Lagrangian mean curvature flow for entire Lipschitz graphs

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in ${{\mathbb R}^{2n}}$ , we show that the parabolic Eq. 1.1 has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t?=?0. In particular, under the mean curvature flow (1.2) the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as t → ∞. Our assumption on the Lipschitz norm is equivalent to the underlying Lagrangian potential u being uniformly convex with its Hessian bounded in L ^∞. As an application of this result we provide conditions under which an entire Lipschitz Lagrangian graph converges after rescaling to a self-expanding solution to the mean curvature flow.     

The graphs of Lipschitz functions and minimal surfaces on Carnot groups

We study and solve a new problem for the class of Lipschitz mappings (with respect to sub-Riemannian metrics) on Carnot groups. We introduce the new concept of graph for the functions on a Carnot group, and then the new concept of sub-Riemannian differentiability generalizing hc-differentiability. We prove that the mapping-??graphs?? are almost everywhere differentiable in the new sense. For these mappings we define a concept of intrinsic measure and obtain an area formula for calculating this measure. By way of application, we find necessary and sufficient conditions on the class of surface-??graphs?? under which they are minimal surfaces (with respect to the intrinsic measure of a surface).     

Mass transportation and rough curvature bounds for discrete spaces

We introduce and study rough (approximate) lower curvature bounds for discrete spaces and for graphs. This notion agrees with the one introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press] and [K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65-131], in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature ?K will have curvature ?K in the sense of [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65-131]. Moreover, in the converse direction, discretizations of metric measure spaces with curvature ?K will have rough curvature ?K. We apply our results to concrete examples of homogeneous planar graphs.     

Compactness and gradient bounds for solutions of the mean curvature system in two independent variables

We establish here a priori estimates for the gradient of solutions of the minimal surface system in two independent variables and for the curvature of their graphs. With the intent of extending these results to graphs with nonzero mean curvature vectors, we then analyze the compactness properties of smooth (C ²) solutions of the mean curvature system. Using a geometric measure theory approach we are able to classify the possible behaviors of a sequence {u _?(x)} of such solutions, under the assumption that a uniform bound on the area of the graphs holds and suitable hypotheses on the length of the mean curvature vectorH(x). In particular, this implies the existence of an a priori gradient bound depending on the oscillation of the solutionu(x).     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.     

On the Dirichlet problem for the prescribed mean curvature equation over general domains

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.     

Generalized gradients of monotone type

We examine the properties of the subdifferential in the sense of Clarke of certain locally Lipschitz, quasi-convex functions. We prove that, even if they may not possess a pseudomonotone-type subdifferential, if we consider the operator A+∂f, where A is an operator of type (S)₊, then the sum is pseudomonotone. A new type of subdifferential for Lipschitz functions is also presented. We prove some calculus rules and we establish that in the context of reflexive Banach spaces is an operator of type (M).     

On the splitting problem for selections

We investigate when does the Repovš-Semenov splitting problem for selections have an affirmative solution for continuous set-valued mappings in finite-dimensional Banach spaces. We prove that this happens when images of set-valued mappings or even their graphs are P-sets (in the sense of Balashov) or strictly convex sets. We also consider an example which shows that there is no affirmative solution of this problem even in the simplest case in R³. We also obtain affirmative solution of the approximate splitting problem for Lipschitz continuous selections in the Hilbert space.     

Diseguaglianza di Harnack per equazioni quasilineari ellittiche sopra superficie di curvatura media assegnata

Summary In this paper we prove a Harnack type inequality for non-negative solutions and supersolutions of second order quasilinear elliptic equations on hypersurfaces (inR ⁿ) of L^p prescribed mean curvature, with p>n. In the last section an application to non-parametric surfaces of Lipschitz mean curvature is given.

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Entrata in Redazione il 13 giugno 1977.     

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if M ⁿ is simply connected and the k-th Ricci curvature of M ⁿ is bounded below by a quantity involving the mean curvature of M ⁿ and the curvature of the ambient manifold, then M ⁿ is diffeomorphic to the standard sphere ${\mathbb{S}^n}$ . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold M ⁿ is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.     

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

Invariant measures for dichotomous stochastic differential equations in Hilbert spaces

We study existence of invariant measures for semilinear stochastic differential equations in Hilbert spaces. We consider infinite dimensional noise that is white in time and colored in space and we assume that the nonlinearities are Lipschitz continuous. We show that if the equation is dichotomous in the sense that the semigroup generated by the linear part is hyperbolic and the Lipschitz constants of the nonlinearities are not too large, then existence of a solution with bounded mean squares implies existence of an invariant measure. To cite this article: O. Van Gaans, S. Verduyn Lunel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1083–1088.     

On extrinsic symmetric spaces with zero mean curvature in Minkowski space-time

For an extrinsic symmetric space M in Minkowski space-time, we prove that if M is spacelike with zero mean curvature, then it is totally geodesic and if M is timelike with zero mean curvature, then it is totally geodesic or it is a flat hypersurface.     

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.     

Regularity of the free boundary in two-phase problems for linear elliptic operators

In this paper we complete the study of the regularity of the free boundary in two-phase problems for linear elliptic operators started in [M.C. Cerutti, F. Ferrari, S. Salsa, Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are C1,γ, Arch. Ration. Mech. Anal. 171 (2004) 329-348]. In particular we prove that Lipschitz and flat free boundaries (in a suitable sense) are smooth. As byproduct, we prove that Lipschitz free boundaries are smooth in the case of quasilinear operators of the form div(A(x,u)∇u) with Lipschitz coefficients.     

Harmonic forms on manifolds with non-negative Bakry–Émery–Ricci curvature

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry–Émery–Ricci curvature if the space of weighted L ² harmonic one-forms is non-trivial, then the weighted volume of the manifold is finite and the universal cover of the manifold splits isometrically as the product of the real line with a hypersurface.     

On the recovery of a curve isometrically immersed in a Euclidean space

It is known from differential geometry that one can reconstruct a curve with n?1 prescribed curvature functions, if these functions can be differentiated a certain number of times in the usual sense and if the first n?2 functions are strictly positive. We establish here that this result still holds under the assumption that the curvature functions belong to some Sobolev spaces, by using the notion of derivative in the distributional sense. We also show that the mapping that associates with such prescribed curvature functions the reconstructed curve is of class $\text{C}{}^{\mathrm{\infty }}$. To cite this article: M. Szopos, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A Two-Phase Parabolic Free Boundary Problem with Coefficients Below the Lipschitz Threshold

We study the regularity of a parabolic free boundary problem of two-phase type with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the optimal ${C_x^{1,1}\cap C_t^{0,1}}$ -regularity of the solution and that the free boundary is, near the so-called branching points, the union of two graphs that are Lipschitz in time and C ¹ in space. In our case, the same monotonicity formula does not apply in the same way. Instead we use scaling arguments similar to the ones used for the elliptic case in Edquist et al. (Ann Inst Henri Poincareé, Anal Non Linéaire 26(6):2359?C2372, 2009) to prove the optimal regularity. However, whenever the spatial gradient does not vanish on the free boundary, we are in the parabolic setting faced with some extra difficulties, that forces us to strain our assumptions slightly.     

The prescribed mean curvature equation for nonparametric surfaces

We study the prescribed mean curvature equation for nonparametric surfaces, obtaining existence and uniqueness results in the Sobolev space W^2,p. We also prove that under appropriate conditions the set of surfaces of mean curvature H is a connected subset of W^2,p. Moreover, we obtain existence results for a boundary value problem which generalizes the one-dimensional periodic problem for the mean curvature equation.