Authors:Nicola Soave; Hugo Tavares; Susanna Terracini; Alessandro Zilio Pages: 743 - 772 Abstract: We consider a class of variational problems for densities that repel each other at a distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional $$\begin{aligned} D(\mathbf{u}) = \sum_{i=1}^k \int_{\Omega} \nabla u_i ^2 \quad \text{or} \quad R(\mathbf{u}) = \sum_{i=1}^k \frac{\int_{\Omega} \nabla u_i ^2}{\int_{\Omega} {u_i^2}}, \end{aligned}$$ minimized in the class of \({H^1(\Omega,\mathbb{R}^k)}\) functions attaining some boundary conditions on ∂Ω, and subjected to the constraint $$\begin{aligned} \text{dist} (\{u_i > 0\}, \{u_j > 0\}) \ge 1 \quad \forall i \neq j. \end{aligned}$$ For these problems, we investigate the optimal regularity of the solutions, prove a free-boundary condition, and derive some preliminary results characterizing the free boundary \({\partial \{\sum_{i=1}^k u_i > 0\}}\) . PubDate: 2018-06-01 DOI: 10.1007/s00205-017-1204-2 Issue No:Vol. 228, No. 3 (2018)

Authors:Boris Buffoni; Mark D. Groves; Erik Wahlén Pages: 773 - 820 Abstract: Fully localised solitary waves are travelling-wave solutions of the three- dimensional gravity–capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as ‘lumps’), and a mathematically rigorous existence theory for strong surface tension (Bond number \({\beta}\) greater than \({\frac{1}{3}}\) ) has recently been given. In this article we present an existence theory for the physically more realistic case \({0 < \beta < \frac{1}{3}}\) . A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey–Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set. PubDate: 2018-06-01 DOI: 10.1007/s00205-017-1205-1 Issue No:Vol. 228, No. 3 (2018)

Authors:Andrea Malchiodi; Rainer Mandel; Matteo Rizzi Pages: 821 - 866 Abstract: In this paper we construct entire solutions to the phase field equation of Willmore type \({-\Delta(-\Delta u+W^{\prime}(u))+W^{\prime\prime}(u)(-\Delta u+W^{\prime}(u))=0}\) in the Euclidean plane, where W(u) is the standard double-well potential \({\frac{1}{4} (1-u^2)^2}\) . Such solutions have a non-trivial profile that shadows a Willmore planar curve, and converge uniformly to \({\pm 1}\) as \({x_2 \to \pm \infty}\) . These solutions give a counterexample to the counterpart of Gibbons’ conjecture for the fourth-order counterpart of the Allen–Cahn equation. We also study the x 2-derivative of these solutions using the special structure of Willmore’s equation. PubDate: 2018-06-01 DOI: 10.1007/s00205-017-1206-0 Issue No:Vol. 228, No. 3 (2018)

Authors:Antonin Chambolle; Sergio Conti; Gilles A. Francfort Pages: 867 - 889 Abstract: Linear fracture mechanics (or at least the initiation part of that theory) can be framed in a variational context as a minimization problem over an SBD type space. The corresponding functional can in turn be approximated in the sense of \({\Gamma}\) -convergence by a sequence of functionals involving a phase field as well as the displacement field. We show that a similar approximation persists if additionally imposing a non-interpenetration constraint in the minimization, namely that only nonnegative normal jumps should be permissible. PubDate: 2018-06-01 DOI: 10.1007/s00205-017-1207-z Issue No:Vol. 228, No. 3 (2018)

Authors:Codina Cotar; Gero Friesecke; Claudia Klüppelberg Pages: 891 - 922 Abstract: We prove rigorously that the exact N-electron Hohenberg–Kohn density functional converges in the strongly interacting limit to the strictly correlated electrons (SCE) functional, and that the absolute value squared of the associated constrained search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi-marginal optimal transport problem with Coulomb cost associated to the SCE functional. This extends our previous work for N = 2 (Cotar etal. in Commun Pure Appl Math 66:548–599, 2013). The correct limit problem has been derived in the physics literature by Seidl (Phys Rev A 60 4387–4395, 1999) and Seidl, Gorigiorgi and Savin (Phys Rev A 75:042511 1-12, 2007); in these papers the lack of a rigorous proofwas pointed out.We also give amathematical counterexample to this type of result, by replacing the constraint of given one-body density—an infinite dimensional quadratic expression in the wavefunction—by an infinite-dimensional quadratic expression in the wavefunction and its gradient. Connections with the Lawrentiev phenomenon in the calculus of variations are indicated. PubDate: 2018-06-01 DOI: 10.1007/s00205-017-1208-y Issue No:Vol. 228, No. 3 (2018)

