Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Maximizing the L∞ norm of the gradient of solutions to the Poisson equation

We find the maximum of ¦Du _f ¦ _L∞ when u_f is the solution, which vanishes at infinity, of the Poisson equation Δu =f on ? ⁿ in terms of the decreasing rearrangement off. Hence, we derive sharp estimates for ¦Du _f ¦ _L∞ in terms of suitable Lorentz orL ^p norms off. We also solve the problem of maximizing ¦Du _f ^B (0)¦ whenu _f ^B is the solution, vanishing on?B, to the Poisson equation in a ballB centered at 0 and the decreasing rearrangement off is assigned.     

A gradient estimate in the Calabi–Yau theorem

We prove a C ¹-estimate for the complex Monge–Ampère equation on a compact Kähler manifold directly from the C ⁰-estimate, without using a C ²-estimate. This was earlier done only under additional assumption of non-negative bisectional curvature.     

On the sufficient descent property of the Shanno’s conjugate gradient method

Satisfying in the sufficient descent condition is a strength of a conjugate gradient method. Here, it is shown that under the Wolfe line search conditions the search directions generated by the memoryless BFGS conjugate gradient algorithm proposed by Shanno satisfy the sufficient descent condition for uniformly convex functions.     

On the ?ojasiewicz-Simon gradient inequality

We prove a general version of the ?ojasiewicz-Simon inequality, and we show how to apply the abstract result to study energy functionals E of the form

     

The Lewy–Stampacchia inequality for the obstacle problem with quadratic growth in the gradient

In this paper we prove that at least one solution of the obstacle problem for a linear elliptic operator perturbed by a nonlinearity having quadratic growth in the gradient satisfies the Lewy–Stampacchia inequality.

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Résumé Nous démontrons dans cet article qu’au moins une solution du problème de l’obstacle pour un opérateur elliptique linéaire perturbé par une non linéarité à croissance quadratique par rapport au gradient vérifie l’inégalité de Lewy–Stampacchia.

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Mathematics Subject Classification (2000) 35J85, 47J20     

Existence and symmetry for elliptic equations in ℝn with arbitrary growth in the gradient

We define the Polish space R of non-degenerate rank-1 systems. Each non-degenerate rank-1 system can be viewed as a measure-preserving transformation of an atomless, σ-finite measure space and as a homeomorphism of a Cantor space. We completely characterize when two non-degenerate rank-1 systems are topologically isomorphic. We also analyze the complexity of the topological isomorphism relation on R, showing that it is \({F_\sigma }\) as a subset of R× R and bi-reducible to E₀. We also explicitly describe when a non-degenerate rank-1 system is topologically isomorphic to its inverse.     

Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient

It is proved that if Ω ⊂ Rⁿ {R^n}  is a bounded Lipschitz domain, then the inequality || u ||₁ \leqslant c(n)\textdiam( W)ò_W | e^D(u) | {\left\| u \right\|_1} \leqslant c(n){\text{diam}}\left( \Omega \right)\int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} is valid for functions of bounded deformation vanishing on ∂Ω. Here e^D(u) {\varepsilon^D}(u) denotes the deviatoric part of the symmetric gradient and ò_W | e^D(u) | \int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} stands for the total variation of the tensor-valued measure e^D(u) {\varepsilon^D}(u) . Further results concern possible extensions of this Poincaré-type inequality. Bibliography: 27 titles.     

A hybrid conjugate gradient method based on a quadratic relaxation of the Dai–Yuan hybrid conjugate gradient parameter

To take advantage of the attractive features of the Hestenes–Stiefel and Dai–Yuan conjugate gradient (CG) methods, we suggest a hybridization of these methods using a quadratic relaxation of a hybrid CG parameter proposed by Dai and Yuan. In the proposed method, the hybridization parameter is computed based on a conjugacy condition. Under proper conditions, we show that our method is globally convergent for uniformly convex functions. We give a numerical comparison of the implementations of our method and two efficient hybrid CG methods proposed by Dai and Yuan using a set of unconstrained optimization test problems from the CUTEr collection. Numerical results show the efficiency of the proposed hybrid CG method in the sense of the performance profile introduced by Dolan and Moré.     

