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In this paper, we study the asymptotic profiles, uniqueness and orbital stability of McCann's uniformly rotating binary stars (Houston J Math 32(2):603–631, 2006) governed by the Euler–Poisson system. A new uniqueness result will be importantly used in stability analysis. Moreover, we apply our framework to the study of uniformly rotating binary galaxies of the Vlasov–Poisson system through Rein's reduction (Handbook of differential equations: evolutionary equations, vol III, pp 383–476, 2007). [ABSTRACT FROM AUTHOR]

In this paper, we develop dispersive PDE techniques for the Fermi–Pasta–Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg–de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity requirement in the previous study (Schneider and Wayne, In: International conference on differential equations, Berlin, 1999, World Sci. Publ, River Edge, NJ, Vol 1, 2, pp 390–404, 2000). [ABSTRACT FROM AUTHOR]

We are concerned with the asymptotic behavior of a solution to the initial value problem for a system of hyperbolic conservation laws coupled with elliptic equations. This kind of problem was first considered in our previous paper. In the present paper, we generalize the previous results to a broad class of hyperbolic-elliptic coupled systems. Assuming the existence of the entropy function and the stability condition, we prove the global existence and the asymptotic decay of the solution for small initial data in a suitable Sobolev space. Then, it is shown that the solution is well approximated, for large time, by a solution to the corresponding hyperbolic-parabolic coupled system. The first result is proved by deriving a priori estimates through the standard energy method. The spectral analysis with the aid of the a priori estimate gives the second result. [ABSTRACT FROM AUTHOR]

In this paper, we study a Hamiltonian structure of the Vlasov-Poisson system, first mentioned by Fröhlich et al. (Commun Math Phys 288:1023-1058, 2009). To begin with, we give a formal guideline to derive a Hamiltonian on a subspace of complex-valued L2 integrable functions α on the one particle phase space RZ2d; s.t. f=α2 is a solution of a collisionless Boltzmann equation. The only requirement is a sufficiently regular energy functional on a subspace of distribution functions f∈L1. Secondly, we give a full well-posedness theory for the obtained system corresponding to Vlasov-Poisson in d≧3 dimensions. Finally, we adapt the classical globality results (Lions and Perthame in Invent Math 105:415-430, 1991; Pfaffelmoser in J Differ Equ 95:281-303, 1992; Schaeffer in Commun Partial Differ Equ 16(8-9):1313-1335, 1991) for d = 3 to the generalized system. [ABSTRACT FROM AUTHOR]

We discuss old and new results on the mathematical justification of Boltzmann’s equation. The classical result along these lines is a theorem which was proven by Lanford in the 1970s. This paper is naturally divided into three parts.I. Classical. We give new proofs of both the uniform bounds required for Lanford’s theorem, as well as the related bounds due to Illner and Pulvirenti for a perturbation of vacuum. The proofs use a duality argument and differential inequalities, instead of a fixed point iteration.II. Strong chaos. We introduce a new notion of propagation of chaos. Our notion of chaos provides for uniform error estimates on a very precise set of points; this set is closely related to the notion of strong (one-sided) chaos and the emergence of irreversibility.III. Supplemental. We announce and provide a proof (in “Appendix A”) of the propagation of partial factorization at some phase-points where complete factorization is impossible. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence, uniqueness, and global stability of traveling waves in discrete periodic media for a system of ordinary differential equations exhibiting bistable dynamics. The main tools used to prove the uniqueness and asymptotic stability of traveling waves are the comparison principle, spectrum analysis, and constructions of super/subsolutions. To prove the existence of traveling waves, the system is converted to an integral equation which is common in the study of monostable dynamics but quite rare in the study of bistable dynamics. The main purpose of this paper is to introduce a general framework for the study of traveling waves in discrete periodic media. [ABSTRACT FROM AUTHOR]