Authors:Dongsheng Li; Kai Zhang Pages: 923 - 967 Abstract: In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise C α, C 1,α and C 2,α regularity. As byproducts, we also prove the A–B–P maximum principle, Harnack inequality, uniqueness and solvability of the equations. PubDate: 2018-06-01 DOI: 10.1007/s00205-017-1209-x Issue No:Vol. 228, No. 3 (2018)

Authors:Yuan Cai; Zhen Lei Pages: 969 - 993 Abstract: This paper studies the Cauchy problem of the incompressible magnetohydro dynamic systems with or without viscosity ν. Under the assumption that the initial velocity field and the displacement of the initialmagnetic field froma non-zero constant are sufficiently small in certain weighted Sobolev spaces, the Cauchy problem is shown to be globally well-posed for all ν ≧ 0 and all spaces with dimension n ≧ 2. Such a result holds true uniformly in nonnegative viscosity parameters. The proof is based on the inherent strong null structure of the systems introduced by Lei (Commun Pure Appl Math 69(11):2072–2106, 2016) and the ghost weight technique introduced by Alinhac (Invent Math 145(3):597–618, 2001). PubDate: 2018-06-01 DOI: 10.1007/s00205-017-1210-4 Issue No:Vol. 228, No. 3 (2018)

Authors:Nejla Nouaili; Hatem Zaag Pages: 995 - 1058 Abstract: We construct a solution for the Complex Ginzburg–Landau equation in a critical case which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows us to prove the stability of the constructed solution. PubDate: 2018-06-01 DOI: 10.1007/s00205-017-1211-3 Issue No:Vol. 228, No. 3 (2018)

Authors:Cristian E. Gutiérrez; Ahmad Sabra Pages: 341 - 399 Abstract: We show the existence of a lens, when its lower face is given, such that it refracts radiation emanating from a planar source, with a given field of directions, into the far field that preserves a given distribution of energies. Conditions are shown under which the lens obtained is physically realizable. It is shown that the upper face of the lens satisfies a pde of Monge-Ampère type. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1196-y Issue No:Vol. 228, No. 2 (2018)

Authors:Alessio Figalli; Connor Mooney Pages: 401 - 429 Abstract: A developable cone (“d-cone”) is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance \({\varepsilon}\) . Starting from a nonlinear model depending on the thickness h > 0 of the sheet, we prove a \({\Gamma}\) -convergence result as \({h \rightarrow 0}\) to a fourth-order obstacle problem for curves in \({\mathbb{S}^2}\) . We then describe the exact shape of minimizers of the limit problem when \({\varepsilon}\) is small. In particular, we rigorously justify previous results in the physics literature. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1195-z Issue No:Vol. 228, No. 2 (2018)

Authors:Aifang Qu; Wei Xiang Pages: 431 - 476 Abstract: In this paper, we study the stability of the three-dimensional jet created by a supersonic flow past a concave cornered wedge with the lower pressure at the downstream. The gas beyond the jet boundary is assumed to be static. It can be formulated as a nonlinear hyperbolic free boundary problem in a cornered domain with two characteristic free boundaries of different types: one is the rarefaction wave, while the other one is the contact discontinuity, which can be either a vortex sheet or an entropy wave. A more delicate argument is developed to establish the existence and stability of the square jet structure under the perturbation of the supersonic incoming flow and the pressure at the downstream. The methods and techniques developed here are also helpful for other problems involving similar difficulties. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1197-x Issue No:Vol. 228, No. 2 (2018)

Authors:L. Caffarelli; D. De Silva; O. Savin Pages: 477 - 493 Abstract: We prove Lipschitz continuity of solutions to a class of rather general two-phase anisotropic free boundary problems in 2D and we classify global solutions. As a consequence, we obtain \({C^{2,1}}\) regularity of solutions to the Bellman equation in 2D. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1198-9 Issue No:Vol. 228, No. 2 (2018)

Authors:Dominic Breit; Sebastian Schwarzacher Pages: 495 - 562 Abstract: We study the Navier–Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter’s elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies \({\gamma > \frac{12}{7}}\) ( \({\gamma >1 }\) in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in Lengeler and Růžičkaka (Arch Ration Mech Anal 211(1):205–255, 2014) on incompressible Navier–Stokes equations. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1199-8 Issue No:Vol. 228, No. 2 (2018)

