Sharp Sobolev Research Articles

Sharp Sobolev inequalities in critical dimensions

They discovered the following remarkable phenomenon: the qualitative behavior of the set of solutions of (1) and (2) is highly sensitive to N, the dimension of the space. To state their result precisely, let us denote by λ1 > 0 the first eigenvalue of −1 in . Brezis and Nirenberg showed for K = 1 that: (a) in dimension N ≥ 4, there exists a positive solution of (1) and (2) if and only if λ ∈ (0, λ1); while (b) in dimension N = 3 and when  = B1 is the unit ball, there exists a positive solution of (1) and (2) if and only if λ∈ (λ1/4, λ1). Pucci and Serrin [13] later considered the general polyharmonic problem (1) with K ≥ 1 and with homogenous Dirichlet boundary conditions given by Du = 0 on ∂ for k = 0, . . . , K − 1. (3)

Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and 2 2 (M) be the Sobolev space consisting of functions in L2(M) whose derivatives up to the order two are also in L2(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H 2 2 (M), $$\left\| u \right\|_{2^\sharp }^2 \leqslant A\left\| {\Delta _g u} \right\|_2^2 + B\left\| u \right\|_{H_1^2 }^2 $$ where 2#=2n/(n−4) is critical, and\(\left\| \cdot \right\|_{H_1^2 } \) is the usual norm on the Sobolev space H 1 2 (M) consisting of functions in L2(M) whose derivatives of order one are also in L2(M). The sharp constant A in this inequality is K 0 2 where K0, an explicit constant depending only on n, is the sharp constant for the Euclidean Sobolev inequality\(\left\| u \right\|_{2^\sharp } \leqslant K\left\| {\nabla u} \right\|_2 \). We prove in this article that for any compact Riemannian manifold, A=K 0 2 is attained in the above inequality.

Sharp Sobolev Inequality of Logarithmic Type and the Limiting Regularity Condition to the Harmonic Heat Flow

We show a sharp version of the Sobolev inequality of the Beale--Kato--Majda and the Kozono--Taniuchi type in Lizorkin--Triebel space. As an application of this inequality, the regularity problem under the critical condition to the gradient flow of the harmonic map into a sphere is considered in the class $L^2(0,T;BMO({\mathbb R}^n;{\mathbb S}^{m}))$, where BMO is the class of functions of bounded mean oscillations.

A sharp Sobolev inequality on Riemannian manifolds

Let $(M,g)$ be a smooth compact Riemannian manifold without boundary of dimension $n\ge 6$. We prove that <p align="center"> $||u||_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_{g} u|^2+c(n)R_{g} u^2\}dv_g +A||u||_{L^{2n/(n+2)}(M,g)}^2,$ <p align="left" class="times"> for all $u\in H^1(M)$, where $2^*=2n/(n-2)$, $c(n)=(n-2)/[4(n-1)]$, $R_g$ is the scalar curvature, $K^{-1}=$ inf $\|\nabla u\|_{L^2(\mathbb R^n)}\|u\|_{L^{2n/(n-2)}(\mathbb R^n)}^{-1}$ and $A>0$ is a constant depending on $(M,g)$ only. The inequality is <em>sharp</em in the sense that on any $(M,g)$, $K$ can not be replaced by any smaller number and $R_g$ can not be replaced by any continuous function which is smaller than $R_g$ at some point. If $(M,g)$ is not locally conformally flat, the exponent $2n/(n+2)$ can not be replaced by any smaller number. If $(M,g)$ is locally conformally flat, a stronger inequality, with $2n/(n+2)$ replaced by $1$, holds in all dimensions $n\ge 3$.

Sharp Sobolev type inequalities for higher fractional derivatives

On R n , n⩾1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2 s and q= 2n n−2s any function f∈ H s( R n) satisfies ‖f‖ 2 q⩽S n,s (−Δ) s/2f 2 2, where the operator (− Δ) s in Fourier spaces is defined by (−Δ) sf (k):=(2π|k|) 2s f(k) . To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804.

A sharp Sobolev inequality on Riemannian manifolds

We outline our results in [11] concerning some sharp Sobolev inequalities on Riemannian manifolds. Our inequalities emphasize the role of scalar curvature in this context. To cite this article: Y.Y. Li, T. Ricciardi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 519–524.

Asymptotic behaviour of sharp Sobolev constants in thin domains

In this paper we study the asymptotic behaviour of the constants in Sobolev inequalities in thin domains with respect to the thickness of the domain ε. We prove that the sharp Sobolev constants in thin domains converge to the sharp Sobolev constant on the lower-dimensional domain, as ε tends to zero.

