We investigate the dependence of optimal constants in Poincaré- 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, showing, in particular, that Dirichler 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 $Γ$-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.
We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, prove that the disc, and only the disc maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so-called Auerbach triangle.
We derive weighted Sobolev-Poincaré type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic $A_p$-class, where only the parabolic cubes are involved in the definition.
In this paper we will establish different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class $% A_{p(\cdot ),p^{\ast }(\cdot )}$, where the exponent $p(\cdot )$ satisfies appropriate hypotheses, and in this case we obtain the first order weighted Poincaré inequalities with variable exponents. In the case of Carnot groups we also set up…
We prove an improved version of Poincaré-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space.
The notions of higher-order weighted multilinear Poincaré and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibnitz-type rules in Campanato-Morrey spaces are established.
By using quasi-Banach techniques as key ingredient we prove Poincaré- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of Błocki's integrability conjecture for $m$-subharmonic functions.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
In a previous paper we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in $W_{0}^{1,1}(Ω)$. In this paper we extend our method to Sobolev functions that do not vanish at the boundary.
In this note, we establish a $L^p-$version of the Poincaré--Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. The interest of this result is that it relates both the Poincaré (or Hardy) inequality and the Sobolev inequality with the sharp constant in $\mathbb H^n$. Our approach is based on the comparison of the $L^p-$norm of gradient of the symmetric decreasing rearrangement of a function in both the hyperbolic space and the Euclidean space, and the sharp Sobolev inequalities in Euclidean spaces. This approach also gives the proof of the Poincaré--Gagliardo--Nirenberg and Poincaré--…
A priori estimates for the weak solutions the Dirichlet problem for the uniformly higher-order elliptic equations in a smooth bounded domain $Ω\subset \Rn$ in generalized weighted Sobolev-Morrey spaces are obtained.
We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].
We study the a priori estimates,existence/nonexistence of radial sign changing solution, and the Palais-Smale characterisation of the problem $-\De_{\Bn}u - \la u = |u|^{p-1}u, u\in H^1(\Bn)$ in the hyperbolic space $\Bn$ where $1<p\leq\frac{N+2}{N-2}$. We will also prove the existence of sign changing solution to the Hardy-Sobolev-Mazya equation and the critical Grushin problem.
We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap betwe…
We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions …
Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincaré inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of $X$ and use them to obtain several results on $X$ itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincaré inequalities. We also provide a discussion about possible applications of the completions and extension results to $p$-harmonic functions on noncomplete spaces and show by examples…
We present a unified strategy to derive Hardy-Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. The idea also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.
We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy with the entropy production functional reflect the contraction properties of the flow. Our approach provides a unified framework for the study of the Kolmogorov-Fokker-Planck (KFP) equation.
In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincaré-Sobolev inequalities. We show that minimizers do exist for smooth domains in $\mathbb{R}^d$, an also for some polyhedral domains. On the other hand, we prove the non-existence of minimizers in the rectangular isosceles triangle in $\mathbb{R}^2$.
We introduce a class of weak solutions to the quasilinear equation $-Δ_p u = σ|u|^{p-2}u$ in an open set $Ω\subset\mathbf{R}^n$. Here $p>1$, and $Δ_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $σ$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-Δ_p v = (p-1)|\nabla v|^p + σ$, under the same conditions on $σ$. Our results for this latter equation will allow us to characterize the class of distributions $σ$ which satisfy the Sobolev-Poincare inequality, there…
If $Ω\subset \R^n$ is a smooth bounded domain and $q \in (0, \frac{n}{n-1})$ we consider the Poincare-Sobolev inequality \[ c \Bigl(\int_Ω \abs{u}^\frac{n}{n-1}\Bigr)^{1-\frac{1}{n}} \le \int_Ω \abs{Du}, \] for every $u \in \mathrm{BV}(Ω)$ such that $\int_Ω \abs{u}^{q-1} u = 0$. We show that the sharp constant is achieved. We also consider the same inequality on an $n$--dimensional compact Riemannian manifold $M$. When $n \ge 3$ and the scalar curvature is positive at some point, then the sharp constant is achieved. In the case $n \ge 2$, we need the maximal scalar curvature to satisfy some st…
We extend a Poincaré-type inequality for functions with large zero-sets by Jiang and Lin to fractional Sobolev spaces. As a consequence, we obtain a Hausdorff dimension estimate on the size of zero sets for fractional Sobolev functions whose inverse is integrable. Also, for a suboptimal Hausdorff dimension estimate, we give a completely elementary proof based on a pointwise Poincaré-style inequality.
