Embedding Constants Research Articles

Embedding constants for periodic Sobolev spaces of fractional order

We obtain an explicit expression for the norms of the embedding operators of the periodic Sobolev spaces into the space of continuous functions (the norms of this type are usually called embedding constants). The corresponding formulas for the embedding constants express them in terms of the values of the well-known Epstein zeta function which depends on the smoothness exponent s of the spaces under study and the dimension n of the space of independent variables. We establish that the embeddings under consideration have the embedding functions coinciding up to an additive constant and a scalar factor with the values of the corresponding Epstein zeta function. We find the asymptotics of the embedding constants as s → n/2.

Compact embeddings of broken Sobolev spaces and applications

In this paper, we present several extensions of theoretical tools for the analysis of discontinuous Galerkin (DG) method beyond the linear case. We define broken Sobolev spaces for Sobolev indices in [1, ∞), and we prove generalizations of many techniques of classical analysis in Sobolev spaces. Our targeted application is the convergence analysis for DG discretizations of energy minimization problems of the calculus of variations. Our main tool in this analysis is a theorem which permits the extraction of a ‘weakly’ converging subsequence of a family of discrete solutions and which shows that any ‘weak limit’ is a Sobolev function. As a second application, we compute the optimal embedding constants in broken Sobolev–Poincare inequalities.

Best Constants for Moser type Inequalities in Zygmund Spaces

We first survey some recent results on optimal embeddings for the space of functions with ∆u ∈ L(Ω), where Ω ⊂ R is a bounded domain. The target space in the embeddings turns out to be a Zygmund space and the best constants are explicitly known. Remarkably, the best constant in the case of zero boundary data is twice the best constant in the case of compactly supported functions. Then, following the same strategy, we establish a new version of the celebrated Trudinger–Moser inequality, as embedding into the Zygmund space Z 1/2 0 (Ω), and we prove that, in contrast to the Moser case, here the best embedding constant is not attained.

Reduced Basis Error Bound Computation of Parameter-Dependent Navier–Stokes Equations by the Natural Norm Approach

This work focuses on the a posteriori error estimation for the reduced basis method applied to partial differential equations with quadratic nonlinearity and affine parameter dependence. We rely on natural norms—local parameter-dependent norms—to provide a sharp and computable lower bound of the inf-sup constant. We prove a formulation of the Brezzi–Rappaz–Raviart existence and uniqueness theorem in the presence of two distinct norms. This allows us to relax the existence condition and to sharpen the field variable error bound. We also provide a robust algorithm to compute the Sobolev embedding constants involved in the error bound and in the inf-sup lower bound computation. We apply our method to a steady natural convection problem in a closed cavity, with a Grashof number varying from 10 to $10^7$.

Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO

We consider the generalized Gagliardo–Nirenberg inequality in $${\mathbb{R}}^n$$ in the homogeneous Sobolev space $$\dot{H}^{s, r}({\mathbb{R}}^n)$$ with the critical differential order s = n/r, which describes the embedding such as $$L^p({\mathbb{R}}^n) \cap \dot{H}^{n/r,r}({\mathbb{R}}^n) \subset L^q({\mathbb{R}}^n)$$ for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that $$||u||_{L^{q}({\mathbb{R}}^n)} \leqq C_n q||u||_{L^{p}({\mathbb{R}}^n)}^{\frac{p}{q}}||u||_{BMO}^{1-\frac{p}{q}}$$ with the constant C n depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L ∞-bound is established by means of the BMO-norm and the logarithm of the $$\dot{H}^{s, r}$$ -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.

Existence results for Brezis–Nirenberg problems with Hardy potential and singular coefficients

In this paper, we consider the existence of non-trivial solutions to semi-linear Brezis–Nirenberg type problems with Hardy potential and singular coefficients. First, we shall study the corresponding eigenvalue problem, and obtain some basic properties of eigenvalues and asymptotic estimates of the eigenfunctions and approximating eigenfunctions. Secondly, we consider the extremal functions of the best embedding constant, and get some crucial estimates for the cut-off function of the extremal functions. Thirdly, applying different variational theorems for distinct cases of those parameters appearing in the equation, we obtain two existence results for non-trivial solutions to semi-linear Brezis–Nirenberg type problems. Our existence results are divided into non-resonant and resonant cases.

Best constants and minimizers for embeddings of second order Sobolev spaces

By considering the kernels of the first two traces, four different second order Sobolev spaces may be constructed. For these spaces, embeddings into Lebesgue spaces, the best embedding constant and the possible existence of minimizers are studied. The Euler equation corresponding to some of these minimization problems is a semilinear biharmonic equation with boundary conditions involving third order derivatives: it is shown that the complementing condition is satisfied.

Estimates on Moser Embedding

A convexity relation in terms of dimension is obtained for Gagliardo–Nirenberg embedding constants. Such estimates give bounds with the correct asymptotic dependence on dimension.