Given H:ℝ3→ℝ of class C 1 and bounded, we consider a sequence ( u n ) of solutions of the H-system [InlineMediaObject not available: see fulltext.] in the unit open disc [InlineMediaObject not available: see fulltext.] satisfying the boundary condition u n = γ n on ∂[InlineMediaObject not available: see fulltext.]. In the first part of this paper, assuming that ( u n ) is bounded in H 1([InlineMediaObject not available: see fulltext.],ℝ3) we study the behavior of ( u n ) when the boundary data γ n shrink to zero. We show that either u n →0 strongly in H 1([InlineMediaObject not available: see fulltext.],ℝ3) or u n blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on ℝ2. Under additional assumptions on H, we can obtain more precise information on the blow up. In the second part of this paper we investigate the multiplicity of solutions for the Dirichlet problem on the disc with small boundary datum. We detect a family of nonconstant functions H (even close to a nonzero constant in any reasonable topology) for which the Dirichlet problem cannot admit a ``large'' solution at a mountain pass level when the boundary datum is small. [ABSTRACT FROM AUTHOR]

The paper is concerned with mappings $${h \colon \mathbb{X}}$$ $${{\begin{array}{ll} {\rm onto} \\ \longrightarrow \end{array}}}$$ $${\mathbb{Y}}$$ between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in $${\mathbb{X}}$$) of the energy-minimal mappings is established within the class $${\overline{\fancyscript{H}}_2(\mathbb{X}, \mathbb{Y})}$$ of strong limits of homeomorphisms in the Sobolev space $${\fancyscript{W}^{1,2}(\mathbb{X}, \mathbb{Y})}$$ , a result of considerable interest in the mathematical models of nonlinear elasticity. The inner variation of the independent variable in $${\mathbb{X}}$$ leads to the Hopf differential $${h_{z} \overline{h_{\bar{z}}} {\rm d}z \otimes {\rm d}z}$$ and its trajectories. For a pair of doubly connected domains, in which $${\mathbb{X}}$$ has finite conformal modulus, we establish the following principle: A mapping $${h \in \overline{\fancyscript{H}}_{2} (\mathbb{X}, \mathbb{Y})}$$ is energy-minimal if and only if its Hopf-differential is analytic in $${\mathbb{X}}$$ and real along $${\partial \mathbb{X}}$$ . In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of slits in $${\mathbb{X}}$$ (cognate with cracks). Slits are triggered by points of concavity of $${\mathbb{Y}}$$ . They originate from $${\partial \mathbb{X}}$$ and advance along vertical trajectories of the Hopf differential toward $${\mathbb{X}}$$ where they eventually terminate, so no crosscuts are created. [ABSTRACT FROM AUTHOR]

This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow. [ABSTRACT FROM AUTHOR]

The paper analyzes a Lagrangian system which is controlled by directly assigning some of the coordinates as functions of time, by means of frictionless constraints. In a natural system of coordinates, the equations of motion contain terms which are linear or quadratic with respect to time derivatives of the control functions. After reviewing the basic equations, we explain the significance of the quadratic terms related to geodesics orthogonal to a given foliation. We then study the problem of stabilization of the system to a given point by means of oscillating controls. This problem is first reduced to theweak stability for a related convex-valued differential inclusion, then studied by Lyapunov functions methods. In the last sections, we illustrate the results by means of various mechanical examples. [ABSTRACT FROM AUTHOR]

In this paper we prove global existence for solutions of the Vlasov–Poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we study a free boundary problem obtained as a limit as ε → 0 to the following regularizing family of semilinear equations $${\Delta u = \beta_{\varepsilon}(u) F(\nabla u)}$$ , where β ε approximates the Dirac delta in the origin and F is a Lipschitz function bounded away from 0 and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces that are uniform in ε. This allows to prove that the free boundary of a limit has the “right” weak geometry, in the measure theoretical sense. By the construction of some barriers with curvature, the classification of global profiles of the blow-up analysis is carried out and the limit functions are proven to be viscosity and pointwise solution ( $${\mathcal{H}^{n-1}}$$ almost everywhere) to a free boundary problem. Finally, the free boundary is proven to be a C 1,α surface around $${\mathcal{H}^{n-1}}$$ almost everywhere point. [ABSTRACT FROM AUTHOR]

In this paper we investigate some free boundary problems with space-dependent free boundary velocities. Based on maximum principle-type arguments, we show the uniform convergence of the solutions in the homogenization limit. The main step is to show the uniqueness of the limiting free boundary velocity, which turns out to be a continuous function of the normal direction of the free boundary. [ABSTRACT FROM AUTHOR]