Optimal Sobolev inequalities and extremal functions. The three dimensional case

The existence of extremal functions for sharp Sobolev inequalities on compact manifolds is fairly well understood when the dimension of the manifold is greater than or equal to four. We investigate in this paper the existence of extremal functions for such inequalities when the dimension is three, and, as a byproduct, we investigate also low dimension phenomena that occur in the study of nonlinear elliptic PDEs involving the Laplace operator.

The 𝐴𝐵 program in geometric analysis: sharp Sobolev inequalities and related problems

The 𝐴𝐵 program in geometric analysis: sharp Sobolev inequalities and related problems

From best constants to critical functions

The study of sharp Sobolev inequalities starts with the notion of best constant and leads naturally to the question to know whether or not there exist extremal functions for these inequalities. We restrict ourselves in this paper to the \(H_1^2\)-Sobolev inequality. Then, we extend the notion of best constant to that of critical function, and, with the help of this notion, we answer the question to know whether or not there exist extremal functions for the sharp \(H_1^2\)-Sobolev inequality. Partial answers to the more general question to know whether or not an extremal function always comes with a critical function are also given.

An elementary proof of sharp Sobolev embeddings

We present an elementary unified and self-contained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Brézis and Wainger.

On the Grushin operator and hyperbolic symmetry

Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry.

Sharp Sobolev inequalities with lower order remainder terms

Sharp Sobolev inequalities with lower order remainder terms

A Stability Result for Extremal Functions for Some Sharp Sobolev Inequalities with Critical Exponent

This Paper is devoted to a proof of a stability result on the solutions of the equation on an open bounded set of , where p*stands for the critical exponent for the Sobolev embedding of W 1-pinto Lk spaces, and Δ p for the p-Laplacian operator. In particular, we use the concentration-compactness theorem of P.L.Lions in order to prove that, under some assumptions on the data, the solutions cannot converge to 0.

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 \(\)

Sharp sobolev embeddings and related hardy inequalities: The critical case

AbstractThe paper deals with sharp embeddings of the Sobolev spaces Hsp(IRn) and the Besov spaces Bsp,p(IRn) into rearrangement—invariant spaces and related Hardy inequalities. Here 1 &lt; p &lt; ∞ and s = n/p.

On a Sharp Sobolev-Type Inequality on Two-Dimensional Compact Manifolds

Motivated by the asymptotic analysis of double vortex condensates in the Chern‐Simons‐Higgs theory, we construct a suitable minimizing sequence for a sharp Sobolev inequality “a la Moser” for two‐dimensional compact manifolds. As a consequence, we first obtain a direct proof of the sharp character of such an inequality. Secondly, and more interestingly, we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum.

Some General Forms of Sharp Sobolev Inequalities

In this paper, we establish some general forms of sharp Sobolev inequalities on the upper half space or any compact Riemannian manifold with smooth boundary. These forms extend some previous results Escobar [11], Li and Zhu [18].

Sharp Sobolev Inequalities Involving Boundary Terms

Sharp Sobolev Inequalities Involving Boundary Terms

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Let ( M , g ) (M,g) be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature K ≤ − 1 K\le -1 . If f f is a compactly supported function of bounded variation on M M , then f f satisfies the Sobolev inequality \[ 4 π ∫ M f 2 d A + ( ∫ M | f | d A ) 2 ≤ ( ∫ M ‖ ∇ f ‖ d A ) 2 . 4\pi \int _M f^2\,dA+ \left (\int _M |f|\,dA \right )^2\le \left (\int _M\|\nabla f\|\,dA \right )^2. \] Conversely, letting f f be the characteristic function of a domain D ⊂ M D\subset M recovers the sharp form 4 π A ( D ) + A ( D ) 2 ≤ L ( ∂ D ) 2 4\pi A(D)+A(D)^2\le L(\partial D)^2 of the isoperimetric inequality for simply connected surfaces with K ≤ − 1 K\le -1 . Therefore this is the Sobolev inequality “equivalent” to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces. Under the same assumptions on ( M , g ) (M,g) , if c : [ a , b ] → M c\colon [a,b]\to M is a closed curve and w c ( x ) w_c(x) is the winding number of c c about x x , then the Sobolev inequality implies \[ 4 π ∫ M w c 2 d A + ( ∫ M | w c | d A ) 2 ≤ L ( c ) 2 , 4\pi \int _M w_c^2\,dA+ \left (\int _M|w_c|\,dA \right )^2\le L(c)^2, \] which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature ≤ − 1 \le -1 .