Showing papers in "Annali di Matematica Pura ed Applicata in 2008"

Renormalized solutions of nonlinear parabolic equations with general measure data

Abstract: Let $$\Omega\subseteq \mathbb{R}^n$$ a bounded open set, N ≥ 2, and let p > 1; we prove existence of a renormalized solution for parabolic problems whose model is $$\left\{ \begin{array}{lll} u_t - \Delta _p u = \mu &{\rm in}\,(0,T) \times \Omega , \\ u(0,x) = u_0 &{\rm in}\, \Omega , \\u(t,x) = 0 &{\rm on}\, (0,T) \times \partial \Omega, \\ \end{array} \right$$ where T > 0 is a positive constant, $$\mu\in M(Q)$$ is a measure with bounded variation over $$Q=(0,T) \times \Omega, u_o\in L^1(\Omega)$$ , and $$-\Delta_{p} u=-{\rm div} (| abla u|^{p-2} abla u )$$ is the usual p-Laplacian

Global regularity for almost minimizers of nonconvex variational problems

Abstract: We prove some global, up to the boundary of a domain \(\Omega \subset {\mathbb{R}}^{n}\) , continuity and Lipschitz regularity results for almost minimizers of functionals of the form $${\bf {\rm u}} \mapsto \int_{\Omega} g({\bf {\rm x}}, {\bf {\rm u}}({\bf {\rm x}}), abla{\bf {\rm u}}({\bf {\rm x}}))\,{\rm d}{\bf x}.$$ The main assumption for g is that it be asymptotically convex with respect its third argument. For the continuity results, the integrand is allowed to have some discontinuous behavior with respect to its first and second arguments. For the global Lipschitz regularity result, we require g to be Holder continuous with respect to its first two arguments.

A Dirichlet problem for polyharmonic functions

Abstract: In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given.

The critical hyperbola for a Hamiltonian elliptic system with weights

Abstract: In this paper we look for existence results for nontrivial solutions to the system, \( \left\{ {\begin{array}{*{20}c} { - \Updelta u = \frac{{v^{p} }}{{\left| x \right|^{\alpha } }}} & {{\text{in}}\,\Upomega ,} \\ { - \Updelta v = \frac{{u^{p} }}{{\left| x \right|^{\beta } }}} & {{\text{in}}\,\Upomega ,} \\ \end{array} } \right. \) with Dirichlet boundary conditions, u = v = 0 on ∂Ω and α, β < N. We find the existence of a critical hyperbola in the (p, q) plane (depending on α, β and N) below which there exists nontrivial solutions. For the proof we use a variational argument (a linking theorem).

From Brunn–Minkowski to sharp Sobolev inequalities

Abstract: We present a simple direct proof of the classical Sobolev inequality in \(\mathbb{R}^n\) with best constant from the geometric Brunn–Minkowski–Lusternik inequality.

Pseudo-conformal quaternionic CR structure on (4n + 3)-dimensional manifolds

Abstract: We study an integrable, nondegenerate codimension 3-subbundle $${\mathcal{D}}$$ on a (4n + 3)-manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space It is thought of as a generalization of quaternionic CR structure We single out an $${\mathfrak{s}}{\mathfrak{p}}(1)$$ -valued 1-form ω locally on a neighborhood U such that $${\rm Null}\omega = \mathcal D|U$$ and construct the curvature invariant on (M, ω) whose vanishing gives a uniformization to flat quaternionic CR geometry The invariant obtained on M has the same formula as that of pseudo-quaternionic Kahler 4n-manifolds From this viewpoint, we exhibit a quaternionic analogue of Chern-Moser’s CR structure

Estimates for the Sobolev trace constant with critical exponent and applications

Abstract: In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $$S\|u\|_{L^{p_*}(\partial\Omega)}^p \le \|u\|_{W^{1,p}(\Omega)}^p$$ that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760, 1991) for general p. Here p * : = p(N − 1)/(N − p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernandez Bonder and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.

