Showing papers in "Calculus of Variations and Partial Differential Equations in 2009"

Finsler interpolation inequalities

Abstract: We extend Cordero-Erausquin et al.’s Riemannian Borell–Brascamp–Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.

A priori estimates for integro-differential operators with measurable kernels

Abstract: The aim of this work is to develop a localization technique and to establish a regularity result for non-local integro-differential operators \({\fancyscript{L}}\) of order \({\alpha\in (0,2)}\) . Thereby we extend the De Giorgi–Nash–Moser theory to non-local integro-differential operators. The operators \({\fancyscript{L}}\) under consideration generate strong Markov processes via the theory of Dirichlet forms. As is well known, regularity properties of the resolvents are important for many aspects of the corresponding stochastic process. Therefore, this work is related to probability theory and analysis, especially partial differential equations, at the same time.

Affine Moser–Trudinger and Morrey–Sobolev inequalities

Abstract: An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L n energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L n Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L p energy, with p > n, is replaced by the affine energy.

A new class of transport distances between measures

Abstract: We introduce a new class of distances between nonnegative Radon measures in \({\mathbb{R}^d}\) . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous \({W^{-1,p}_\gamma}\) -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

Abstract: This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M c such that if $${M \in (0, M_c]}$$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for $${M \in (0, M_c)}$$ . While characterising the possible infinite time blowing-up profile for M = M c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two.

Symmetry and monotonicity of least energy solutions

Abstract: We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153–179, 1988) and Maris (Arch Ration Mech Anal, 192:311–330, 2009) and answer questions from Brezis and Lieb (Comm Math Phys 96:97–113, 1984) and Lions (Ann Inst H Poincare Anal Non Lineaire 1:223–283, 1984).

On the volume functional of compact manifolds with boundary with constant scalar curvature

Abstract: We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

Optimal transport and Perelman’s reduced volume

Abstract: We show that a certain entropy-like function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.

Hölder regularity of optimal mappings in optimal transportation

Abstract: It is known that optimal mappings in optimal transportation problems are uniquely determined by corresponding potential functions. In this paper we prove various local properties of potential functions. In particular we obtain the C 1,α regularity of potential functions with optimal exponent α, which improves previous regularity results of Loeper.

Absolute continuity and summability of transport densities: simpler proofs and new estimates

Abstract: The paper presents some short proofs for transport density absolute continuity and L p estimates Most of the previously existing results which were proven by geometric arguments are re-proved through a strategy based on displacement interpolation and on approximation by discrete measures; some of them are partially extended

On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source

Abstract: This paper is concerned with the study of the nonlinear damped wave equation $${u_{tt} - \Delta u+ h(u_t)= g(u) \quad \quad {\rm in}\,\Omega \times ] 0,\infty [,}$$ where Ω is a bounded domain of $${\mathbb{R}^2}$$ having a smooth boundary ∂Ω = Γ. Assuming that g is a function which admits an exponential growth at the infinity and, in addition, that h is a monotonic continuous increasing function with polynomial growth at the infinity, we prove both: global existence as well as blow up of solutions in finite time, by taking the initial data inside the potential well. Moreover, optimal and uniform decay rates of the energy are proved for global solutions.

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

Abstract: We study the existence of different types of positive solutions to problem $$\left\{\begin{array}{lll} -\Delta u - \lambda_1\dfrac{u}{|x|^2}-|u|^{2^*-2}u = u\,h(x)\alpha\,|u|^{\alpha-2}|v|^{\beta}u, &{\rm in}\,{\mathbb{R}}^{N},\\ &\qquad\qquad\qquad\qquad x \in {\mathbb{R}}^N,\quad N \geq 3,\\ -\Delta v - \lambda_2\dfrac{v}{|x|^2}-|v|^{2^*-2}v = u\,h(x)\beta\,|u|^{\alpha}|v|^{\beta-2}v, &{\rm in}\,{\mathbb{R}}^N, \end{array}\right.$$ where $${\lambda_1, \lambda_2 \in (0, \Lambda_N)}$$ , $${\Lambda_N := \frac{(N-2)^2}{4}}$$ , and $${2* = \frac{2N}{N-2}}$$ is the critical Sobolev exponent. A careful analysis of the behavior of Palais-Smale sequences is performed to recover compactness for some ranges of energy levels and to prove the existence of ground state solutions and mountain pass critical points of the associated functional on the Nehari manifold. A variational perturbative method is also used to study the existence of a non trivial manifold of positive solutions which bifurcates from the manifold of solutions to the uncoupled system corresponding to the unperturbed problem obtained for ν = 0.

C1 regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two

Abstract: We prove C1 regularity of c-convex weak Alexandrov solutions of a Monge–Ampere type equation in dimension two, assuming only a bound from above on the Monge–Ampere measure. The Monge–Ampere equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151–183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C1 regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge–Ampere equation det D2u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841–845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].

