Quasi-linear Setting Research Articles

Existence and regularity for a general class of quasilinear elliptic problems involving the Hardy potential

In a very general quasilinear setting, we show that the regularizing effect of a first order term causes the existence of energy solutions for problems involving the Hardy potential and L1 data. In the same setting we study sharp (local and global) integral estimates for the second derivatives of the solutions.

Well-Posedness for the Dispersive Hunter–Saxton Equation

Abstract This article represents a 1st step toward understanding the well-posedness of the dispersive Hunter–Saxton equation, which arises in the study of nematic liquid crystals. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character. Further, the lack of spatial decay obstructs access to dispersive tools, including local smoothing estimates. Here, we give the 1st proof of local and global well-posedness for the Cauchy problem. Secondly, we improve our well-posedness results with respect to the low regularity of the initial data. The key techniques we use include constructing modified energies to realize a normal form analysis in our quasilinear setting, and frequency envelopes to prove continuous dependence with respect to the initial data.

Efficient ex post implementable auctions and English auctions for bidders with non-quasilinear preferences

We study efficient auction design for a single indivisible object when bidders have interdependent values and non-quasilinear preferences. Instead of quasilinearity, we assume only that bidders have positive wealth effects. Our setting nests cases where bidders are ex ante asymmetric, face financial constraints, are risk averse, and/or face ensuing risk. We give necessary and sufficient conditions for the existence of an ex post implementable and (ex post Pareto) efficient mechanism. These conditions differ between the standard case where the auctioneer is a seller and when the auctioneer is a buyer (a procurement auction).When the auctioneer is a seller, there is an efficient ex post implementable mechanism if there is an efficient ex post implementable mechanism in a corresponding quasilinear setting. This result extends established results on efficient ex post equilibria of English auctions with quasilinearity to our non-quasilinear setting. Yet, in the procurement setting there is no mechanism that has an efficient ex post equilibrium if the level of interdependence between bidders is sufficiently strong. This result holds even if bidder costs satisfy standard single crossing conditions that are sufficient for efficient ex post implementation in the quasilinear setting.

Quasilinear convexity and quasilinear stars in the ray space of a supertropical quadratic form

Relying on rays, we search for submodules of a module V over a supertropical semiring on which a given anisotropic quadratic form is quasilinear. Rays are classes of a certain equivalence relation on V, that carry a notion of convexity, which is consistent with quasilinearity. A criterion for quasilinearity is specified by a Cauchy-Schwartz ratio which paves the way to a convex geometry on , supported by a ‘supertropical trigonometry’. Employing a (partial) quasiordering on , this approach allows for producing convex quasilinear sets of rays, as well as paths, which contain a given quasilinear set in a systematic way. Minimal paths are endowed with a surprisingly rich combinatorial structure, delivered to the graph determined by pairs of quasilinear rays – apparently a fundamental object in the theory of supertropical quadratic forms.

Single Object Auctions With Externalities: A Tractable Model

We consider the single object auction model with allocative externalities in a private valuation and quasi-linear setting. We model externalities by assuming that every agent has a private valuation (for the object) and a strict ranking of other agents. The utility for an agent when another agent receives the object is the product of his own valuation and a real number that depends on the rank of this agent in his ranking. When the only private information is the valuation of the agents, we characterise the implementable allocation rules and use these to derive the optimal auction. The optimal auction collects payments from agents who do not receive the object.

Cloaking for a quasi-linear elliptic partial differential equation

In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N=2,3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks though non-degenerate are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.

Efficient Ex Post Implementable Auctions and English Auctions for Bidders with Non-Quasilinear Preferences

We study efficient auction design for a single indivisible object when bidders have interdependent values and non-quasilinear preferences. Instead of quasilinearity, we assume only that bidders have positive wealth effects. Our setting nests cases where bidders are ex ante asymmetric, face financial constrains, are risk averse, and/or face ensuing risk. We give necessary and sufficient conditions for when there exists an ex post implementable and (ex post Pareto) efficient mechanism. Necessary and sufficient conditions for the existence of efficient ex post implementable mechanisms differ between the case where the object is a good and when the object is a bad. In the good setting, there is an efficient ex post implementable mechanism if there is an efficient ex post implementable mechanism in a corresponding quasilinear setting. This result extends established results on efficient ex post equilibria of English auctions with quasilinearity to our non-quasilinear setting. Yet, in the bad setting (i.e. a procurement auction) there is no mechanism that has an ex post efficient equilibrium if the level of interdependence between bidders is sufficiently strong. This result holds even if bidder costs satisfy standard crossing conditions that are sufficient for efficient ex post implementation in the quasilinear setting.

Contests, private provision of public goods and evolutionary stability

We study evolutionary stability for public goods games incentivized by a contest. In a quasi-linear setting, we derive conditions such that evolutionary stability, Nash equilibrium and efficient solution coincide.

The Incompatibility of Pareto Optimality and Dominant-Strategy Incentive Compatibility in Sufficiently-Anonymous Budget-Constrained Quasilinear Settings

We analyze the space of deterministic, dominant-strategy incentive compatible, individually rational and Pareto optimal combinatorial auctions. We examine a model with multidimensional types, nonidentical items, private values and quasilinear preferences for the players with one relaxation; the players are subject to publicly-known budget constraints. We show that the space includes dictatorial mechanisms and that if dictatorial mechanisms are ruled out by a natural anonymity property, then an impossibility of design is revealed. The same impossibility naturally extends to other abstract mechanisms with an arbitrary outcome set if one maintains the original assumptions of players with quasilinear utilities, public budgets and nonnegative prices.