Nonlinear equations with unbounded heat conduction and integrable data

Abstract: We consider a class of quasi-linear diffusion problems involving a matrix A(t,x,u) which blows up for a finite value m of the unknown u. Stationary and evolution equations are studied for L 1 data. We focus on the case where the solution u can reach the value m. For such problems we introduce a notion of renormalized solutions and we prove the existence of such solutions.

Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

Abstract: In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation − Δu = λ u − up in Ω, with the nonlinear boundary condition ∂u/∂ν = ur on ∂Ω. Here Ω is a smooth bounded domain of \({\mathbb{R}}^d\) with outward unit normal ν, λ is a real parameter and p, r > 0. We also give the precise behavior of solutions for large |λ| in the cases where they exist. The proofs are mainly based on bifurcation techniques, sub-supersolutions and variational methods.

Global stress regularity of convex and some nonconvex variational problems

Abstract: We derive a global regularity theorem for stress fields which correspond to minimizers of convex and some special nonconvex variational problems with mixed boundary conditions on admissible domains These are Lipschitz domains satisfying additional geometric conditions near those points, where the type of the boundary conditions changes In the first part it is assumed that the energy densities defining the variational problem are convex but not necessarily strictly convex and satisfy a convexity inequality The regularity result for this case is derived with a difference quotient technique In the second part the regularity results are carried over from the convex case to special nonconvex variational problems taking advantage of the relation between nonconvex variational problems and the corresponding (quasi-) convexified problems The results are applied amongst others to the variational problems for linear elasticity, the p-Laplace operator, Hencky elasto-plasticity with linear hardening and for scalar and vectorial two-well potentials (compatible case)

Invariant surfaces with constant mean curvature in {\mathbb{H}}^2 \times {\mathbb{R}}

Abstract: We classify the profile curves of all surfaces with constant mean curvature in the product space \({\mathbb{H}}^2 \times {\mathbb{R}}\) , which are invariant under the action of a 1-parameter subgroup of isometries.

The no-response approach and its relation to non-iterative methods for the inverse scattering

Abstract: This paper addresses the inverse obstacle scattering problem. In the recent years several non-iterative methods have been proposed to reconstruct obstacles (penetrable or impenetrable) from near or far field measurements. In the chronological order, we cite among others the linear sampling method, the factorization method, the probe method and the singular sources method. These methods use differently the measurements to detect the unknown obstacle and they require the use of many incident fields (i.e. the full or a part of the far field map). More recently, two other approaches have been added. They are the no-response test and the range test. Both of them use few incident fields to detect some informations about the scatterer. All the mentioned methods are based on building functions depending on some parameter. These functions share the property that their behaviors with respect to the parameter change drastically. The surface of the obstacle is located at most in the interface where these functions become large. The goal of this work is to investigate the relation between some of the non-iterative reconstruction schemes regarding the convergence issue. A given method is said to be convergent if it reconstructs a part or the entire obstacle by using few or many incident fields respectively. For simplicity we consider the obstacle reconstruction problem from far field data for the Helmholtz equation.

Anti-selfdual Lagrangians II: unbounded non self-adjoint operators and evolution equations

Abstract: New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non lineaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form \({-Au \in \partial\varphi(u)}\) as well ass dissipative initial value evolutions of the form \({-\dot{u}(t) - Au(t) + \omega{u}(t) \in \partial\varphi(l,u(l))}\) where \({\varphi}\) is a convex potential on an infinite dimensional space, A is a linear operator and \({\omega}\) is any scalar. The framework developed in the above mentioned paper reformulates these problems as \({0\in \bar{\partial}L(u)}\) and \({\dot{u}(t) \in \bar{\partial}L(t,u(t))}\) respectively, where \({\bar{\partial}L}\) is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term.