The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds

Abstract: Given (M, g) a smooth compact Riemannian N-manifold, N ≥ 2, we show that positive solutions to the problem $${-\varepsilon^2\Delta_g u + u = u^{p-1}\,{\rm in}\,M,}$$ are generated by stable critical points of the scalar curvature of g, provided $${\varepsilon}$$ is small enough. Here p > 2 if N = 2 and $${2 < p < 2^{*} = {2N \over N-2}}$$ if N ≥ 3.

Examples of area-minimizing surfaces in the sub-Riemannian Heisenberg group \({\mathbb{H}^1}\) with low regularity

Abstract: We give new examples of entire area-minimizing t-graphs in the sub-Riemannian Heisenberg group \({\mathbb{H}^1}\). They are locally Lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given point. Amongst them, a large family of area-minimizing cones is obtained.

Gamma convergence of an energy functional related to the fractional Laplacian

Abstract: We prove a Γ-convergence result for an energy functional related to some fractional powers of the Laplacian operator, (−Δ) s for 1/2 < s < 1, with two singular perturbations, that leads to a two-phase problem. The case (−Δ)1/2 was considered by Alberti–Bouchitte–Seppecher in relation to a model in capillarity with line tension effect. However, the proof in our setting requires some new ingredients such as the Caffarelli–Silvestre extension for the fractional Laplacian and new trace inequalities for weighted Sobolev spaces.

On the first eigenvalue of a fourth order Steklov problem

Abstract: We prove some results about the first Steklov eigenvalue d1 of the biharmonic operator in bounded domains Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains Next, we study the optimization of d1 for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems Finally, we prove several properties of the ball

Γ-convergence for incompressible elastic plates

Abstract: We derive a two-dimensional model for elastic plates as a Γ-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The resulting model describes plate bending, and is determined from the isochoric elastic moduli of the three-dimensional problem. Without the constraint of incompressibility, a plate theory was first derived by Friesecke et al. (Comm Pure Appl Math 55:1461–1506, 2002). We extend their result to the case of p growth at infinity with p ϵ [1, 2), and to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies the nonlinear constraint pointwise. One main ingredient is the density of smooth isometries in W 2,2 isometries, which was obtained by Pakzad (J Differ Geom 66:47–69, 2004) for convex domains and by Hornung (Comptes Rendus Mathematique 346:189–192, 2008) for piecewise C 1 domains.

On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

Abstract: In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g 0) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g 0) is not conformally equivalent to the unit sphere S n and that the set of all such metrics is compact. When (M, g 0) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2.

Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions

Abstract: We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation −div(D∇u) + div(V u) = f posed in a bounded domain Ω ⊂ RN , with pure Neumann boundary conditions D∇u * n = (V * n) u on ∂Ω. Under the assumption that V ∈ Lp (Ω)N with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u ∈ H 1 (Ω) if Ω f dx = 0, and also that the kernel is generated by a function u ∈ H 1 (Ω), unique up to a multiplicative constant, which satisfies u > 0 a.e. on Ω. We also prove that the equation −div(D∇u) + div(V u) + ν u = f has a unique solution for all ν > 0 and the map f → u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation −div(D T ∇v) − V * ∇v = g. The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.

Infinitely many solutions of some nonlinear variational equations

Abstract: The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem $$ {\bar J}(u)=\int \limits _\Omega {\bar A} (x,u)| abla u|^p dx - \int \limits _\Omega G(x,u) dx$$ in the Banach space $${W^{1,p}_0(\Omega) \cap L^\infty(\Omega)}$$ , being Ω a bounded domain in $${\mathbb {R}^N}$$ . In order to use “classical” theorems, a suitable variant of condition (C) is proved and $${W^{1,p}_0(\Omega)}$$ is decomposed according to a “good” sequence of finite dimensional subspaces.

Regularity of radial minimizers of reaction equations involving the p-Laplacian

Abstract: We consider semi-stable, radially symmetric, and decreasing solutions of − Δ p u = g(u) in the unit ball of $${\mathbb{R}^n}$$ , where p > 1, Δ p is the p-Laplace operator, and g is a locally Lipschitz function. For this class of radial solutions, which includes local minimizers, we establish pointwise, L q , and W 1,q estimates which are optimal and do not depend on the specific nonlinearity g. Among other results, we prove that every radially decreasing and semi-stable solution u belonging to W 1,p (B 1) is bounded whenever n 0 is a parameter, it is proved that the corresponding extremal solution u * is semi-stable, and hence, it enjoys the regularity stated in our main result.

