Nonpositive Constant Research Articles

Gamma-convergence of quadratic functionals perturbed by bounded linear functionals

Given a bounded open set Ω⊂Rn, we study sequences of quadratic functionals on the Sobolev space H01(Ω), perturbed by sequences of bounded linear functionals. We prove that their Γ-limits, in the weak topology of H01(Ω), can always be written as the sum of a quadratic functional, a linear functional, and a non-positive constant. The classical theory of G- and H-convergence completely characterises the quadratic and linear parts of the Γ-limit and shows that their coefficients do not depend on Ω. The constant, which instead depends on Ω and will be denoted by −ν(Ω), plays an important role in the study of the limit behaviour of the energies of the solutions. The main result of this paper is that, passing to a subsequence, we can prove that ν coincides with a non-negative Radon measure on a sufficiently large collection of bounded open sets Ω. Moreover, we exhibit an example that shows that the previous result cannot be obtained for every bounded open set. The specific form of this example shows that the compactness theorem for the localisation method in Γ-convergence cannot be easily improved.

Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $$\Sigma $$ with non-positive Yamabe invariant in a Riemannian n-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that $$\Sigma $$ is locally volume-minimizing in a manifold $$M^n$$ with scalar curvature bounded from below by a non-positive constant and mean convex boundary, we conclude that locally M splits along $$\Sigma $$ . In the case that the scalar curvature of M is at least $$-n(n-1)$$ and $$\Sigma $$ locally minimizes a certain functional inspired by works of Yau [35] and Andersson-Galloway [4], a neighbourhood of $$\Sigma $$ in M is isometric to $$((-\varepsilon , \varepsilon ) \times \Sigma ,\mathrm{{d}}t^{2}+e^{2t}g)$$ , where g is Ricci flat with totally geodesic boundary.

Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces

New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry–Émery Ricci curvature, the Escobar–Lichnerowicz–Reilly type estimates are proved; under the assumption of nonnegative ∞-Bakry–Émery Ricci curvature and the m-Bakry–Émery Ricci curvature bounded from below by a non-positive constant, the Li–Yau type lower bound estimates are given. The weighted p-Bochner formula and the weighted p-Reilly formula are derived as the key tools for the establishment of the above results.

Gradient estimates for doubly nonlinear diffusion equations

In this short note we obtain some gradient estimates for the positive solution to the doubly nonlinear diffusion equation on closed Riemannian manifold with Ricci curvature bounded below by a non-positive constant. As applications, we also derive corresponding Harnack inequalities.

Diffeomorphic theorems for open Riemannian manifolds with curvature decay

In this paper, we study the topology of complete non-compact Riemannian manifolds with curvature decay to a non-positive constant. We show that such a complete open manifold M is diffeomorphic to a Euclidean n-space Rn if it contains enough rays starting from the base point. As applications, we also show that this kind of manifolds with Ricci curvature bounded from below by a non-positive constant are diffeomorphic to Rn if the volumes of geodesic balls in M grow properly. Our results generalize the main theorems of Wang�Xia for manifolds with quadratic curvature decay to zero.

Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds

Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation \[ u_t=\Delta u-\nabla\phi\cdot\nabla u-au\log u-bu, \] where $\phi$ is a $C^2$ function, and $a\neq0$ and $b$ are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).

Rigidity of Compact Riemannian Spin Manifolds with Boundary

In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a nonpositive constant. In particular, we obtain generalizations of a result of Hang–Wang (Pac J Math 232(2):283–288, 2007) based on a conjecture of Schroeder and Strake (Comment Math Helv 64:173–186, 1989).

Rigidity of surfaces whose geodesic flows preserve smooth foliations of codimension 1

Let S S be a closed orientable surface. Assume that there exists a codimension one foliation F \mathcal F of class C 3 C^3 in the unit tangent bundle of S S , whose leaves are invariant under the geodesic flow of S S . Then, the curvature of S S is a nonpositive constant.

On the Essential Spectrum of the Laplacian and Vague Convergence of the Curvature at Infinity

We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval [(n − 1)2 K/4, ∞) if an n-dimensional Rieman-nian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant (n − 1)K converges to zero at infinity.

Geometry of generalized Einstein manifolds

A formula linking the horizontal Laplacian Δ ¯ φ of a function φ on the fibre bundle W of unitary tangent vectors to a Finslerian compact manifold without boundary ( M , g ) , to the square of a symmetric 2-tensor and Finslerian curvature. From it an estimate, under a certain condition, is obtained for the function λ : Δ ¯ φ = λ φ . If λ = n k where k is a positive constant and M simply connected, then M is homeomorphic to an n-sphere. Let F ○ ( g t ) be a deformation of ( M , g ) preserving the volume of W. One proves that the critical points g 0 ∈ F ○ ( g t ) of the integral I ( g t ) of a certain Finslerian scalar curvature on W define a generalized Einstein manifold. One calculates the second variationals at the critical points first in the general case, then, for an infinitesimal conformal deformation and one shows that in certain cases one has I ″ ( g 0 ) ⩾ 0 . We also study the case when the scalar curvature is non-positive constant. To cite this article: H. Akbar-Zadeh, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Integration by parts on the Brownian Meander

We prove infinite-dimensional integration by parts formulae for the laws of the Brownian Meander, of the Bessel Bridge of dimension 3 between z , z ′ ≥ 0 z,z’\geq 0 and of the Brownian Motion on the set of all paths taking values greater than or equal to a nonpositive constant. We give applications to SPDEs with reflection.

Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds

Given (M,g) a smooth compact Riemannian n-manifold, n > 3, we return in this article to the study of the sharp Sobolev-Poincare type inequality (0.1) ∥u∥ 2 2* ≤ K 2 n ∥Δ∥u∥ 2 2 + B∥u∥ 2 1 where 2 * = 2n/(n - 2) is the critical Sobolev exponent, and K n is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that (0.1) is true if n = 3, that (0.1) is true if n > 4 and the sectional curvature of g is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension n is true and the sectional curvature of g is nonpositive, but that (0.1) is false if n > 4 and the scalar curvature of g is positive somewhere. When (0.1) is true, we define B(g) as the smallest B in (0.1). The saturated form of (0.1) reads as (0.2) ∥u∥ 2 2* ≤ K 2 n ∥Δu∥ 2 2 + B(g)∥u∥ 2 1 . We assume in this article that n > 4, and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincare inequality (0.1). We prove that (0.1) is true, and that (0.2) possesses extremal functions when the scalar curvature of g is negative. A fairly complete answer to the question of the validity of (0.1) under the assumption that the scalar curvature is not necessarily negative, but only nonpositive, is also given.

Isoperimetric Inequalities for Extrinsic Balls in Minimal Submanifolds and their Applications

S.-Y. Cheng, P. Li and S.-T. Yau proved comparison theorems for the volume of extrinsic balls in minimal submanifolds of space forms. These results were extended by S. Markvorsen for minimal submanifolds of a riemannian manifold with just an upper bound on the sectional curvature. In the paper an isoperimetric inequality for extrinsic balls in minimal submanifolds of a riemannian manifold N with sectional curvatures bounded from above by a non-positive constant is found. As a corollary of this result an alternative proof is obtained of the comparison for the volume of extrinsic balls stated by the preceding authors, but now the equality case is characterized when the upper bound for the sectional curvatures of the ambient manifold is strictly negative. Finally, when the sectional curvatures of N are bounded from above for any constant (positive or negative), it is proved that the ∞-isoperimetric quotient of the extrinsic balls is bounded from below by the mean curvature of the geodesic spheres in the m-dimensional real space forms.