On the derivatives of the Lempert functions

Abstract: We show that if the Kobayashi–Royden metric of a complex manifold is continuous and positive at a given point and any non-zero tangent vector, then the “derivatives” of the higher order Lempert functions exist and equal the respective Kobayashi metrics at the point. It is a generalization of a result by M. Kobayashi for taut manifolds.

Bochner’s theorem and Stepanov almost periodic functions

Abstract: Motivated by a renewed interest in generalizations of classical almost periodicity (originally due to Harald Bohr), we develop a theorem of Bochner within the framework of almost periodic functions in the sense of Stepanov. As a result we establish some conditions that guarantee the existence of Stepanov almost periodic solutions to differential equations with Stepanov almost periodic coefficients. Finally, we extend a now classic theorem of Favard originally stated for classical almost periodic functions to the Stepanov almost periodic case.

Friction boundary conditions on thermal incompressible viscous flows

Abstract: This work adresses an unsteady heat flow problem involving friction and convective heat transfer behaviors on a part of the boundary. The problem is constituted by a variational motion inequality with energy dependent coefficients, and the energy equation in the framework of L1-theory for the dissipative term. Using the duality theory of convex analysis, it also envolves the existence of Lagrange multipliers. Weak solutions of an approximate coupled system are proven by a fixed point argument for multivalued mappings and compactness methods. Then the existence result for the initial coupled system is proven by the passage to the limit.

On the multiplicities of eigenvalues of a Hermitian matrix whose graph is a tree

Abstract: A different approach is given to recent results due mainly to R. C. Johnson and A. Leal Duarte on the multiplicities of eigenvalues of a Hermitian matrix whose graph is a tree. The techniques developed are based on some results of matching polynomials and used a work by O. L. Heilmann and E. H. Lieb on an apparently unrelated topic.

Lyapunov exponents of continuous Schrödinger cocycles over irrational rotations

Abstract: We consider the Lyapunov exponent of those continuous SL\((2,\mathbb{R})\)-valued cocycles over irrational rotations that appear in the study of Schrodinger operators and prove generic results related to large coupling asymptotics and uniform convergence.

A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II

Abstract: In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197–220, 2008) where we developed a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory. There we proved local monotonicity formulae for solutions $${{u^\varepsilon}}$$ to the singular perturbation problem and for $${u=\lim{u^\varepsilon}}$$ , assuming that both $${{u^\varepsilon}}$$ and u were defined in an arbitrary domain $${\mathcal{D}}$$ in $${\mathbb{R}^{N+1}}$$ . In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while $${{u^\varepsilon}}$$ are not. We derive such global formulae from a local one that we prove here. In particular, we obtain a global monotonicity formula for blow up limits u 0 of limit functions u that are not globally defined. As a consequence of this formula, we characterize blow up limits u 0 in terms of the value of a density at the blow up point. We also present applications of the results in this paper to the study of the regularity of ∂{u > 0} (the flame front in combustion models). The fact that our results hold for the inhomogeneous singular perturbation problem allows a very wide applicability, for instance to problems with nonlocal diffusion and/or transport.

Koksma–Hlawka type inequalities of fractional order

Abstract: The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order $${\alpha > 0}$$ is in $${\mathcal{L}_p}$$ . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality to functions whose partial fractional derivatives are in $${\mathcal{L}_p}$$ . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.

Degenerate resonances and branching of periodic orbits

Abstract: In this paper we establish results on the existence of Lyapunov families of periodic orbits of reversible systems in \(\mathbb{R}^6\) around an equilibrium that presents a 0:1:1-resonance. The main proofs are based on a combined use of normal form theory, Lyapunov–Schmidt reduction and elements of symbolic computation.

