Poincare Sobolev Inequalities Research Articles

Convergence rates for linear elasticity systems on perforated domains

In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was $$L^{\frac{2d}{d-1-\tau }}$$ -error estimates $$O\big (\varepsilon ^{1-\frac{\tau }{2}}\big )$$ for all $$\tau \in (0,1)$$ in a bounded smooth domain, which is new even for homogenization problems on unperforated domains. It followed from weighted Hardy–Sobolev’s inequalities (given by Lehrback and Vahakangas in J Funct Anal 271(2):330–364, 2016) and a suboptimal error estimate for the square function of the first-order approximating corrector (earliest investigated by Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036, 2012) under additional regularity assumption on coefficient). The new approach relied on the weighted quenched Calderon–Zygmund estimate (initially appeared in Gloria et al. work Milan J Math 88(1):99–170, 2020 for a quantitative stochastic homogenization theory). The second effort was $$L^2$$ -error estimates $$O\big (\varepsilon ^{\frac{5}{6}} \ln ^{\frac{2}{3}}(1/\varepsilon )\big )$$ for a Lipschitz domain, followed from a duality scheme coupled with interpolation inequalities. Also, we developed a new weighted extension theorem and local-type Sobolev–Poincare inequalities on perforated domains. Throughout the paper, we do not impose any smoothness assumption on the coefficients.

Existence and non-existence of minimizers for Poincaré–Sobolev inequalities

In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincare–Sobolev inequalities. We show that minimizers do exist for smooth domains in $${\mathord {{\mathbb {R}}}}^d$$, an also for some polyhedral domains. On the other hand, we prove the non-existence of minimizers in the rectangular isosceles triangle in $${\mathord {{\mathbb {R}}}}^2$$.

Dirichlet conditions in Poincaré–Sobolev inequalities: the sub-homogeneous case

We investigate the dependence of optimal constants in Poincare–Sobolev inequalities of planar domains on the region where the Dirichlet condition is imposed. More precisely, we look for the best Dirichlet regions, among closed and connected sets with prescribed total length L (one-dimensional Hausdorff measure), that make these constants as small as possible. We study their limiting behaviour as $$L\rightarrow +\infty $$ , showing, in particular, that Dirichlet regions homogenize inside the domain with comb-shaped structures, periodically distribuited at different scales and with different orientations. To keep track of these information we rely on a $$\Gamma $$ -convergence result in the class of varifolds. This also permits applications to reinforcements of anisotropic elastic membranes. At last, we provide some evidences for a conjecture.

On the First Eigenvalue of the Degenerate $$\varvec{p}$$ p -Laplace Operator in Non-convex Domains

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate p-Laplace operator, $$p>2$$ , in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincare–Sobolev inequalities. On this base we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors-type domains (i.e. quasidiscs). This class of domains includes some snowflake-type domains with fractal boundaries.

Pointwise estimates to the modified Riesz potential

In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev–Poincare inequality. We show that this method gives a Sobolev–Poincare inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev–Poincare inequality depends on the domain. The Sobolev–Poincare inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy–Littlewood maximal operator.

Spectral estimates of the p-Laplace Neumann operator and Brennan’s conjecture

In this paper we obtain lower estimates for the first non-trivial eigenvalue of the p-Laplace Neumann operator in bounded simply connected planar domains \(\varOmega \subset {\mathbb {R}}^2\). This study is based on a quasiconformal version of the universal two-weight Poincare–Sobolev inequalities obtained in our previous papers for conformal weights and its non weighted version for so-called K-quasiconformal \(\alpha \)-regular domains. The main technical tool is the geometric theory of composition operators in relation with the Brennan’s conjecture for (quasi)conformal mappings.

Fractional Sobolev-Poincaré inequalities in irregular domains

This paper is devoted to the study of fractional (q, p)-Sobolev-Poincare in- equalities in irregular domains. In particular, the author establishes (essentially) sharp fractional (q, p)-Sobolev-Poincare inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional (q, p)-Sobolev-Poincare inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P., Sobolev-Poincare implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.

Existence theorem and global solution for semilinear edge-degenerate hypoelliptic equations

In this article, we use the edge-type of Sobolev inequality,Hardy inequlity and Poincare inequality to prove the existence theorem for a class of semilinear degenerate hypoelliptic equation on manifolds with conical singularities. In this paper we shall find the existence theorem for the problem 1.1 in cone Sobolev space \({\mathcal {H}}^{1,\frac{N}{2}}_{2,0}({\mathbb {E}}).\) Finally, we obtain existence result of global solutions with exponential decay and show the blow-up in finite time of solutions.

