Convex Planar Domain Research Articles

Birkhoff attractors of dissipative billiards

Abstract We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with ‘pinched’ curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that, generically, the Birkhoff attractor is complicated, both from the topological and the dynamical points of view.

On the shape of solutions to elliptic equations in possibly non convex planar domains

On the shape of solutions to elliptic equations in possibly non convex planar domains

A comparison between Neumann and Steklov eigenvalues

This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue |\Omega| \mu_1(\Omega) for a Lipschitz open set \Omega in the plane and the normalized first (non-trivial) Steklov eigenvalue P(\Omega) \sigma_1(\Omega) . More precisely, we study the ratio F(\Omega):=|\Omega| \mu_1(\Omega)/P(\Omega) \sigma_1(\Omega) . We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets \Omega . Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke–Santaló diagrams (x,y)=(|\Omega| \mu_1(\Omega), P(\Omega) \sigma_1(\Omega) ) .

Convergence of a regularized finite element discretization of the two-dimensional Monge–Ampère equation

This paper proposes a regularization of the Monge–Ampère equation in planar convex domains through uniformly elliptic Hamilton–Jacobi–Bellman equations. The regularized problem possesses a unique strong solution u ε u_\varepsilon and is accessible to the discretization with finite elements. This work establishes uniform convergence of u ε u_\varepsilon to the convex Alexandrov solution u u to the Monge–Ampère equation as the regularization parameter ε \varepsilon approaches 0 0 . A mixed finite element method for the approximation of u ε u_\varepsilon is proposed, and the regularized finite element scheme is shown to be uniformly convergent. The class of admissible right-hand sides are the functions that can be approximated from below by positive continuous functions in the L 1 L^1 norm. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions u u .

Convex Hulls of Random Order Types

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n -point order type, chosen uniformly at random from all such order types, is on average 4+ o (1). For labeled order types, this number has average \(4- \mbox{$\frac{8}{n^2 - n +2}$}\) and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2 d + o (1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős-Szekeres theorem: for any fixed k , as n → ∞, a proportion 1 - O (1/ n ) of the n -point simple order types contain a triangle enclosing a convex k -chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A 4 , S 4 , or A 5 (and each case is possible). These are the finite subgroups of SO (3) and our proof follows the lines of their characterization by Felix Klein.

On intersection probabilities of four lines inside a planar convex domain

AbstractLet $n\geq 2$ random lines intersect a planar convex domain D. Consider the probabilities $p_{nk}$ , $k=0,1, \ldots, {n(n-1)}/{2}$ that the lines produce exactly k intersection points inside D. The objective is finding $p_{nk}$ through geometric invariants of D. Using Ambartzumian’s combinatorial algorithm, the known results are instantly reestablished for $n=2, 3$ . When $n=4$ , these probabilities are expressed by new invariants of D. When D is a disc of radius r, the simplest forms of all invariants are found. The exact values of $p_{3k}$ and $p_{4k}$ are established.

Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains

Abstract We prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either the Dirichlet energy E f ⁢ ( Ω ) {E_{f}(\Omega)} of the Laplacian in the domain Ω or the first eigenvalue λ 1 ⁢ ( Ω ) {\lambda_{1}(\Omega)} of the Dirichlet-Laplacian. Usually, one considers minimization of such functionals (often with measure constraint), as for example for the famous Saint-Venant and Faber-Krahn inequalities. By adding the convexity constraint (and possibly other natural constraints), we instead consider the rather unusual and difficult question of maximizing these functionals. This paper follows a series of papers by the authors, where the leading idea is that a certain concavity property of the shape functional that is minimized leads optimal shapes to locally saturate their convexity constraint, which geometrically means that they are polygonal. In these previous papers, the leading term in the shape functional was usually the opposite of the perimeter, for which the aforementioned concavity property was rather easy to obtain through computations of its second order shape derivative. By carrying classical shape calculus, a similar concavity property can be observed for the opposite of E f ⁢ ( Ω ) {E_{f}(\Omega)} or λ 1 ⁢ ( Ω ) {\lambda_{1}(\Omega)} when shapes are smooth and convex. The main novelty in the present paper is the proof of a weak convexity property of E f ⁢ ( Ω ) {E_{f}(\Omega)} and λ 1 ⁢ ( Ω ) {\lambda_{1}(\Omega)} among planar convex shapes, namely rather nonsmooth shapes. This involves new computations and estimates of the second order shape derivatives of E f ⁢ ( Ω ) {E_{f}(\Omega)} and λ 1 ⁢ ( Ω ) {\lambda_{1}(\Omega)} interesting for themselves.

On the number of critical points of the second eigenfunction of the Laplacian in convex planar domains

In this paper we consider the second eigenfunction of the Laplacian with Dirichlet boundary conditions in convex domains. If the domain has large eccentricity then the eigenfunction has exactly two nondegenerate critical points (of course they are one maximum and one minimum). The proof uses some estimates proved by Jerison ([13]) and Grieser-Jerison ([10]) jointly with a topological degree argument. Analogous results for higher order eigenfunctions are proved in rectangular-like domains considered in [11].

Geometric computation of Christoffel functions on planar convex domains

For an arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelogram containing the domain is constructed.As an application we obtain a new proof that every planar convex domain possesses optimal polynomial meshes.

