Showing papers in "Calculus of Variations and Partial Differential Equations in 2005"

Singular limits in liouville-type equations

Abstract: We consider the boundary value problem $ \Delta u + \varepsilon ^{2} k{\left( x \right)}e^{u} = 0$ in a bounded, smooth domain $\Omega$ in $ \mathbb{R}^{{\text{2}}} $ with homogeneous Dirichlet boundary conditions. Here $$ \varepsilon > 0,k(x) $$ is a non-negative, not identically zero function. We find conditions under which there exists a solution $ u_{\varepsilon } $ which blows up at exactly m points as $ \varepsilon \to 0 $ and satisfies $ \varepsilon ^{2} {\int_\Omega {ke^{{u_{\varepsilon } }} \to 8m\pi } }% $ . In particular, we find that if $k\in C^2(\bar\Omega)$ , $ \inf _{\Omega } k > 0 $ and $\Omega$ is not simply connected then such a solution exists for any given $m \ge 1$

Existence results for energetic models for rate-independent systems

Abstract: We consider mechanical models which are driven by an external loading on a time scale much slower than any internal time scale (like viscous relaxation times) but still much faster than the time needed to find the thermo-dynamical equilibrium. Typical phenomena involve dry friction, elasto-plasticity, certain hysteresis models for shape-memory alloys and quasistatic delamination or fracture. The main feature is the rate-independency of the system response, which means that a loading with twice (or half) the speed will lead to a response with exactly twice (or half) the speed. We refer to [BrS96, KrP89, Vis94, Mon93] for approaches to these phenomena involving either differential inclusions or abstract hysteresis operators. Our method is different, as we avoid time derivatives and use energy principles instead. As is well-known from dry friction, such systems will not necessarily relax into a complete equilibrium, since friction forces do not tend to 0 for vanishing velocities. One way to explain this phenomenon on a purely energetic basis is via so-called “wiggly energies”, where the macroscopic energy functional has a super-imposed fluctuating part with many local minimizers. Only after reaching a certain activation energy it is possible to leave these local minima and generate macroscopic changes, cf. [ACJ96, Jam96, Men02]. Here we use a different approach which involves a dissipation distance which locally behaves homogeneous of degree 1, in contrast to viscous dissipation which is homogeneous of degree 2. This approach was introduced in [MiT99, MiT03, MTL02, GMH02] for models for shape-memory alloys and is now generalized to many other rate-independent systems. See [Mie03a] for a general setup for rate-independent material models in the framework of “standard generalized materials”. To be more specific we consider the following continuum mechanical model. Let Ω ⊂ R be the undeformed body and t ∈ [0, T ] the slow process time. The deformation or displacement φ(t) : Ω → R is considered to lie in the space F of admissible deformations containing suitable Dirichlet boundary conditions. The internal variable z(t) : Ω → Z ⊂ R describes the internal state which may involve plastic deformations, hardening variables, magnetization or phase indicators. The elastic (Gibbs) stored energy is given

PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians

Abstract: We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus \(\mathbb{T}^N\)) just coercive, continuous and quasiconvex, for which a Hamiltonian flow cannot necessarily be defined. The analysis is focused on the family of Hamilton-Jacobi equations \(H(x,Du) = a\) with a real parameter, and in particular on the unique equation of the family, corresponding to the so-called critical value a = c, for which there is a viscosity solution on \(\mathbb{T}^N\). We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians.

Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures

Abstract: We define a notion of quasistatic evolution for the elliptic approximation of the Mumford-Shah functional proposed by Ambrosio and Tortorelli. Then we prove that this regular evolution converges to a quasi static growth of brittle fractures in linearly elastic bodies.

Non radial positive solutions for the Hénon equation with critical growth

Abstract: We study the Dirichlet problem in a ball for the Henon equation with critical growth and we establish, under some conditions, the existence of a positive, non radial solution. The solution is obtained as a minimizer of the quotient functional associated to the problem restricted to appropriate subspaces of H 0 1 invariant for the action of a subgroup of ${\bf O}(N)$ . Analysis of compactness properties of minimizing sequences and careful level estimates are the main ingredients of the proof.