We establish several Poincaré--Sobolev type inequalities for the Lapalce--Beltrami operator $Δ_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or sharp Sobolev inequality in $\mathbb H^n$. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for $Δ_g$ in $\mathbb H^n$. As a consequence, we obtain the Poincaré--Sobolev inequality for the seco…
In this article we continue our study of higher Sobolev regularity of flexible convex integration solutions to differential inclusions arising from applications in materials sciences. We present a general framework yielding higher Sobolev regularity for Dirichlet problems with affine data in $\text{int}(K^{lc})$. This allows us to simultaneously deal with linear and nonlinear differential inclusion problems. We show that the derived higher integrability and differentiability exponent has a lower bound, which is independent of the position of the Dirichlet boundary data in $\text{int}(K^{lc})$.…
Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $ρ$ be a defining function of $M$ and $σ\in(0,1)$. In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of $W^{1,2}(ρ^{1-2 σ}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 σ}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(ρ^{1-2σ}, M)$, $$ \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq μ^{-1} \int_{M} ρ^{1-2 σ}|\nabla_{g} u|^…
We study the sharp constant for the embedding of $W^{1,p}_0(Ω)$ into $L^q(Ω)$, in the case $2<p<q$. We prove that for smooth connected sets, when $q>p$ and $q$ is sufficiently close to $p$, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of $p-$Laplace--type equations by L. Damascelli an…
In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincaré-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincaré-Sobolev type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the fact that we keep track of the behaviour of the constants involved when the fractional parameter approaches to $1$. Our main method is based on techniques coming from harmonic analysis related to the self-i…
We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scattering matrix for an elliptic operator of 0-type, modified by some differential operators. We study the scattering matrix by using the 0-calculus and generalize a result of Graham for the case of the standard hyperbolic metric on a ball.
This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approach…
We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones $K\subset \mathbb{R}^3$ in the specific scale $\ B^α_{τ,τ}, \ \frac{1}τ=\fracα{d}+\frac{1}{p}\ $ of Besov spaces. The regularity of the solution in these spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractiona…
We study extremal functions for a family of Poincaré-Sobolev-type inequalities. These functions minimize, for subcritical or critical $p\geq 2$, the quotient ${\|\nabla u\|_2}/{\|u\|_p}$ among all $u \in H^1(B)\setminus\{0\}$ with $\int_{B}u=0$. Here $B$ is the unit ball in $\mathbb{R}^N$. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of $B$. We also prove that, for…
We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications…
Contains a further development of the V.G. Maz'ya concept of polynomial capacities.A different different approach is given together with new results. The primary application is for Hardy inequalities. This is treated in a later report. Here is given an application which consists of a reformulation of the Spectral synthesis question in terms of polynomial capacities. However so far this does not give a new proof, though a similar but different question for the nonnegative cone in Sobobolev space of order 2 is resolved.
We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $\det Du = f$, where $f$ is integrable and bounded away from zero. In particular, we take $f\in L^p$, where $p > 1$, or in $L\log L$. We prove that for a Baire-generic $f$ in either space there are no solutions with the expected regularity.
In this paper we prove the existence of a solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. This improves a celebrated theorem obtained by S. Hildebrandt, H. Kaul and K. Widman in 1977. In particular no curvature assumptions on the target are required. Our proof relies on a careful analysis of the Sobolev spaces of maps involved in the variational process, and on a deformation result which permits to glue a suitable Euclidean end to the geodesic ball.
Our main result is an estimate for a sharp maximal function, which implies a Keith-Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces.
We prove Sobolev-Poincaré and Poincaré inequalities in variable Lebesgue spaces $L^{p(\cdot)}(Ω)$, with $Ω\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have been used previously. In particular, we require $p(\cdot)$ to satisfy a new \emph{boundary $\log$-Hölder condition} that imposes some logarithmic decay on the oscillation of $p(\cdot)$ towards the boundary of the domain. Some control over the interior oscillation of $p(\cdot)$ is also needed, but it is given by a very general condition that allows $p(\cdot)$ to be discontinu…
We discuss the solvability of Dirichlet problems of the type $- Δ_{p, w} u = σ$ in $Ω$; $u = 0$ on $\partial Ω$, where $Ω$ is a bounded domain in $\mathbb{R}^{n}$, $Δ_{p, w}$ is a weighted $(p, w)$-Laplacian and $σ$ is a nonnegative locally finite Radon measure on $Ω$. We do not assume the finiteness of $σ(Ω)$. We revisit this problem from a potential theoretic perspective and provide criteria for the existence of solutions by $L^{p}(w)$-$L^{q}(σ)$ trace inequalities or capacitary conditions. Additionally, we apply the method to the singular elliptic problem $- Δ_{p, w} u = σu^{- γ}$ in $Ω$; $…
In this paper we study several inequalities of log-Sobolev type for Dunkl operators. After proving an equivalent of the classical inequality for the usual Dunkl measure $μ_k$, we also study a number of inequalities for probability measures of Boltzmann type of the form $e^{-|x|^p} dμ_k$. These are obtained using the method of $U$-bounds. Poincaré inequalities are obtained as consequences of the log-Sobolev inequality. The connection between Poincaré and log-Sobolev inequalities is further examined, obtaining in particular tight log-Sobolev inequalities. Finally, we study application to exponen…
Let $Ω$ be an arbitrary bounded domain of $\R^n$. We study the right invertibility of the divergence on $Ω$ in weighted Lebesgue and Sobolev spaces on $Ω$, and rely this invertibility to a geometric characterization of $Ω$ and to weighted Poincaré inequalities on $Ω$. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when $Ω$ is Lipschitz or, more generally, when $Ω$ is a John domain, and focus on the case of $s$-John domains.
