finally, we present numerical experiments, including fractional magnetic Taylor-Green and Orszag-Tang vortices, which confirm the theoretical convergence rates, demonstrate monotone energy decay, and highlight the efficiency of the adaptive time-stepping strategy.

We establish the generalized parametric logarithmic Sobolev inequalities in the Gagliardo-Nirenberg form for variable exponential space with log Holder exponential function. Employing the generalized parametric logarithmic Sobolev inequalities, we establish the existence of weak solutions to the boundary problem for the hyperbolic equation with logarithmic nonlinearity and involving variable exponents.

. We develop a quantitative contraction framework for Schrödinger and Sinkhorn bridges based on transportation–cost inequalities and Riccati matrix difference equations. Our approach combines logarithmic Sobolev and Talagrand-type inequalities to obtain explicit entropy and Wasserstein contraction bounds for Sinkhorn bridge measures, entropic optimal transport plans, and the associated Markov transport maps. A key feature of the analysis is the interplay between transport-cost inequalities and matrix Riccati difference equations arising in filtering and stochastic control. The results are established under local regularity assumptions on the reference transition, formulated in terms of curvature, Lipschitz continuity, and Fisher-information bounds. Within this general setting, we derive quantitative stability and convergence estimates for Schrödinger bridges and Sinkhorn iterates that are robust with respect to the choice of reference measure. As a main application, we specialize the theory to linear-Gaussian reference transitions, where the Gaussian structure permits sharp constants, refined exponential decay rates, and continuity estimates for Schrödinger bridges, Sinkhorn iterates, barycentric projections, conditional covariances, and proximal sampler semigroups. In this setting, we recover and extend several known contraction results for entropic and Wasserstein distances and obtain new quantitative bounds that improve previously available rates. Our results provide a unified probabilistic framework for stability, regularity, and convergence of Sinkhorn algorithms. We illustrate the impact of our results on regularized entropic transport, proximal samplers, and diffusion-based generative models, as well as on diffusion flow-matching.

We provide a complete characterization of compactness of Sobolev embeddings of radially symmetric functions on the entire space $\mathbb{R}^n$ in the general framework of rearrangement-invariant function spaces. We avoid any unnecessary restrictions and cover also embeddings of higher order, providing a complete picture within this framework. To achieve this, we need to develop new techniques because the usual techniques used in the study of compactness of Sobolev embeddings in the general framework of rearrangement-invariant function spaces are limited to domains of finite measure, which is essential for them to work. Furthermore, we also study certain weighted Sobolev embeddings of radially symmetric functions on balls, where the weight is a nonnegative power of the distance from the origin. We completely characterize their compactness and also describe optimal target rearrangement-invariant function spaces in these weighted Sobolev embeddings.

The best constants of two types of discrete $\ell ^p$ Sobolev inequalities on a complete graph

Sobolev inequalities for canceling operators

An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding

Sobolev inequalities for nonlinear Dirichlet forms