On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formulae

Abstract: In this paper we give an unified approach to some questions arising in different fields of nonlinear analysis, namely: (a) the study of the structure of the Fucik spectrum and (b) possible variants and extensions of the monotonicity formula by Alt-Caffarelli-Friedman [1]. In the first part of the paper we present a class of optimal partition problems involving the first eigenvalue of the Laplace operator. Beside establishing the existence of the optimal partition, we develop a theory for the extremality conditions and the regularity of minimizers. As a first application of this approach, we give a new variational characterization of the first curve of the Fucik spectrum for the Laplacian, promptly adapted to more general operators. In the second part we prove a monotonicity formula in the case of many subharmonic components and we give an extension to solutions of a class of reaction-diffusion equation, providing some Liouville-type theorems.

Compactness of solutions to the Yamabe problem. II

Abstract: We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.

A minimum problem with free boundary for a degenerate quasilinear operator

Abstract: In this paper we prove $C^{1,\alpha}$ regularity (near flat points) of the free boundary $\partial\{u > 0\}\cap\Omega$ in the Alt-Caffarelli type minimum problem for the p-Laplace operator: $$J(u)=\int_\Omega\left( | abla u|^p + \lambda^p\chi_{\{u>0\}}\right)dx\rightarrow \min\qquad (1

Infinitely many bound states for some nonlinear scalar field equations

Abstract: In this paper we consider the problem \(-\Delta u + a(x)u = \vert u\vert ^{p-2}u\) in \(\mathbb{R}^N\), where p > 2 and \(p 2. Assuming that the potential a(x) is a regular function such that \(\liminf_{\vert x\vert\rightarrow + \infty} a(x) = a_\infty > 0\) and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.

Generalized Cheeger sets related to landslides

Abstract: We study the maximization problem, among all subsets X of a given domain $\Omega$ , of the quotient of the integral in X of a given function f by the integral on the boundary of X of another function g. This is a generalization of the well-known Cheeger problem corresponding to constant functions f,g. The non-constant case is motivated by applications to landslides modeling where the the supremum given by a variational blocking problem appears as a safety coefficient. We prove that this coefficient is equal to the supremum of the shape optimization problem formerly mentioned. For constant data, this amounts to studying the first eigenvalue of the 1-laplacian operator. We prove existence of optimal sets, and give some differential characterization of their internal boundary. We study their symmetry properties using the Steiner symmetrization. In dimension two, we give explicit solutions for data depending only on one variable.

C 1,α -solutions to non-autonomous anisotropic variational problems

Abstract: We establish several smoothness results for local minimizers of non-autonomous variational integrals with anisotropic growth conditions.

Some improved Caffarelli-Kohn-Nirenberg inequalities

Abstract: For 1 < p < N and $-\infty < \gamma < \frac{N-p}{p}$ we obtain the following improved Hardy-Sobolev Inequalities $$ \int\limits_\Omega \vert abla \phi\vert^p\vert x\vert^{-p\gamma}dx -\left(\frac{N-p(\gamma +1)}{p}\right)^p \int\limits_\Omega \frac{\vert\phi\vert^p}{\vert x\vert^{p(\gamma + 1)}}dx $$ $$ \ge C(p,q,r,\gamma,\vert\Omega \vert)\left(\int\limits_\Omega \vert abla\phi\vert^q\vert x\vert^{-r\gamma}dx\right)^{\frac{p}{q}}, $$ where 1 < q < p and $q\le r < \infty$ if $\gamma \le 0$ , $1\le r < p + \rho(N,p,q,\gamma)$ if $\gamma > 0$ , for some positive constant $\rho(N,p,q,\gamma)$ . Also we give an alternative proof of the optimal improved inequality for p = 2 by Wang-Willem in [16].

On the higher eigenvalues for the $\infty$ -eigenvalue problem

Abstract: We study the higher eigenvalues and eigenfunctions for the so-called $\infty$ -eigenvalue problem. The problem arises as an asymptotic limit of the nonlinear eigenvalue problems for the p-Laplace operators and is very closely related to the geometry of the underlying domain. We are able to prove several properties that are known in the linear case p = 2 of the Laplacian, but are unknown for other values of p. In particular, we establish the validity of the Payne-Polya-Weinberger conjecture regarding the ratio of the first two eigenvalues and the Payne nodal conjecture, which deals with the zero set of a second eigenfunction. The limit problem also exhibits phenomena that are not encountered for any $1 < p < \infty$ .

Lipschitz continuity of state functions in some optimal shaping

Abstract: We prove local Lipschitz continuity of the solution to the state equation in two kinds of shape optimization problems with constraint on the volume: the minimal shaping for the Dirichlet energy, with no sign condition on the state function, and the minimal shaping for the first eigenvalue of the Laplacian. This is a main first step for proving regularity of the optimal shapes themselves.