Analytic non-integrable Hamiltonian systems and irregular singularity

Abstract: We study a C ∞-Liouville-integrable and analytic non-integrable Hamiltonian system We will show that an irregular singular character plays a crucial role in the analytic non-integrability of the system

A spectral approach to ill-posed problems for wave equations

Abstract: In this article, we consider the problem of finding a solution to ill-posed problems for abstract wave equations in a Hilbert space, of the form $$\frac{{\rm d}^{2}u}{{\rm d}t^{2}}\left( t\right) + Au \left(t\right) = 0,\quad t \in \left(0, T\right),\quad u \left(0\right) = 0,\quad u \left(T\right) = u_{0},$$ when A is a general linear selfadjoint operator. We study issues like existence, uniqueness and continuance dependance of data and stability for this problem. Under precise constraint conditions on T, we make such problems well posed and in effect, generalize known results about these equations.

Analytic general solutions of nonlinear difference equations

Abstract: There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions of the following nonlinear second order difference equation, $$u(t+2)=f(u(t),u(t+1)),$$ where f(x,y) is an entire function of x, y. The main work of the present paper is obtaining representations of analytic general solutions of the difference equation with new methods of complex analysis.

Linear elliptic problems with non-H−1 data and pathological solutions

Abstract: In the theory of linear elliptic problems with data not belonging to H −1 two cases can be distinguished. When the right hand side in the equation is a summable function we point out that the a priori estimates can be attained very quickly by symmetrization methods. On the other side, when the datum includes a distributional term, different and subtler tools have to be used. We deal with an equation in the plane, whose right hand side is a functional on a space of Holder continuous functions with a suitable exponent. We obtain a priori bounds via duality arguments; these, in addition, show Serrin pathological solution (see Serrin in Ann. Scuola Norm. Sup. Pisa 18(3), 385–387, 1964) in its true light.

Submanifolds of positive Ricci curvature in a Euclidean space

Abstract: In this paper we study the role of constant vector fields on a Euclidean space R n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R n+p with mean curvature vector field H and a constant vector field $${\xi }$$ on R n+p, the smooth function $${\varphi =\left\langle H,\xi \right\rangle }$$ is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).

A higher integrability result for nondivergence elliptic equations

Abstract: The aim of this paper is to establish a higher integrability result of the second derivatives of solutions to nondivergence elliptic equations of the type � n i,j aij ∂ 2 u ∂xi∂xj = h. We assume that the coefficients aij are bounded and have small BMO-norm.

Isometric deformations of surfaces preserving the third fundamental form

Abstract: We study the following problem: To what extend is a surface in the Euclidean space \(\mathbb{R}^{4}\) determined by the third fundamental form? We prove the existence of families of surfaces in \(\mathbb{R}^{4}\) which allow isometric deformations with isometric but not congruent Gaussian images. In particular, we provide a method which gives locally all surfaces in \(\mathbb{ R}^{4}\) with conformal Gauss map that allow such deformations. As a consequence, we have a way for constructing non-spherical pseudoumbilical surfaces in \(\mathbb{R}^{4}.\)

On t 2 -like solutions of certain second order differential equations

Abstract: Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation $${\frac{{\rm d}^{2}x}{{\rm d}t^{2}}=a(t)x^{\lambda}}$$ , t ≥ t 0 > 0, with solutions x such that $${x(t)\sim c\cdot t^{2}}$$ as $${t\rightarrow +\infty}$$ , $${c ot= 0}$$ , and the equation $${\frac{{\rm d}^{2}y}{{\rm d}u^{2}}=b(u)e^{y}}$$ , u > 0, with solutions y such that $${\lim sup_{u\searrow k}\frac{y(u)}{\ln (u-k)}=c_{1} < 0}$$ for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation $${\Delta U=K(\left| x\right|)e^U}$$ , $${\left| x\right| > x_0 > 0}$$ , that blow up as $${\left| x\right|\searrow x_0}$$ in the two dimensional case.

A class of one-dimensional geometric variational problems with energy involving the curvature and its derivatives: a geometric measure-theoretic approach

Abstract: The notions of Legendrian and Gaussian towers are defined and investigated. Then applications in the context of one-dimensional geometric variational problems with the energy involving the curvature and its derivatives are provided. Particular attention is paid to the case when the functional is defined on smooth boundaries of plane sets.