Sobolev Trace Inequality Research Articles

Explicit forms for extremals of sharp Sobolev trace inequalities on the unit balls

Explicit forms for extremals of sharp Sobolev trace inequalities on the unit balls

Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds

In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev inequalities, we also give the stability of the higher and fractional order Sobolev inequalities with the lower bounds. This extends to some extent the stability of the first order Sobolev inequalities with the explicit lower bounds established by Dolbeault, Esteban, Figalli, Frank and Loss in [18] to the higher and fractional order case. Our proofs are based on the competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the dual stability theory. As another application of the stability of the HLS inequality, we also establish the stability of Beckner's [4] restrictive Sobolev inequalities of fractional order s with 0<s<n2 on the flat sub-manifold Rn−1 and the sphere Sn−1 with the explicit lower bound. When s=1, this implies the explicit lower bound for the stability of Escobar's first order Sobolev trace inequality [19] which has remained unknown in the literature.

Sobolev trace-type inequalities via time-space fractional heat equations

Abstract This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of $\dot {W}^{\nu /2}_p(\mathbb {R}^n)$ in terms of the solution $u(x,t)$ . Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to $L^p-$ logarithmic Sobolev inequalities for $\dot {W}^{\nu /2}_{p}(\mathbb R^{n})$ . As a byproduct, Sobolev affine trace-type inequalities for $\dot {H}^{-\nu /2}(\mathbb {R}^n)$ and local Sobolev-type trace inequalities for $Q_{\nu /2}(\mathbb {R}^n)$ are established.

Sobolev capacities and trace inequalities related to $ \alpha- $Hermite operators with application to $ p $-Laplace type equations

Sobolev capacities and trace inequalities related to $ \alpha- $Hermite operators with application to $ p $-Laplace type equations

Almost sharp weighted Sobolev trace inequalities in the unit ball under constraints

In this paper, we establish some improved weighted Sobolev trace inequalities [Formula: see text] under the zero higher order moments constraint via the concentration compactness principle, where [Formula: see text] is a defining function of [Formula: see text] and [Formula: see text]. This relates to the fractional (conformal) Laplacians and related problems in conformal geometry. We construct some test functions and show that the inequality is almost optimal when [Formula: see text].

A note on the Sobolev trace inequality

Consider the classical Sobolev trace inequality ‖ ∇ φ ‖ L 2 ( R + n ) ≥ K ‖ φ ‖ L 2 ( n − 1 ) n − 2 ( ∂ R + n ) \begin{equation*} \|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)} \geq K\|\varphi \|_{L^{\frac {2(n-1)}{n-2}}(\partial \mathbb {R}^n_+)} \end{equation*} for all φ ∈ W 0 1 , 2 ( R + n ) \varphi \in W^{1,2}_0(\mathbb {R}^n_+) , where K K is the best constant. Here, W 0 1 , 2 ( R + n ) W^{1,2}_0(\mathbb {R}^n_+) is the space obtained by taking the completion in the norm ‖ ∇ φ ‖ L 2 ( R + n ) \|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)} of the set of all smooth functions with support contained in the closure of R + n \mathbb {R}^n_+ , and n ≥ 3 n\geq 3 . Let M \mathcal {M} be the set of functions for which we have equality in the Sobolev trace inequality above. In this note, we show that there is a positive constant α \alpha such that ‖ ∇ φ ‖ L 2 ( R + n ) 2 − K 2 ‖ φ ‖ L 2 ( n − 1 ) n − 2 ( ∂ R + n ) 2 ≥ α d ( φ , M ) 2 \begin{equation*} \|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)}^2-K^2 \|\varphi \|_{L^{\frac {2(n-1)}{n-2}}(\partial \mathbb {R}^n_+)}^2\geq \alpha d(\varphi ,\mathcal {M})^2 \end{equation*} for all φ ∈ W 0 1 , 2 ( R + n ) \varphi \in W^{1,2}_0(\mathbb {R}^n_+) , where d d is the distance in the Sobolev space W 0 1 , 2 ( R + n ) W^{1,2}_0(\mathbb {R}^n_+) .

A generalization of the Escobar–Riemann mapping-type problem to smooth metric measure spaces

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization of Sobolev Trace Inequality deduced by Bolley at. al. This trace inequality allows us to introduce an Escobar quotient and its infimum. This infimum we call the Escobar weighted constant. The Escobar-Riemann mapping type problem for smooth metric measure spaces in manifolds with boundary consists of finding a function which attains the Escobar weighted constant. Furthemore, we resolve the later problem when Escobar weighted constant is negative. Finally, we get an Aubin type inequality connecting the weighted Escobar constant for compact smooth metric measure space and the optimal constant for the trace inequality due to Bolley at. al.

Liouville-type theorems on manifolds with nonnegative curvature and strictly convex boundary

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.

Towards a Fully Nonlinear Sharp Sobolev Trace Inequality

Towards a Fully Nonlinear Sharp Sobolev Trace Inequality

Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

We establish a family of sharp Sobolev trace inequalities involving the Wk,2(R+n+1,ya)-norm. These inequalities are closely related to the realization of fractional powers of the Laplacian on Rn=∂R+n+1 as generalized Dirichlet-to-Neumann operators associated to powers of the weighted Laplacian in upper half space, generalizing observations of Caffarelli–Silvestre and of Yang.

