Sobolev Trace Inequality Research Articles

Logarithmic Sobolev trace inequalities

We prove a logarithmic Sobolev trace inequality and we study the trace operator in the weighted Sobolev space $W^{1,p} (\Omega , \gamma)$ for sufficiently regular domain, where $\gamma$ is the Gauss measure. Applications to PDE are also considered.

Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms

In this paper, we establish the Gagliardo–Nirenberg inequality under Lorentz norms for fractional Laplacian. Based on special cases of this inequality under Lebesgue norms, we prove the Lp-logarithmic Gagliardo–Nirenberg and Sobolev inequalities. Motivated by the L2-logarithmic Sobolev inequality, we obtain a fractional logarithmic Sobolev trace inequality in terms of the restriction τku of u from Rn to Rn−k. Finally, we prove the fractional Hardy inequality under Lorentz norms.

Sharp trace inequalities on fractional Sobolev spaces

AbstractThe best constant and extremal functions for Sobolev trace inequalities on fractional Sobolev spaces are achieved by a simple argument. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

The sharp Sobolev trace inequality in a limiting case

The sharp Sobolev trace inequality in a limiting case

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $S\|u\|^p_{L^{p_*}(\partial\Omega) \hookrightarrow \|u\|^p_{W^{1,p}(\Omega)}$ that are independent of $\Omega$. This estimates generalized those of [3] for general $p$. Here $p_* := p(N-1)/(N-p)$ is the critical exponent for the immersion and $N$ is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of [16]. Finally, we study an optimal design problem with critical exponent.

Estimates of the best Sobolev constant of the embedding of [formula omitted] into [formula omitted] and related shape optimization problems

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality λ 1 ( Ω ) ‖ u ‖ L 1 ( ∂ Ω ) ≤ ‖ u ‖ W 1 , 1 ( Ω ) that are independent of Ω . These estimates generalize those of [J. Fernandez Bonder, N. Saintier, Estimates for the Sobolev trace constant with critical exponent and applications, Ann. Mat. Pura. Aplicata (in press)] concerning the p -Laplacian to the case p = 1 . We apply our results to prove the existence of an extremal for this embedding. We then study an optimal design problem related to λ 1 , and eventually compute the shape derivative of the functional Ω → λ 1 ( Ω ) .

Best constants in Sobolev inequalities on manifolds with boundary in the presence of symmetries and applications

In this paper we establish the best constants for a Sobolev inequality and a Sobolev trace inequality on compact Riemannian manifolds with boundary, the functions being invariant under the action of a compact subgroup G of the isometry group I ( M , g ) and we give applications to some nonlinear PDEs with upper critical Sobolev exponent.

Best constant in Sobolev trace inequalities on the half-space

Using a mass transportation method, we study optimal Sobolev trace inequalities on the half-space and prove a conjecture made by Escobar in 1988 about the minimizers.

A sharp Sobolev trace inequality for the fractional-order derivatives

We establish a sharp Sobolev trace inequality for the fractional-order derivatives. As a close connection with this best estimate, we show a fractional-order logarithmic Sobolev trace inequality with the asymptotically optimal constant, but also sharpen the Poincaré embedding for the conformal invariant energy and BMO spaces.

ON THE EXISTENCE OF EXTREMALS FOR THE SOBOLEV TRACE EMBEDDING THEOREM WITH CRITICAL EXPONENT

In this paper, the existence problem is studied for extremals of the Sobolev trace inequality W1,p(ω)→Lp*(∂Ω), where Ω is a bounded smooth domain in RN, p*=p(N−1)/(N−p), is the critical Sobolev exponent, and 1 < p < N. 2000 Mathematics Subject Classification 35J65 (primary), 35B33 (secondary).

Logarithmic Sobolev trace inequality

A logarithmic Sobolev trace inequality is derived. Bounds on the best constant for this inequality from above and below are investigated using the sharp Sobolev inequality and the sharp logarithmic Sobolev inequality.

Best constants in Sobolev trace inequalities

In this paper we establish the best constant for a Sobolev trace inequality on compact Riemannian manifolds with boundary. More specifically, let 1< p< n and 1 K ̄ (n,p) = inf ∇u∈L p( R + n) u∈L p ̄ ∗ (∂ R + n)⧹{0} ∫ R + n | ∇u| p ( ∫ ∂ R + n |u| p ̄ ∗ ) p/ p ̄ ∗ , where p ̄ ∗=p(n−1)/(n−p) . We prove that for any compact n-dimensional Riemannian manifold with boundary ( M, g), for any ε>0 there exists A ε >0 such that ||u|| L p ̄ ∗ (∂M) p⩽( K ̄ (n,p)+ε)|| ∇ gu|| L p(M) p+A ε||u|| L p(∂M) p for all u∈ H 1, p ( M).

Sharp Sobolev inequalities with interior norms

In this paper, we establish some new sharp Sobolev inequalities on any smooth bounded domain \(\Omega \subset{\Bbb R}^n\). Let \(S_1\) and S be the sharp constants corresponding to the Sobolev embedding and trace inequalities respectively. If \(n\ge 4\), there exist constants \(A(\Omega)\), \(A_1(\Omega)>0\) such that \(\forall u \in H^1(\Omega)\)\(\) and \(\) If \(n=3\), for any \(k_3 >3\), there exist constants \(A(\Omega, k_3), A_1(\Omega, k_3)>0 \) such that \(\forall u \in H^1(\Omega)\)\(\) and \(\)

The Zeta Functional Determinants on Manifolds with Boundary

This is a continuing research of our previous work (S.-Y. A. Chang and J. Qing (1997),J. Funct. Anal.147, 327–362). In this paper we showW2,2-compactness of isospectral set within a subclass of conformal metrics, and discuss extremal properties of the zeta functional determinants, for certain elliptic boundary value problems on 4-manifolds with smooth boundary. To do so we establish some sharp Sobolev trace inequalities.

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

In this paper, we establish some sharp Sobolev trace inequalities on n-dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2(n - 1)/(n - 2), 1/S = inf {∫ |∇u|2 : ∇u ∈ L2(R+n), ∫ |u|q = 1}. We establish for any Riemannian manifold with a smooth boundary, denoted as (M, g), that there exists some constant A = A(M, g) > 0, (∫dM|u|qdsg)2/q < or = to S ∫M |∇gu|2dvg + A ∫dMu2dsg, for all u ∈ H1 (M). The inequality is sharp in the sense that the inequality is false when S is replaced by any smaller number. © 1997 John Wiley & Sons, Inc.