We study the initial-boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the Rankine-Hugoniot relations about an unsteady piecewise smooth solution. This solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity (current-vortex sheet). Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev space W21, σ. Despite the fact that the constant coefficients linearized problem does not meet the uniform Kreiss-Lopatinskii condition, the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current-vortex sheets. The result of this paper is a necessary step in proving the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of magnetohydrodynamics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an entropy condition for hyperbolic systems of balance laws. Under this condition, we use the Chapman-Enskog expansion to derive the corresponding viscous conservation laws. Further structural conditions are discussed in order to develop (local and global) existence theories for the balance laws and viscous conservation laws. [ABSTRACT FROM AUTHOR]

We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation where $${\Omega \subset \mathbb{R}^{n}}$$ is a ring-shaped domain, a and μ are given positive constants, $${\mathbb{H}(\cdot)}$$ is the Heaviside maximal monotone graph: $${\mathbb{H}(s) = 1}$$ if s > 0, $${\mathbb{H}(0) = [0, 1],\, \mathbb{H}(s) = 0}$$ if s < 0. Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as combustion. We show that under certain conditions on the initial data the level sets $${\it \Gamma}_{\mu }=\{(\mathbf{x}, t) :u(\mathbf{x}, t)=\mu\}$$ are n-dimensional hypersurfaces in the ( x, t)-space and show that the dynamics of Γ μ is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential equation expresses the velocity of advancement of the level surface Γ μ through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory of each of the fluid particles. [ABSTRACT FROM AUTHOR]

A new type of differential equations for probability measures on Euclidean spaces, called measure differential equations (briefly MDEs), is introduced. MDEs correspond to probability vector fields, which map measures on an Euclidean space to measures on its tangent bundle. Solutions are intended in weak sense and existence, uniqueness and continuous dependence results are proved under suitable conditions. The latter are expressed in terms of the Wasserstein metric on the base and fiber of the tangent bundle. MDEs represent a natural measure-theoretic generalization of ordinary differential equations via a monoid morphism mapping sums of vector fields to fiber convolution of the corresponding probability vector Fields. Various examples, including finite-speed diffusion and concentration, are shown, together with relationships to partial differential equations. Finally, MDEs are also natural mean-field limits of multi-particle systems, with convergence results extending the classical Dobrushin approach. [ABSTRACT FROM AUTHOR]

We consider the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We prove a persistence result for such relative equilibria, present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and show how it allows one to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of relative equilibria of nonlinear Schrödinger and Manakov equations. We provide a comparison of our approach to the one by Grillakis-Shatah-Strauss. [ABSTRACT FROM AUTHOR]

The stationary solutions are triples f,ρ,U of three functions: the distribution function f=fx,v, the potential U=Ux and the local density ρ=ρx, x,v∈R2, which are linked by the Vlasov-Poisson system. We prove the existence of wide classes of spherically symmetric stationary solutions with the property that ρ depends on x=r and f on the energy E:=Ux+v22. First we answer the question of which given functions ρ are the local density of a stationary solution (inverse problem). Our result is (up to technicalities) that every ρ≥0 which is strictly decreasing on an interval 0,R and zero on its complement (R≦∞) belongs to this class. Second, we ask: which given functions q induce distribution functions f of the form f=q-E0-E (E0≧0) of a stationary solution? (direct problem). This question is answered for many q which are positive for positive and vanish for negative arguments in an approximative and constructive way which is based on numerical methods. [ABSTRACT FROM AUTHOR]

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in Rn+1 up to sets of null Hn-1 measure. In particular, we prove local finiteness of the (n−1)-dimensional Hausdorff measure of the free boundary,Hn-1-rectifiability of the free boundary,classification of the frequencies up to a set of Hausdorff dimension at most (n−2) and classification of the blow-ups at Hn-1 almost every free boundary point. [ABSTRACT FROM AUTHOR]

Best possible second-order regularity is established for solutions to p-Laplacian type equations with p∈(1,∞) and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L2-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all. [ABSTRACT FROM AUTHOR]

Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero. [ABSTRACT FROM AUTHOR]

We consider the existence of Beltrami fields with a nonconstant proportionality factor f in an open subset U of $${\mathbb{R}^3}$$ . By reformulating this problem as a constrained evolution equation on a surface, we find an explicit differential equation that f must satisfy whenever there is a nontrivial Beltrami field with this factor. This ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the C topology, $${k\geqq 7}$$ . In particular, there are no nontrivial Beltrami fields whenever f has a regular level set diffeomorphic to the sphere. This provides an explanation of the helical flow paradox of Morgulis et al. (Commun Pure Appl Math 48:571-582, ). [ABSTRACT FROM AUTHOR]

