Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

Isoperimetric problems are of importance in engineering applications, where it is often desirable to maximize or minimize some physical variable by shape variation, subject to geometrical constraints, such as keeping an area or volume constant. The calculus of variations can offer a powerful tool for the solution of such problems where there is a governing variational minimum or maximum principle, e.g. Helmholtz's principle in slow viscous flow. In these problems the well-known Euler equations derived by the calculus of variations are supplemented with additional boundary conditions arising from the shape variation, as well as the usual physical boundary conditions. The exact solution of such unknown boundary problems can be difficult to find. A good approach then is to apply a complementary extremum principle that offers an algorithm for determining bounds on the exact extremal value of the original functional. This paper shows how this may be done in the case of the fundamental problem of the calculus of variations with variable endpoints. We apply this approach to a simple engineering problem of a stretched spring.

We study quantum Hall states on surfaces with conical singularities. We show that the electronic fluid at the cone tip possesses an intrinsic angular momentum, which is due solely to the gravitational anomaly. We also show that quantum Hall states behave as conformal primaries near singular points, with a conformal dimension equal to the angular momentum. Finally, we argue that the gravitational anomaly and conformal dimension determine the fine structure of the electronic density at the conical point. The singularities emerge as quasiparticles with spin and exchange statistics arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces.

We discuss various representations of planar $p$-harmonic systems of equations and their solutions. For coordinate functions of $p$-harmonic maps we analyze signs of their Hessians, the Gauss curvature of $p$-harmonic surfaces, the length of level curves as well as we discuss curves of steepest descent. The isoperimetric inequality for the level curves of coordinate functions of planar $p$-harmonic maps is proven. Our main techniques involve relations between quasiregular maps and planar PDEs. We generalize some results due to P. Lindqvist, G. Alessandrini, G. Talenti and P. Laurence.

We investigate the logarithmic and power-type convexity of the length of the level curves for a-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the p-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} regularity. The results are complemented by a number of estimates for the derivatives L′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L'$$\\end{document} and L′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L''$$\\end{document} of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes results due to Alessandrini, Longinetti, Talenti and Lewis in the Euclidean setting, as well as a recent article of ours devoted to the harmonic case on surfaces.

The interconnection between the maximum principle [13] and dynamic programming [3], the two main methods in the theory of optimal control and differential games, is now understood much better due to the theory of viscosity solutions of firstand second-order nonlinear PDEs [2, 8, 14]. The value functions in optimal control and differential games have been proved to be generalized (viscosity) solutions of the corresponding Hamilton–Jacobi–Bellman–Isaacs (HJBI) equations. The classical method of characteristics (MC), which reduces the solution of a PDE to the integration of an ODE (characteristic) system, is one of the attractive and powerful tools for solving nonlinear first-order PDEs arising in control theory and mathematical physics [5]. Characteristics are of special interest in control theory, since they represent optimal paths [10]. This is clear in regarding the regular characteristics. The nonsmoothness of the generalized (e.g., viscosity) solution and/or of the Hamiltonian (left-hand-side function of the PDE), i.e., the presence of singular surfaces, is often referred to as an obstacle to the implementation of the MC. On the other hand, the construction of singular surfaces and lines is an essential and interesting part of the solution of a problem in control theory. The method of singular characteristics (MSC) shows that one can overcome this obstacle by using the same notion of characteristics suitably modified [11,12]. In this paper, a new notion of singular characteristics (SC) is suggested, which, together with the classical (regular) ones, form generalized characteristics. SC are effective for the construction of singular lines, surfaces, and manifolds of a nonsmooth solution (to firstor second-order PDEs), which carry essential information on the corresponding control or physical problem or phenomenon. Singular characteristics were found owing to investigation of singular paths in differential games and optimal control [4,9,10]. Regular paths in these domains are known to be governed by a Hamiltonian ODE system, the characteristic system for the HJBI-equation. In many cases, singular paths are described by similar equations by using the so-called singular controls. The attempt to eliminate singular controls from these equations has led to the discovery of singular characteristics, which appear to be inherent not only to game or control problems but also to the general nonlinear first-order PDEs. Owing to the notion of SC, many singular lines and surfaces known in control theory have received their invariant description in terms of general (abstract) PDEs. The possible interconnection between singular paths and characteristics (in some generalized sense) was mentioned in [7]. This general mathematical insight into the nature of a singularity, as a rule, simplifies the solution procedure of a game or control problem. Correspondingly, the experience accumulated during the construction of solutions to the latter problems is highly useful for the understanding the structure of singularities of the viscosity solutions to general PDEs. The considerations of the present paper are given in general mathematical terms rather than in terms specific for control theory. The ODE systems of singular characteristics are derived for several types of well-known singular surfaces in optimal control and differential games. The solutions of particular problems are presented.