Asymptotic estimates for Gagliardo-Nirenberg embedding constants

Sharp asymptotic information is determined for the Gagliardo–Nirenberg embedding constants in high dimension. This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality. Moreover, one sees here that Hardy's inequality can be interpreted as the asymptotic limit of the logarithmic Sobolev inequality.

Critical dimensions and higher order Sobolev inequalities with remainder terms

Pucci and Serrin [21] conjecture that certain space dimensions behave ''critically'' in a semilinear polyharmonic eigenvalue problem. Up to now only a considerably weakened version of this conjecture could be shown. We prove that exactly in these dimensions an embedding inequality for higher order Sobolev spaces on bounded domains with an optimal embedding constant may be improved by adding a ''linear'' remainder term, thereby giving further evidence to the conjecture of Pucci and Serrin from a functional analytic point of view. Thanks to Brezis-Lieb [5] this result is already known for the space \( H_0^1 \) in dimension n=3; we extend it to the spaces \( H_0^K \) (K>1) in the ''presumably'' critical dimensions. Crucial tools are positivity results and a decomposition method with respect to dual cones.

On the Caffarelli–Kohn–Nirenberg inequalities

Consider the following inequalities due to Caffarelli, Kohn and Nirenberg [3]: (∫RN|x|−bp|u|pdx)2/p≤Ca,b∫RN|x|−2a|∇u|2dx, where for N≥3: −∞<a<N−22, a≤b≤a+1, and p=2NN−2+2(b−a). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. We also study the bound state solutions of the corresponding Euler equations and construct positive solutions having prescribed symmetry for certain parameter region. On considère les inégalités suivantes de Caffarelli, Kohn et Nirenberg [3] : (∫RN|x|−bp|u|pdx)2/p≤Ca,b∫RN|x|−2a|∇u|2dx, où N≥3: −∞<a<N−22, a≤b≤a+1 et p=2NN−2+2(b−a). Nous répondons à quelques questions fondamentales concernant ces inégalités, telles que meilleures constantes d'injection, existence ou non-existence des fonctions extrémales et leur propriétés qualitatives. On étudie aussi les solutions à état borné qui correspondent à l'équation d'Euler et on construit des solutions positives avec une symétrie prescrite dans certaines régions de l'espace des paramètres.

On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions

Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6] where, for N ≥ 3, −∞ < a < (N − 2)/2, a ≤ b ≤ a + 1, and p = 2N/(N − 2 + 2(b − a)). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case a ≥ 0 has been studied extensively and a complete solution is known, little has been known for the case a < 0. Our results for the case a < 0 reveal some new phenomena which are in striking contrast with those for the case a ≥ 0. Results for N = 1 and N = 2 are also given. © 2001 John Wiley & Sons, Inc.

Embedding functions and their role in interpolation theory

The embedding functions of an intermediate space A into a Banach couple (A0, A1) are defined as its embedding constants into the couples , ∀α, β &gt; 0. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak‐type interpolation to any Banach couple to obtain new interpolation theorems.

Counting Peaks of Solutions to Some Quasilinear Elliptic Equations with Large Exponents

We consider the asymptotic behavior of certain solutions to a quasilinear problem with large exponent in the nonlinearity. Starting with the investigation of a Sobolev embedding, we get a sharp estimate for the embedding constant. Then we obtain a crucial L1-estimate for the N-Laplacian operators in RN. Using these estimates we prove that the solutions obtained by the standard variational method will develop a spiky pattern of peaks as the nonlinear exponent gets large, and we also have an upper bound depending on N only of the number of peaks. Stronger results for some special convex domains acid some special solutions are also achieved.

Recursive enumeration of clusters in general dimension on hypercubic lattices.

A recursive method for enumerating clusters on a hypercubic lattice in d spatial dimensions is presented from which the weak embedding constants are determined as polynomials in d. A tabulation for all clusters having no free ends is available for nb≤15, where nb is the number of bonds. As illustrated here and elsewhere, this tabulation can be used to generate many series expansions. A novel method of checking the enumeration with an algebraic calculation is presented.

Positive solutions of nonlinear elliptic equations—existence and nonexistence of solutions with radial symmetry in 𝐿_{𝑝}(𝑅^{𝑁})

It is shown that when r r is nonincreasing, radially symmetric, continuous and bounded below by a positive constant, the solution set of the nonlinear elliptic eigenvalue problem \[ − Δ u = λ u + r u 1 + σ , u &gt; 0 on R N , u → 0 as |x| → ∞ - \Delta u = \lambda u + r{u^{1 + \sigma }},\qquad u &gt; 0\qquad {\text {on}}\,{\mathbf {R}^N},\qquad u \to 0\qquad {\text {as}}\,{\text {|x|}} \to \infty \] , contains a continuum C \mathcal {C} of nontrivial solutions which is unbounded in R × L p ( R N ) \mathbf {R}\, \times \,{L_p}({\mathbf {R}^N}) for all p ≥ 1 p \geq 1 . Various estimates of the L p {L_p} norm of u u are obtained which depend on the relative values of σ \sigma and p p , and the Pohozaev and Sobolev embedding constants.