We develop a theory of invariant manifolds for the steady Boltzmann equation and apply it to the study of boundary layers and nonlinear waves. The steady Boltzmann equation is an infinite dimensional differential equation, so the standard center manifold theory for differential equations based on spectral information does not apply here. Instead, we employ a time-asymptotic approach using the pointwise information of Green's function for the construction of the linear invariant manifolds. At the resonance cases when the Mach number at the far field is around one of the critical values of −1, 0 or 1, the truly nonlinear theory arises. In such a case, there are wave patterns combining the fast decaying Knudsen-type and slow varying fluid-like waves. The key Knudsen manifolds consisting of only Knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical Mach numbers. Around Mach number ± 1, the fluidlike waves are compressive and expansive waves; and around the Mach number 0, they are linear thermal layers. The quantitative analysis of the fluid-like waves is done using the reduction of dimensions to the center manifolds.Two-scale nonlinear dynamics based on those on the Knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns. There are striking bifurcations in the transition of evaporation to condensation and in the transition of the Milne's problem with a subsonic far field to one with a supersonic far field. The analysis of these wave patterns allows us to understand the Sone Diagram for the study of the complete condensation boundary value problem. The monotonicity of the Boltzmann shock profiles, a problem that initially motivated the present study, is shown as a consequence of the quantitative analysis of the nonlinear fluid-like waves. [ABSTRACT FROM AUTHOR]

We analyze the extendability of the solutions to a certain second order differential equation on a Riemannian manifold ( M, g), which is defined by a general class of forces (both prescribed on M or depending on the velocity). The results include the general time-dependent anholonomic case, and further refinements for autonomous systems or forces derived from a potential are obtained. These extend classical results for Lagrangian and Hamiltonian systems. Several examples show the optimality of the assumptions as well as the utility of the results, including an application to relativistic pp-waves. [ABSTRACT FROM AUTHOR]

We establish the L resolvent estimates for the Stokes operator in Lipschitz domains in $${\mathbb{R}^d}$$, $${d\geqq 3}$$ for $${|\frac{1}{p}-\frac{1}{2}| < \frac{1}{2d} +\varepsilon}$$. The result implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in L for (3/2) − ε < p < 3 + ε. This gives an affirmative answer to a conjecture of M. Taylor (Progr. Nonlinear Differential Equations Appl., vol. 42, pp. 320-334). [ABSTRACT FROM AUTHOR]

We use elliptic theory to prove the existence of steady two-dimensional periodic waterwaves of large amplitude in a flowwith an arbitrary bounded but discontinuous vorticity. This is achieved by developing a local and global bifurcation construction of weak solutions of the elliptic partial differential equations that are relevant to this hydrodynamical context. [ABSTRACT FROM AUTHOR]

We consider a Cauchy problem for a parabolic-elliptic system of drift-diffusion type. The problem is formally of the form This system describes a mass-conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis. Our concern is the asymptotic behavior of blowup solutions when the blowup is type I, in the sense that its blowup rate is the same as the corresponding ordinary differential equation y = y (up to a multiple constant). It is shown that all type I blowup is asymptotically (backward) self-similar, provided that the solution is radial, nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three. [ABSTRACT FROM AUTHOR]

We consider a discrete mechanical system with a non-trivial mass matrix, subjected to perfect unilateral constraints described by the geometrical inequalities $${f_{\alpha} (q) \geqq 0, \alpha \in \{1, \dots, \nu\} (\nu \geqq 1)}$$. We assume that the transmission of the velocities at impact is governed by Newton's Law with a coefficient of restitution e = 0 (so that the impact is inelastic). We propose a time-discretization of the second order differential inclusion describing the dynamics, which generalizes the scheme proposed in Paoli (J Differ Equ 211:247-281, 2005) and, for any admissible data, we prove the convergence of approximate motions to a solution of the initial-value problem. [ABSTRACT FROM AUTHOR]