Inverse problems and rigidity questions in billiard dynamics

AbstractA Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

Nodal line estimates for the second Dirichlet eigenfunction

We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser–Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side lengths. We also discuss applications of our results to Courant-sharp eigenfunctions and spectral partitioning.

Blaschke--Santaló Diagram for Volume, Perimeter, and First Dirichlet Eigenvalue

We are interested in the study of Blaschke-Santalo diagrams describing the possible inequalities involving the first Dirichlet eigenvalue, the perimeter and the volume, for different classes of sets. We give a complete description of the diagram for the class of open sets in R d , basically showing that the isoperimetric and Faber-Krahn inequalities form a complete system of inequalities for these three quantities. We also give some qualitative results for the Blaschke-Santalo diagram for the class of planar convex domains: we prove that in this case the diagram can be described as the set of points contained between the graphs of two continuous and increasing functions. This shows in particular that the diagram is simply connected, and even horizontally and vertically convex. We also prove that the shapes that fill the upper part of the boundary of the diagram are smooth (C 1,1), while those on the lower one are polygons (except for the ball). Finally, we perform some numerical simulations in order to have an idea on the shape of the diagram; we deduce both from theoretical and numerical results some new conjectures about geometrical inequalities involving the functionals under study in this paper.

Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation

We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium enhancement in the random diffusion of criminal agents may exert visible relaxation effects. It is shown that sufficient regularity of the non-negative source terms in the system and a sufficiently strong nonlinear enhancement ensure that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is globally bounded under mild additional conditions on the source terms. These results are supplemented by numerical evidence which illustrates smoothing effects in solutions with sharply structured initial data in the presence of such porous medium-type diffusion and support the existence of singular structures in the linear diffusion case, which is the type of diffusion proposed in Ref. 29.

Regularity of optimal transport between planar convex domains

For $0<p<+\infty$, we prove a global $W^{2,p}$-estimate for potentials of optimal transport maps between convex domains in the plane. Among the tools developed for that purpose are obliqueness in general convex domains and estimates for the growth of eccentricity of sections of the potentials.

Asymptotics of the Solution of a Singular Optimal Distributed Control Problem with Essential Constraints in a Convex Domain

We consider an optimal distributed control problem in a convex planar domain with a quadratic performance functional and a small parameter multiplying the higher derivatives. Further, the characteristics of the limit equation of the problem are parallel to the $$y$$-axis. Using the method of matching asymptotic expansions in conjunction with the auxiliary parameter method, we derive a complete asymptotic expansion (up to any power of the small parameter) of the optimal state of the controlled system and the optimal control.

The Torsion Function of Convex Domains of High Eccentricity

The torsion function of a convex planar domain Ω has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function when the eccentricity of the domain is large. We obtain an approximation for the torsion function on any convex planar domain by viewing the domain as a perturbation of a rectangle in order to define an approximate Green’s function for the Laplacian. For a class of convex domains we use this approximation to establish sharp bounds on the Hessian and the infinitesimal shape of the level sets around its maximum. We also use these results to construct examples demonstrating contrasting behaviour of the torsion function and the first eigenfunction of the Dirichlet Laplacian around their respective maxima.

Chord shortening flow and a theorem of Lusternik and Schnirelmann

We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord which is not orthogonal to the boundary would shrink to a point in finite time under the flow.

Prescribing Capacitary Curvature Measures on Planar Convex Domains

For $p\in (1,2]$ and a bounded, convex, nonempty, open set $\Omega\subset\mathbb R^2$ let $\mu_p(\bar{\Omega},\cdot)$ be the $p$-capacitary curvature measure (generated by the closure $\bar{\Omega}$ of $\Omega$) on the unit circle $\mathbb S^1$. This paper shows that such a problem of prescribing $\mu_p$ on a planar convex domain: "Given a finite, nonnegative, Borel measure $\mu$ on $\mathbb S^1$, find a bounded, convex, nonempty, open set $\Omega\subset\mathbb R^2$ such that $d\mu_p(\bar{\Omega},\cdot)=d\mu(\cdot)$" is solvable if and only if $\mu$ has centroid at the origin and its support $\mathrm{supp}(\mu)$ does not comprise any pair of antipodal points. And, the solution is unique up to translation. Moreover, if $d\mu_p(\bar{\Omega},\cdot)=\psi(\cdot)\,d\ell(\cdot)$ with $\psi\in C^{k,\alpha}$ and $d\ell$ being the standard arc-length element on $\mathbb S^1$, then $\partial\Omega$ is of $C^{k+2,\alpha}$.

Prime lattice points in ovals

We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.

Asymptotics of the Solution of a Singular Optimal Distributed Control Problem in a Convex Domain

We consider an optimal distributed control problem in a planar convex domain with smooth boundary and a small parameter at the highest derivatives of an elliptic operator. The zero Dirichlet condition is given on the boundary of the domain, and the control is included additively in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. Solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with a coefficient. This structure of the optimality criterion makes it possible to strengthen, if necessary, the role of either the first or the second term of the criterion. In the first case, it is more important to achieve the desired state, while, in the second case, it is preferable to minimize the resource consumption. We study in detail the asymptotics of the problem generated by the sum of the Laplace operator with a small coefficient and a first-order differential operator. A feature of the problem is the presence of the characteristics of the limit operator which touch the boundary of the domain. We obtain a complete asymptotic expansion of the solution of the problem in powers of the small parameter in the case where the optimal control is an interior point of the set of admissible controls.