Lower bounds for the scalar curvature of noncompact gradient solitons of List’s flow

In this paper, we show that a noncompact gradient shrinking soliton to the List flow has at most quadratic decay for ${S = R-\alpha_n| \nabla \varphi|^2}$ . Moreover, we prove a similar result for certain noncompact gradient steady solitons to the List flow. These generalize the results of Chow et al. [4].     

Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient

The aim of this work is to prove \({\mathcal{C}^{1,\gamma}}\) regularity up to the boundary for solutions of some fully nonlinear degenerate elliptic equations with a “sublinear” Hamiltonian term.     

𝒞1 continuous discretization of gradient elasticity

For the modeling of size effects, gradient continua can be applied. In this contribution, strain gradients are used in a nonlinear hyperelastic material model. The resulting partial differential equations are solved numerically in terms of a Galerkin scheme requiring 𝒞¹ continuity of the shape functions. The performance of three 𝒞¹ continuous finite elements and the 𝒞¹ Natural Element method is compared. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

S. Bernstein’s idea for bounding the gradient of solutions to the quasi-linear Dirichlet problem

S. Bernstein’s idea is outlined to estimate ${\| u_{x} \|_{0,0,\Omega}}$ of a solution u(x) to the quasi-linear elliptic differential equation, especially in the case of n > 2. Up to about 1956, investigations of nonlinear problems have been dealing with the case of n = 2. A large number of methods were developed, but none of them were applicable for n > 2. The maximum-minimum principle had been a powerful tool to find bounds, however in the case of n > 2 this tool is not available for the gradient of the solution. In 1956, Cordes [Math. Ann. 130 (1956), 278–312] gave estimates for the case n > 2. Later, Ladyzhenskaya and Ural’tseva [Linear and Quasilinear Elliptic Equations. Academic Press, New York, 1968] established estimates in the Sobolev space ${W^{2}_{2} (\Omega)}$ . In their line of reasoning they used the idea of S. Bernstein and investigated the transformed function υ(x) with u(x) = ?(υ(x)). In this paper, we give a more simpler proof of those results in the classical Banach space C _2,α (Ω).     

On the sphericity of the subgroup PSL2(ℝ) in the group of diffeomorphisms of the circle

It is shown that the group PSL₂(?) is a spherical subgroup in the group of C³-diffeomorphisms of the circle.     

Correction to «Holder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions»

Summary In the proof of [1, Theorem 1.2] an incorrect assertion was made. Using the notation of that work, we claimed that [D(u)–v] is finite, which may not be true. We give here a correct, simpler proof of the theorem.     

On the constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem

We study the rate of convergence in von Neumann’s ergodic theorem. We obtain constants connecting the power rate of convergence of ergodic means and the power singularity at zero of the spectral measure of the corresponding dynamical system (these concepts are equivalent to each other). All the results of the paper have obvious exact analogs for wide-sense stationary stochastic processes.     

Global indeterminacy of the equilibrium in the Chamley model of endogenous growth in the vicinity of a Bogdanov–Takens bifurcation

This paper studies the dynamics implied by the Chamley (1993) model, a variant of the two-sector model with an implicit characterization of the learning function. We first show that under some “regularity” conditions regarding the learning function, the model has (a) one steady state, (b) no steady states or (c) two steady states (one saddle and one non-saddle). Moreover, via the Bogdanov–Takens theorem, we prove that for critical regions of the parameters space, the dynamics undergoes a particular global phenomenon, namely the homoclinic bifurcation. Because these findings imply the existence of a continuum of equilibrium trajectories, all departing from the same initial value of the predetermined variable, the model exhibits global indeterminacy.