We prove the existence of a weak solution to the steady Navier–Stokes problem in a three dimensional domain Ω, whose boundary ∂Ω consists of M unbounded components Γ, . . . , Γ and N − M bounded components Γ, . . . , Γ. We use the inhomogeneous Dirichlet boundary condition on ∂Ω. The prescribed velocity profile α on ∂Ω is assumed to have an L-extension to Ω with the gradient in L(Ω). We assume that the fluxes of α through the bounded components Γ, . . . , Γ of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ, . . . , Γ. [ABSTRACT FROM AUTHOR]

We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983): where Ω is a bounded smooth domain of R N ( N ≧ 7) and 2* is the critical Sobolev exponent. We show that, for each fixed λ > 0, this problem has infinitely many sign-changing solutions. In particular, if λ ≧ λ1, the Brézis–Nirenberg problem has and only has infinitely many sign-changing solutions except zero. The main tool is the estimates of Morse indices of nodal solutions. [ABSTRACT FROM AUTHOR]

We consider n-tuples of differential 1-forms in the Euclidean n-space that satisfy a quasiconformality condition and an asymptotic closedness condition. We show that renormalized sequences of such tuples have subsequences converging to differentials of quasiregular maps. We then use these maps to show that the tuples carry topological information. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behaviour of positive solutions u( t, x) of the fast diffusion equation $${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}$$ posed for $${x\in\mathbb R^d}$$, t > 0, with a precise value for the exponent m = ( d − 4)/( d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (B lanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold $${(\mathbb R^d,{\bf g})}$$, with a metric g which is conformal to the standard $${\mathbb R^d}$$ metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m. [ABSTRACT FROM AUTHOR]

In this article we show that for the homogenization of multiple integrals, the quasiconvexification of the cell formula is different from the asymptotic formula in general. To this aim, we construct three examples in three different settings: the homogenization of a discrete model, the homogenization of a composite material and the homogenization of a homogeneous material on a perforated domain. [ABSTRACT FROM AUTHOR]

The authors consider boundary value problems for the Navier–Stokes system in a polyhedral domain, where different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are arbitrarily combined on the faces of the polyhedron. They prove existence and regularity theorems for weak solutions in weighted (and nonweighted) L p Sobolev and Hölder spaces with sharp integrability and smoothness parameters. [ABSTRACT FROM AUTHOR]

The article describes the reflector problem in anisotropic media to study reflecting surfaces to reshape electromagnetic radiation in a prescribed manner. It reviews the Snell law of refraction in vector form and discusses surfaces having the uniform refraction property. The differential equation corresponding to the refractor problem is also set up.

We study the limit of the hyperbolic–parabolic approximation The function $${\tilde {\ss}}$$ is defined in such a way as to guarantee that the initial boundary value problem is well posed even if $${\tilde {B}}$$ is not invertible. The data $${\bar {g}}$$ and $${\bar {v}_{0}}$$ are constant. When $${\tilde {B}}$$ is invertible, the previous problem takes the simpler formAgain, the data $${\bar {v}_b}$$ and $${\bar {v}_0}$$ are constant. The conservative case is included in the previous formulations. Convergence of the $${v^{\varepsilon}}$$ , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is, one eigenvalue of $${\tilde {A}}$$ can be 0. Second, as pointed out before, we take into account the possibility that $${\tilde {B}}$$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if this condition is not satisfied, then pathological behaviors may occur. [ABSTRACT FROM AUTHOR]

We discuss quasistatic evolution processes for capacitary measures and shapes in order to model debonding membranes. Minimizing movements as well as rate-independent processes are investigated and some models are described, together with a series of open problems. [ABSTRACT FROM AUTHOR]

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. Using techniques from ordinary differential equation theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods. [ABSTRACT FROM AUTHOR]

We consider the boundary value problem [InlineMediaObject not available: see fulltext.] where Ω is a smooth and bounded domain in ℝ2 and λ > 0. We prove that for any integer k ≧ 1 there exist at least two solutions u λ with the property that the boundary flux satisfies up to subsequences λ → 0, [InlineMediaObject not available: see fulltext.] where the ξ j are points of ∂Ω ordered clockwise in j. [ABSTRACT FROM AUTHOR]