On the existence in the large of solutions to the one-dimensional, isentropic hydrodynamic equations in a bounded domain.- Initial-boundary value and scattering problems in mathematical physics.- On shape optimization of a turbine blade.- Free boundary problems for the Navier-Stokes equations.- A geometric maximum principle, plateau's problem for surfaces of prescribed mean curvature, and the two dimensional analogue of the catenary.- Finite Elements for the Beltrami operator on arbitrary surfaces.- Comparison principles in capillarity.- Remarks on diagonal elliptic systems.- Quasiconvexity, growth conditions and partial regularity.- The monotonicity formula in geometric measure theory, and an application to a partially free boundary problem.- Isoperimetric problems having continua of solutions.- Harmonic maps - Analytic theory and geometric significance.- Asymptotic behavior of solutions of some quasilinear elliptic systems in exterior domains.- Decomposition theorems and their application to non-linear electro- and magneto-static boundary value problems.- Initial boundary value problems in thermoelasticity.- Applications of variational methods to problems in the geometry of surfaces.- Open problems in the degree theory for disc minimal surfaces spanning a curve in ?3.- On a modified version of the free geodetic boundary-value problem.

Collection of Solved Nonlinear Problems for Remote Shaping and Patterning of Liquid Structures on Flat and Curved Substrates by Electric and Thermal Fields

We obtain a generalized version of an inequality, first derived by C. Bandle in the analytic setting, for weak subsolutions of a singular Liouville-type equation. As an application we obtain a new proof of the Alexandrov isoperimetric inequality on singular abstract surfaces. Interestingly enough, motivated by this geometric problem, we obtain a seemingly new characterization of local metrics on Alexandrov's surfaces of bounded curvature. At least to our knowledge, the characterization of the equality case in the isoperimetric inequality in such a weak framework is new as well.

We characterize least-perimeter enclosures of prescribed area on some piecewise smooth manifolds, including certain polyhedra, double spherical caps, and cylindrical cans.

Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

CHAPTER 1 - FOURIER SERIES AND THE ISOPERIMETRIC PROBLEM

Completely boundary-free minimum and maximum principles for neutron transport and their least-squares and Galerkin equivalents

Let G be a k-step Carnot group of homogeneous dimension Q. Later on we shall present some of the results recently obtained in [32] and, in particular, an intrinsic isoperimetric inequality for a C2-smooth compact hypersurface S with boundary @S. We stress that S and @S are endowed with the homogeneous measures n????1 H and n????2 H , respectively, which are actually equivalent to the intrinsic (Q - 1)-dimensional and (Q - 2)-dimensional Hausdor measures with respect to a given homogeneous metric % on G. This result generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [29] and Allard [1], independently. One may also deduce some related Sobolev-type inequalities. The strategy of the proof is inspired by the classical one and will be discussed at the rst section. After reminding some preliminary notions about Carnot groups, we shall begin by proving a linear isoperimetric inequality. The second step is a local monotonicity formula. Then we may achieve the proof by a covering argument. We stress however that there are many dierences, due to our non-Euclidean setting. Some of the tools developed ad hoc are, in order, a \blow-up theorem, which holds true also for characteristic points, and a smooth Coarea Formula for the HS-gradient. Other tools are the horizontal integration by parts formula and the 1st variation formula for the H-perimeter n????1 H already developed in [30, 31] and then generalized to hypersurfaces having non-empty characteristic set in [32]. These results can be useful in the study of minimal and constant horizontal mean curvature hypersurfaces in Carnot groups.

Hyperbolic ends with particles and grafting on singular surfaces

We prove two related results. The first is an “earthquake theorem” for closed hyperbolic surfaces with cone singularities where the total angle is less than π: any two such metrics in are connected by a unique left earthquake. The second result is that the space of “globally hyperbolic” AdS manifolds with “particles” – cone singularities (of given angle) along time-like lines – is parametrized by the product of two copies of the Teichmüller space with some marked points (corresponding to the cone singularities). The two statements are proved together.