Spectral estimates of the [formula omitted]-Laplace Neumann operator in conformal regular domains

In this paper we study spectral estimates of the p-Laplace Neumann operator in conformal regular domains Ω⊂R2. This study is based on (weighted) Poincaré–Sobolev inequalities. The main technical tool is the theory of composition operators in relation with the Brennan’s conjecture. We prove that if the Brennan’s conjecture holds for any p∈(4/3,2) and r∈(1,p/(2−p)) then the weighted (r,p)-Poincare–Sobolev inequality holds with the constant depending on the conformal geometry of Ω. As a consequence we obtain classical Poincare–Sobolev inequalities and spectral estimates for the first nontrivial eigenvalue of the p-Laplace Neumann operator for conformal regular domains.

A mass transportation approach for Sobolev inequalities in variable exponent spaces

In this paper we provide a proof of the Sobolev–Poincare inequality for variable exponent spaces by means of mass transportation methods, in the spirit of Cordero-Erausquin et al. (Adv Math 182(2):307–332, 2004). The importance of this approach is that the method is flexible enough to deal with different inequalities. As an application, we also deduce the Sobolev-trace inequality improving the result of Fan (J Math Anal Appl 339(2):1395–1412, 2008) by obtaining an explicit dependence of the exponent in the constant.

On improved fractional Sobolev–Poincaré inequalities

We study a certain improved fractional Sobolev–Poincare inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincare inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincare inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

Stability of Solutions to Generalized Forchheimer Equations of any Degree

The dynamics of generalized Forchheimer equations for slightly compressible fluids are tudied by means of initial boundary value problem for the pressure. We prove that the solutions continuously depend on the boundary data and the Forchheimer polynomials. New bounds for the solutions are established in the Lα-norm for all α ≥ 1, and then are used to improve estimates for their spatial and time derivatives. New Poincare–Sobolev inequalities and nonlinear Gronwall type estimates for nonlinear differential inequalities are utilized to achieve better asymptotic bounds. The methods developed can be applied to other degenerate parabolic equations. Bibliography: 25 titles.

Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

The aim of this paper is to prove an isoperimetric inequality relative to a convex domain $\Omega\subset\mathbb{R}^d$ intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius $r>0$ and the position $y\in\overline{\Omega}$ of the center of the ball. For this, uniform Sobolev, Poincare and Poincare-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension $d,$ the diameter of the domain and the integrability exponent $p\in[1,d)$.

FRACTIONAL SOBOLEV–POINCARÉ AND FRACTIONAL HARDY INEQUALITIES IN UNBOUNDED JOHN DOMAINS

We prove fractional Sobolev–Poincare inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.

Hölder continuity for nonlinear sub-elliptic systems with sub-quadratic growth

This paper is concerned with partial regularity of weak solutions to nonlinear sub-elliptic systems under sub-quadratic natural growth conditions. We begin with establishing a Sobolev-Poincare type inequality associated with Hormander’s vector fields for $$u\in HW^{1,m}(\Omega , \mathbb {R}^N)$$ with $$1<m<2$$ . Then $$\mathcal {A}$$ -harmonic approximation method is applied, and partial Holder continuity with optimal local Holder exponent for gradients of weak solutions to the systems is established.

Geometric PDEs in the Grushin Plane: Weighted Inequalities and Flatness of Level Sets

A geometric Sobolev-Poincare inequality for stable solutions of semilinear partial differential equations (PDEs) in the Grushin plane will be obtained. Such inequality will bound the weighted L 2 -norm of a test function by a weighted L 2 -norm of its gradient, and the weights will be interesting geometric quantities related to the level sets of the solution. From this, we shall see that a geometric PDE holds on the level sets of stable solutions. We shall study in detail the particular case of local minimizers of a Ginzburg-Landau-Allen-Cahn-type phase transition model and provide for them some one-dimensional symmetry results.

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.

A correction to: ``Some characterizations of space-forms"

In [1], for the application to Theorem 4.3, one needs a version of Proposition 3.1 assuming that the manifold supports an inhomogeneous Poincare-Sobolev inequality of the type (0.1) below. Indeed, an easy modification of the original argument yields the following Proposition 3.1 bis. Let (M, 〈 , 〉) be a complete manifold and assume that, for some 0 ≤ α < 1 and some non-negative function h, the inhomogeneous Sobolev– Poincare type inequality

Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds

We investigate p-harmonic maps, p ≥ 2, from a complete non-compact manifold into a non-positively curved target. First, we establish a uniqueness result for the p-harmonic representative in the homotopy class of a constant map. Next, we derive a Caccioppoli inequality for the energy density of a p-harmonic map and we prove a companion Liouville type theorem, provided the domain manifold supports a Sobolev–Poincare inequality. Finally, we obtain energy estimates for a p-harmonic map converging, with a certain speed, to a given point.

Quasihyperbolic growth conditions and compact embeddings of Sobolev spaces

In this paper, will always denote an open and connected subset of the Euclidean space R, n ≥ 2. Recall that the Sobolev space W 1,p( ) consists of the functions in L( ) whose first-order distributional derivatives all belong to L( ) also. One of the fundamental inequalities associated with Sobolev functions is the Sobolev–Poincare inequality, according to which (∫ |u− u |np/(n−p) dx )(n−p)/np ≤ C (∫ |∇u|p dx )1/p (1)