We show here the impact on the initial-boundary value problem, and on the evolution of viscoelastic systems of the use of a new definition of state based on the stress-response (see, e.g., [48, 16, 41]). Comparisons are made between this new approach and the traditional one, which is based on the identification of histories and states. We shall refer to a stress-response definition of state as the minimal state [29]. Materials with memory and with relaxation are discussed. The energetics of linear viscoelastic materials is revisited and new free energies, expressed in terms of the minimal state descriptor, are derived together with the related dissipations. Furthermore, both the minimum and the maximum free energy are recast in terms of the minimal state variable and the current strain. The initial-boundary value problem governing the motion of a linear viscoelastic body is re-stated in terms of the minimal state and the velocity field through the principle of virtual power. The advantages are (i) the elimination of the need to know the past-strain history at each point of the body, and (ii) the fact that initial and boundary data can now be prescribed on a broader space than resulting from the classical approach based on histories. These advantages are shown to lead to natural results about well-posedness and stability of the motion. Finally, we show how the evolution of a linear viscoelastic system can be described through a strongly continuous semigroup of (linear) contraction operators on an appropriate Hilbert space. The family of all solutions of the evolutionary system, obtained by varying the initial data in such a space, is shown to have exponentially decaying energy. [ABSTRACT FROM AUTHOR]

We consider the Green's function of the Laplace operator in domains with spherical holes (particles). Under natural assumptions on the distribution of particles we show that the Green's function decays exponentially over distances larger than the screening length. This result is fundamental for example when deriving effective equations for coarsening systems in unbounded domains. [ABSTRACT FROM AUTHOR]

We prove the existence of global smooth solutions near a given steady state of the hydrodynamic model of the semiconductors in a bounded domain with physical boundary conditions. The steady state and the doping profile are permitted to be of large variation but the initial velocity must be small. Two cases are considered. In the first one the problem is three-dimensional, the boundary conditions are insulating and the steady state velocity vanishes. In the second one, the problem is one-dimensional, the boundary is of contact type and the steady state velocity does not vanish. [ABSTRACT FROM AUTHOR]

We study the existence of strong solutions to the three-dimensional Navier-Stokes initial-boundary value problem in the domain, O, exterior to a rigid body that rotates with constant angular velocity,?. We show that when the initial data,u0, are prescribed in an appropriate functional class, a strong solution exists at least in some finite time interval. Moreover, the solution exists for all times, providedu0, in suitable norm, and the magnitude of?do not exceed a certain constant depending only on the kinematic viscosity and on the regularity of O. In this latter case, we also show that the velocity field converges to the velocity field of the corresponding steady-state solution. [ABSTRACT FROM AUTHOR]

By minimizing the energy for the Vlasov-Poisson system under a constraint, Guo and Rein have constructed a large class of isotropic, spherically symmetric steady states. They have shown that an isolated minimizer is automatically dynamically stable under general (i.e., not necessarily symmetric) perturbations. The main result of this work is to remove the assumption that the minimizer must be isolated, so minimizers are stable even if they are not isolated. It is also shown that the Lagrange multipliers associated with all minimizers have the same value. Finally, an example where two distinct minimizers exist is studied numerically. [ABSTRACT FROM AUTHOR]

We prove C1,γ regularity of Lipschitz free boundaries of two-phase problems for linear elliptic operators with Hölder continuous coefficients. [ABSTRACT FROM AUTHOR]

We prove that nonsmooth quasilinear parabolic systems admit a local solution in Lp strongly differentiable with respect to time over a bounded three-dimensional polyhedral space domain. The proof rests essentially on new elliptic regularity results for polyhedral Laplace interface problems for anisotropic materials. These results are based on sharp pointwise estimates for Green’s function, which are also of independent interest. To treat the nonlinear problem, we then apply a classical theorem of Sobolevskii for abstract parabolic equations and recently obtained resolvent estimates for elliptic operators and interpolation results. As applications we have in mind primarily reaction-diffusion systems. The treatment of such equations in an Lp context seems to be new and allows (by Gauss’ theorem) the proper definition of the normal component of currents across the boundary. [ABSTRACT FROM AUTHOR]

We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in Rd, by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions. [ABSTRACT FROM AUTHOR]

It is shown that a function u satisfying |∂t+Δu|≦M(|u|+|∇u|), |u(x, t)|≦MeM|x|2 in (ℝn \ (BR) × [0, T] and u(x, 0) = 0 for x?ℝn \ BR must vanish identically in ℝn \ BR×[0, T]. [ABSTRACT FROM AUTHOR]