The sharp affine L2 Sobolev trace inequality and affine energy in the fractional Sobolev spaces

We establish an affine invariant version of the sharp L2 Sobolev trace inequality which contains a recent result of De Nápoli et al. [9] as special case. We also give a notion of the so-called affine fractional energy for functions in the fractional homogeneous Sobolev space H˙s(Rn) for s∈(0,1) and n>2s. We investigate some properties of this affine fractional energy such as a representation formula and the affine fractional Pólya–Szegö principle. Finally, we obtain a sharp affine fractional L2 Sobolev inequality in the fractional homogeneous Sobolev space H˙s(Rn) for s∈(0,1) and n>2s.

Higher order Sobolev trace inequalities on balls revisited

Inspired by a recent sharp Sobolev trace inequality of order four on the balls Bn+1 found by Ache and Chang (2017) [2], we propose a different approach to reprove Ache–Chang's trace inequality. To further illustrate this approach, we reprove the classical Sobolev trace inequality of order two on Bn+1 and provide sharp Sobolev trace inequalities of orders six and eight on Bn+1. To obtain all these inequalities up to order eight, and possibly more, we first establish higher order sharp Sobolev trace inequalities on R+n+1, then directly transferring them to the ball via a conformal change. As the limiting case of the Sobolev trace inequalities, Lebedev–Milin type inequalities of order up to eight are also considered.

Boundary Operators Associated With the Sixth-Order GJMS Operator

AbstractWe describe a set of conformally covariant boundary operators associated with the 6th-order Graham--Jenne--Mason--Sparling (GJMS) operator on a conformally invariant class of manifolds that includes compactifications of Poincaré–Einstein manifolds. This yields a conformally covariant energy functional for the 6th-order GJMS operator on such manifolds. Our boundary operators also provide a new realization of the fractional GJMS operators of order one, three, and five as generalized Dirichlet-to-Neumann operators. This allows us to prove some sharp Sobolev trace inequalities involving the interior $W^{3,2}$-seminorm, including an analogue of the Lebedev–Milin inequality on six-dimensional manifolds.

Sobolev trace inequalities of order four

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean 4-balls.

The sharp affine $$L^2$$ L 2 Sobolev trace inequality and variants

We establish a sharp affine $$L^p$$ Sobolev trace inequality by using the $$L_p$$ Busemann–Petty centroid inequality. For $$p = 2$$ , our affine version is stronger than the famous sharp $$L^2$$ Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all extremizers in this case. For this new inequality, no Euclidean geometric structure is needed.

Some energy inequalities involving fractional GJMS operators

Under a spectral assumption on the Laplacian of a Poincar\'e--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2\gamma\in(0,2)$ or $2\gamma\in(2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption is necessary and sufficient for such an inequality to hold. We prove the energy inequalities by introducing conformally covariant boundary operators associated to the weighted conformal Laplacian and weighted Paneitz operator which generalize the Robin operator. As an application, we establish a new sharp weighted Sobolev trace inequality on the upper hemisphere.

A Liouville type theorem for Lane–Emden systems involving the fractional Laplacian

We establish a Liouville type theorem for the fractional Lane–Emden system:{(−Δ)αu=vqin RN,(−Δ)αv=upin RN,where , and p, q are positive real numbers and in an appropriate new range. To prove our result we will use the local realization of fractional Laplacian, which can be constructed as a Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre (2007 Commun. PDE 32 1245–60). Our proof is based on a monotonicity argument for suitable transformed functions and the method of moving planes in a half infinite cylinder (, where is the half unit sphere in ) based on maximum principles which are obtained by barrier functions and a coupling argument using a fractional Sobolev trace inequality.

Some trace Hardy type inequalities and trace Hardy–Sobolev–Maz'ya type inequalities

We prove a trace Hardy type inequality with the best constant on the polyhedral convex cones which generalizes recent results of Alvino et al. and of Tzirakis on the upper half space. We also prove some trace Hardy–Sobolev–Maz'ya type inequalities which generalize the recent results of Filippas et al. In applications, we derive some Hardy type inequalities and Hardy–Sobolev–Maz'ya type inequalities for fractional Laplacian. Finally, we prove the logarithmic Sobolev trace inequalities and logarithmic Hardy trace inequalities on the upper half spaces. The best constants in these inequalities are explicitly computed in the radial case.

The role of the mean curvature in a Hardy–Sobolev trace inequality

The Hardy–Sobolev trace inequality can be obtained via harmonic extensions on the half-space of the Stein and Weiss weighted Hardy–Littlewood–Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy–Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.

A sharp Sobolev trace inequality involving the mean curvature on Riemannian manifolds

In this paper, we examine the boundary $L^2$ term of the sharp Sobolev trace inequality $\|u\|_{L^{q}(\pa M)}^2\leq S \|\nabla_g u\|_{L^2(M)}^2 +A(M,g)\|u\|^2_{L^2(\pa M)}$ on Riemannian manifolds $(M,g)$ with boundaries $\pa M$, where $q=\frac{2(n-1)}{n-2}$, $S$ is the best constant and $A(M,g)$ is some positive constant depending only on $M$ and $g$. We obtain a sharp trace inequality involving the mean curvature in a remainder term, which would fail in general once the mean curvature is replaced by any smaller function.