Singularly perturbed nonlinear elliptic problems on manifolds

Abstract: Let \({\cal M}\) be a connected compact smooth Riemannian manifold of dimension \(n \ge 3\) with or without smooth boundary \(\partial {\cal M}.\) We consider the following singularly perturbed nonlinear elliptic problem on \({\cal M}\) $$ \varepsilon^2 \Delta_{{\cal M}} u - u + f(u)=0, \ \ u > 0 \quad {\rm on} \quad {\cal M}, \quad \frac{\partial u}{\partial u}=0 {\rm on } \partial {\cal M} $$ where \(\Delta_{{\cal M}}\) is the Laplace-Beltrami operator on \({\cal M} \), \( u\) is an exterior normal to \(\partial {\cal M}\) and a nonlinearity \(f\) of subcritical growth. For certain \(f,\) there exists a mountain pass solution \(u_\varepsilon\) of above problem which exhibits a spike layer. We are interested in the asymptotic behaviour of the spike layer. Without any non-degeneracy condition and monotonicity of \(f(t)/t,\) we show that if \(\partial {\cal M} =\emptyset(\partial {\cal M} e \emptyset),\) the peak point \(x_\varepsilon\) of the solution \(u_\varepsilon\) converges to a maximum point of the scalar curvature \(S\) on \({\cal M}\)(the mean curvature \(H\) on \(\partial {\cal M})\) as \(\varepsilon \to 0,\)respectively.

Embedded isolated singularities of flat surfaces in hyperbolic 3-space

Abstract: We give a complete description of the flat surfaces in hyperbolic 3-space that are regularly embedded around an isolated singularity. Specifically, we show that there is a one-to-one explicit correspondence between this class and the class of regular analytic convex Jordan curves in the 2-sphere. Previously, the only known examples of such surfaces were rotational ones. To achieve this result, we first solve the geometric Cauchy problem for flat surfaces in hyperbolic 3-space.

The obstacle problem for Monge Ampere equation

Abstract: We consider the following obstacle problem for Monge-Ampere equation \(\det D^2 u = f \chi_{\{u > 0\}}\) and discuss the regularity of the free boundary \(\partial \{u = 0 \}\). We prove that \(\partial \{u = 0 \}\) is \(C^{1,\alpha}\) if f is bounded away from 0 and \(\infty\), and it is C1,1 if \(f \equiv 1\).

On the approximation of the elastica functional in radial symmetry

Abstract: We prove a result concerning the approximation of the elastica functional with a sequence of second order functionals, under radial symmetry assumptions. This theorem is strictly related to a conjecture of De Giorgi [8].

Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions

Abstract: We construct heteroclinic the global minimizers of a nonlocal free energy functional that van der Waals derived in 1893. We study the case where the nonlocality satisfies only a weakened type of ellipticity, which precludes the use of comparison methods. In the interesting case when the local part of the energy is nonconvex, we construct a classical the global minimizer by studying a relaxed functional corresponding to the convexification of the local part and exclude the possibility of minimizers of the relaxed functional having rapid oscillations. We also construct examples where the global minimizer is not monotonic.

A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications

Abstract: The system of partial differential equations � −div (v Du) = f in � |Du |− 1 = 0i n{ v> 0} arises in the analysis of mathematical models for sandpile growth and in the con- text of the Monge-Kantorovich optimal mass transport theory. A representation formula for the solutions of a related boundary value problem is here obtained, extending the previous two-dimensional result of the first two authors to arbitrary space dimension. An application to the minimization of integral functionals of the form �

Rank-one convex functions on 2×2 symmetric matrices and laminates on rank-three lines

Abstract: We construct a function on the space of symmetric 2× 2 matrices in such a way that it is convex on rank-one directions and its distributional Hessian is not a locally bounded measure. This paper is also an illustration of a recently proposed technique to disprove L1 estimates by the construction of suitable probability measures (laminates) in matrix space. From this point of view the novelty is that the support of the laminate, besides satisfying a convex constraint, needs to be contained on a rank-three line, up to arbitrarily small errors.