The limit as p →∞ in a nonlocal p-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles

Abstract: In this paper, we study the nonlocal ∞-Laplacian type diffusion equation obtained as the limit as p → ∞ to the nonlocal analogous to the p-Laplacian evolution, $$u_t (t,x) = \int_{\mathbb{R}^N} J(x-y)|u(t,y) - u(t,x)|^{p-2}(u(t,y)- u(t,x)) \, dy.$$ We prove exist ence and uniqueness of a limit solution that verifies an equation governed by the subdifferential of a convex energy functional associated to the indicator function of the set \({K = \{ u \in L^2(\mathbb{R}^N) \, : \, | u(x) - u(y)| \le 1, \mbox{ when } x-y \in {\rm supp} (J)\}}\) . We also find some explicit examples of solutions to the limit equation. If the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞(0, T; L2 (Ω)) to the limit solution of the local evolutions of the p-Laplacian, vt = Δpv. This last limit problem has been proposed as a model to describe the formation of a sandpile. Moreover, we also analyze the collapse of the initial condition when it does not belong to K by means of a suitable rescale of the solution that describes the initial layer that appears for p large. Finally, we give an interpretation of the limit problem in terms of Monge–Kantorovich mass transport theory.

Further PDE methods for weak KAM theory

Abstract: We introduce and make estimates for several new approximations that in appropriate asymptotic limits yield the key PDE for weak KAM theory, namely a Hamilton–Jacobi type equation for a potential u and a coupled transport equation for a measure σ. We revisit as well a singular variational approximation introduced in Evans (Calc Vari Partial Differ Equ 17:159–177, 2003) and demonstrate “approximate integrability” of certain phase space dynamics related to the Hamiltonian flow. Other examples include a pair of strongly coupled PDE suggested by the Lions–Lasry theory (Lasry and Lions in Japan J Math 2:229–260, 2007) of mean field games and a new and extremely singular elliptic equation suggested by sup-norm variational theory.

Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation

Abstract: For the Neumann sinh-Gordon equation on the unit ball \({B \subset \mathbb {R}^2}\) $$\left\{ \begin{array}{ll} -\Delta u = \lambda^+ \left( \frac{e^u}{\int_B e^u}-\frac{1}{\pi} \right)-\lambda^- \left( \frac{e^{-u}}{\int_B e^{-u}}-\frac{1}{\pi} \right) & {\rm in}\,B\\ \frac{\partial u}{\partial u}=0 & {\rm on}\, \partial B \end{array} \right.$$ we construct sequence of solutions which exhibit a multiple blow up at the origin, where λ ± are positive parameters. It answers partially an open problem formulated in Jost et al. [Calc Var Partial Diff Equ 31(2):263–276].

Isoperimetric comparison theorems for manifolds with density

Abstract: We give several isoperimetric comparison theorems for manifolds with density, including a generalization of a comparison theorem from Bray and Morgan. We find for example that in the Euclidean plane with radial density exp(r α ) for α ≥ 2, discs about the origin minimize perimeter for given area, by comparison with Riemannian surfaces of revolution.

Positive mass theorem for the Paneitz–Branson operator

Abstract: We prove that under suitable assumptions, the constant term in the Green function of the Paneitz–Branson operator on a compact Riemannian manifold (M, g) is positive unless (M, g) is conformally diffeomorphic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann and Humbert (Geom Func Anal 15(3):567–576, 2005 [1]).

Linking solutions for quasilinear equations at critical growth involving the "1-Laplace" operator

Abstract: We show that the problem at critical growth, involving the 1-Laplace operator and obtained by relaxation of \({-\Delta_1 u=\lambda |u|^{-1}u+|u|^{1^*-2} u}\) , admits a nontrivial solution \({u \in BV(\Omega)}\) for any λ ≥ λ1. Nonstandard linking structures, for the associated functional, are recognized.

Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs

Abstract: We characterize intrinsic regular submanifolds in the Heisenberg group as intrinsic differentiable graphs.

A mixed problem for the infinity Laplacian via Tug-of-War games

Abstract: In this paper we prove that a function \({ u\in\mathcal{C}(\overline{\Omega})}\) is the continuous value of the Tug-of-War game described in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear) if and only if it is the unique viscosity solution to the infinity Laplacian with mixed boundary conditions $$\left\{ \begin{aligned}-\Delta_{\infty}u(x)=0 \quad & {\rm in} \, \Omega,\\ \frac{\partial u}{\partial n}(x)=0 \quad \quad & {\rm on} \, \Gamma_N,\\ u(x)=F(x) \quad & {\rm on}\, \Gamma_D. \end{aligned} \right.$$ By using the results in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear), it follows that this viscous PDE problem has a unique solution, which is the unique absolutely minimizing Lipschitz extension to the whole \({\overline{\Omega}}\) (in the sense of Aronsson (Ark. Mat. 6:551–561, 1967) and Y. Peres et al. (J. Am. Math. Soc., 2008, to appear)) of the Lipschitz boundary data \({F:\Gamma_D \to \mathbb R }\).