Authors:Luis C. García-Naranjo; James Montaldi Pages: 563 - 602 Abstract: We consider nonholonomic systems with symmetry possessing a certain type of first integral which is linear in the velocities. We develop a systematic method for modifying the standard nonholonomic almost Poisson structure that describes the dynamics so that these integrals become Casimir functions after reduction. This explains a number of recent results on Hamiltonization of nonholonomic systems, and has consequences for the study of relative equilibria in such systems. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1200-6 Issue No:Vol. 228, No. 2 (2018)

Authors:Mitsuo Higaki; Yasunori Maekawa; Yuu Nakahara Pages: 603 - 651 Abstract: We study the two-dimensional stationary Navier–Stokes equations describing the flows around a rotating obstacle. The unique existence of solutions and their asymptotic behavior at spatial infinity are established when the rotation speed of the obstacle and the given exterior force are sufficiently small. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1201-5 Issue No:Vol. 228, No. 2 (2018)

Authors:Johannes Elschner; Guanghui Hu Pages: 653 - 690 Abstract: Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be any number in \({(0,2\pi)\backslash\{\pi\}}\) . We prove that such an obstacle scatters any incoming wave non-trivially (that is, the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1202-4 Issue No:Vol. 228, No. 2 (2018)

Authors:Alessandro Morando; Yuri Trakhinin; Paola Trebeschi Pages: 691 - 742 Abstract: We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal compressible magnetohydrodynamics (MHD) for two dimensional planar flows provided that the Rayleigh–Taylor sign condition \({[\partial p/\partial N] <0 }\) on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity. MHD contact discontinuities are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. This paper is a natural completion of our previous analysis (Morando et al. in J Differ Equ 258:2531–2571, 2015) where the well-posedness in Sobolev spaces of the linearized problem was proved under the Rayleigh–Taylor sign condition satisfied at each point of the unperturbed discontinuity. The proof of the resolution of the nonlinear problem given in the present paper follows from a suitable tame a priori estimate in Sobolev spaces for the linearized equations and a Nash–Moser iteration. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1203-3 Issue No:Vol. 228, No. 2 (2018)

Authors:Chanwoo Kim; Donghyun Lee Abstract: The basic question about the existence, uniqueness, and stability of the Boltzmann equation in general non-convex domains with the specular reflection boundary condition has been widely open. In this paper, we consider cylindrical domains whose cross section is generally non-convex analytic bounded planar domain. We establish a global well-posedness and asymptotic stability of the Boltzmann equation with the specular reflection boundary condition. Our method consists of the delicate construction of \({\epsilon}\) -tubular neighborhoods of billiard trajectories which bounce infinitely many times or hit the boundary tangentially at some moment, and sharp estimates of the size of such neighborhoods. PubDate: 2018-04-26 DOI: 10.1007/s00205-018-1241-5

Authors:Trevor M. Leslie; Roman Shvydkoy Abstract: The potential failure of energy equality for a solution u of the Euler or Navier–Stokes equations can be quantified using a so-called ‘energy measure’: the weak- \(*\) limit of the measures \({ u(t) ^2{\rm d}x}\) as t approaches the first possible blowup time. We show that membership of u in certain (weak or strong) \({L^q L^p}\) classes gives a uniform lower bound on the lower local dimension of \({\mathcal{E}}\) ; more precisely, it implies uniform boundedness of a certain upper s-density of \({\mathcal{E}}\) . We also define and give lower bounds on the ‘concentration dimension’ associated to \({\mathcal{E}}\) , which is the Hausdorff dimension of the smallest set on which energy can concentrate. Both the lower local dimension and the concentration dimension of \({\mathcal{E}}\) measure the departure from energy equality. As an application of our estimates, we prove that any solution to the 3-dimensional Navier–Stokes Equations which is Type-I in time must satisfy the energy equality at the first blowup time. PubDate: 2018-04-18 DOI: 10.1007/s00205-018-1250-4

Authors:Matteo Focardi; Emanuele Spadaro Abstract: We provide a thorough description of the free boundary for the lower dimensional obstacle problem in \({\mathbb{R}^{n+1}}\) up to sets of null \({\mathcal{H}^{n-1}}\) measure. In particular, we prove local finiteness of the (n−1)-dimensional Hausdorff measure of the free boundary, \({\mathcal{H}^{n-1}}\) -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 \({\mathcal{H}^{n-1}}\) almost every free boundary point. PubDate: 2018-04-05 DOI: 10.1007/s00205-018-1242-4