Deforming metrics with negative curvature by a fully nonlinear flow

Abstract: By studying a fully nonlinear flow deforming conformal metrics on compact and connected manifold, we prove that for $\lambda < 1$ , any metric g with its modified Schouten tensor $A^\lambda_{g}\in \Gamma_k^-$ always can be deformed in a natural way to a conformal metric with constant $\sigma_k$ -scalar curvature at exponential rate.

Energy asymptotics for Type II superconductors

Abstract: We study the Ginzburg–Landau functional in the parameter regime describing ‘Type II superconductors’. In the exact regime considered minimizers are localized to the boundary — i.e. the sample is only superconducting in the boundary region. Depending on the relative size of different parameters we describe the concentration behavior and give leading order energy asymptotics. This generalizes previous results by Lu–Pan, Helffer–Pan, and Pan.

On the regularity of the polar factorization for time dependent maps

Abstract: We consider the polar factorization of vector valued mappings, introduced in [3], in the case of a family of mappings depending on a parameter. We investigate the regularity with respect to this parameter of the terms of the polar factorization by constructing some a priori bounds. To do so, we consider the linearization of the associated Monge-Ampere equation.

Higher integrability of minimizing Young measures

Abstract: We prove local higher integrability with large exponents for minimizers and Young measure minimizers of variational integrals of the form \( \int_{\Omega} \! F(x, abla u(x)) dx \) where F is a Caratheodory integrand that resembles the p-Dirichlet integrand at infinity. The result yields existence of minimizing sequences with higher equi-integrability properties locally in \(\Omega\).

A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds

Abstract: An important component of Hamilton’s program for the Ricci flow on compact 3-manifolds is the classification of singularities which form under the flow for certain initial metrics. In particular, Type I singularities, where the evolving metrics have curvatures whose maximums are inversely proportional to the time to blow-up, are modelled on the 3-sphere and the cylinder S × R and their quotients. On the other hand, Type II singularities (the complementary case) are much more difficult to understand. Despite this, it is known from the work of Hamilton that their singularity models are stationary solutions to the Ricci flow. This uses several techniques, including Harnack inequalities of Li-Yau-Hamilton type, the strong maximum principle for systems, dimension reduction, and the study of the geometry at infinity of noncompact stationary solutions (see§§1426 of [ H2].) In terms of Hamilton’s program, at least two obstacles remain: obtaining an injectivity radius estimate for Type II solutions and ruling out the so-called cigar soliton (the unique complete stationary solution on a surface with positive curvature) as the dimension reduction of a Type II singularity model. ∗Research partially supported by NSF grant DMS-9971891. 1In fact, Hamilton and Yau have announced informally that these are the only two obstacles and that they both would follow from obtaining a suitable differential matrix Harnack inequality of Li-Yau-Hamilton type for arbitrary solutions of the Ricci flow on compact 3manifolds. 2Added in proof: Very recently, Perelman [P1] has given a proof of the Little Loop Conjecture in all dimensions without curvature restriction. See also [P2] for some further developments.

Stability remarks to the obstacle problem for p-Laplacian type equations

Abstract: We extend some convergence and L 1 stability results for the coincidence set to the p-obstacle problem under natural nondegeneracy conditions and without restrictions on p, $1\! < \!p\! < \!\infty$ . We rely on the local $C^{1,\lambda}$ regularity of the solution and, as an application, we show the existence of a solution to the thermal membrane problem, and in a limit nonlocal case also its uniqueness for small data.

Nonlinear degenerate curvature flows for weakly convex hypersurfaces

Abstract: We study the evolution of closed, weakly convex hypersurfaces in $\mathbb{R}^{n + 1}$ in direction of their normal vector, where the speed equals a quotient of successive elementary symmetric polynomials of the principal curvatures. We show that there exists a solution for these weakly convex surfaces at least for some short time if the elementary symmetric polynomial in the denominator of the quotient is positive. The results for this nonlinear, degenerate flow are obtained by a cylindrically symmetric barrier construction.

Regularity and relaxed problems of minimizing biharmonic maps into spheres

Abstract: For n >= 5 and k >= 4, we show that any minimizing biharmonic map from Omega subset of R-n to S-k is smooth off a closed set whose Hausdorff dimension is at most n - 5. When n = 5 and k = 4, for a parameter lambda is an element of [0, 1] we introduce lambda-relaxed energy H-lambda of the Hessian energy for maps in W-2,W-2 (Omega; S-4) so that each minimizer u(lambda) of H-lambda is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of H-lambda for lambda is an element of [0, 1).