A note on the incompatibility of strategy-proofness and Pareto-optimality in quasi-linear settings with public budgets

We show that any deterministic mechanism, for allocating identical items that are complements to budget-constrained bidders, cannot simultaneously satisfy individual-rationality, strategy-proofness, Pareto-efficiency, and no-positive-transfers. This holds even for two bidders, two items, and commonly-known budgets, and generalizes to richer settings.

Competitive equilibrium in two sided matching markets with general utility functions

Two sided matching markets are among the most studied models in market design. There is a vast literature on the structure of competitive equilibria in these markets, yet most of it is focused on quasilinear settings. General (non-quasilinear) utilities can, for instance, model smooth budget constraints as a special case. Due to the difficulty of dealing with arbitrary non-quasilinear utilities, most of the existing work on non-quasilinear utilities is limited to the special case of hard budget constraints in which the utility of each agent is quasilinear as long as her payment is within her budget limit and is negative infinity otherwise. Most of the work on competitive equilibria with hard budget constraints rely on some form of ascending auction. For general non-quasilinear utilities, such ascending auctions may not even converge in finite time. As such, almost all of the existing work on general non-quasilinear utilities have resorted to non-constructive proofs based on fixed point theorems or discretization. We present the first direct characterization of competitive equilibria in such markets. Our approach is constructive and solely based on induction. Our characterization reveals striking similarities between the payments at the lowest competitive equilibrium for general utilities and VCG payments for quasilinear utilities. We also show that lowest competitive equilibrium is group strategyproof for the agents on one side of the market (e.g., for buyers).

A simple status quo that ensures participation (with application to efficient bargaining)

We consider Bayesian incentive-compatible mechanisms with independent types and either private values or interdependent values that satisfy a form of “congruence.” We show that in these settings, interim participation constraints are satisfied when the status quo is the randomized allocation that has the same distribution as the equilibrium allocation in the mechanism. Moreover, when utilities are convex in the allocation, we can instead satisfy participation constraints with the deterministic status quo equal to the expected equilibrium allocation in the mechanism. For quasilinear settings, these observations imply the possibility of efficient bargaining when the status quo specifies the expected efficient decision provided that the total surplus is convex in the decision.

Expressive markets for donating to charities

When donating money to a (say, charitable) cause, it is possible to use the contemplated donation as a bargaining chip to induce other parties interested in the charity to donate more. Such negotiation is usually done in terms of matching offers, where one party promises to pay a certain amount if others pay a certain amount. However, in their current form, matching offers allow for only limited negotiation. For one, it is not immediately clear how multiple parties can make matching offers at the same time without creating circular dependencies. Also, it is not immediately clear how to make a donation conditional on other donations to multiple charities when the donor has different levels of appreciation for the different charities. In both these cases, the limited expressiveness of matching offers causes economic loss: it may happen that an arrangement that all parties (donors as well as charities) would have preferred cannot be expressed in terms of matching offers and will therefore not occur. In this paper, we introduce a bidding language for expressing very general types of matching offers over multiple charities. We formulate the corresponding clearing problem (deciding how much each bidder pays, and how much each charity receives), and show that it cannot be approximated to any ratio in polynomial time unless P = NP, even in very restricted settings. We give a mixed integer program formulation of the clearing problem, and show that for concave bids, the program reduces to a linear program. We then show that the clearing problem for a subclass of concave bids is at least as hard as the decision variant of linear programming. We also consider the case where each charity has a target amount, and biddersʼ willingness-to-pay functions are concave. Here, we show that the optimal surplus can be approximated to a ratio m, the number of charities, in polynomial time (and no significantly better approximation is possible in polynomial time unless P = NP); no polynomial-time approximation ratio is possible for maximizing the total donated, unless P = NP. Subsequently, we show that the clearing problem is much easier when bids are quasilinear—for maximizing surplus, the problem decomposes across charities, and for maximizing the total donated, a greedy approach is optimal if the bids are concave (although this latter problem is weakly NP-hard when the bids are not concave). For the quasilinear setting, we study the mechanism design question. We show that an ex-post efficient mechanism is impossible even with only one charity and a very restricted class of bids. We also show that there can be benefits to linking the charities from a mechanism design standpoint. Finally, we discuss an experiment in which we used this methodology to collect money for victims of the 2004 Indian Ocean Tsunami.

A Wave Amplitude Transition in a Quasi-Linear Model with Radiative Forcing and Surface Drag

Abstract A quasi-linear two-layer quasigeostrophic β-plane model of the interaction between a baroclinic jet and a single zonal wavenumber perturbation is used to study the mechanics leading to a wave amplitude bifurcation—in particular, the role of the critical surfaces in the upper-tropospheric jet flanks. The jet is forced by Newtonian heating toward a radiative equilibrium state, and Ekman damping is applied at the surface. When the typical horizontal scale is approximately the Rossby radius of deformation, the waves equilibrate at a finite amplitude that is comparable to the mean flow. This state is obtained as a result of a wave-induced temporary destabilization of the mean flow, during which the waves grow to their finite-equilibrium amplitude. When the typical horizontal scale is wider, the model also supports a state in which the waves equilibrate at negligible amplitudes. The transition from small to finite-amplitude waves, which occurs at weak instabilities, is abrupt as the parameters of the system are gradually varied, and in a certain range of parameter values both equilibrated states are supported. The simple two-layer quasi-linear setting of the model allows a detailed examination of the temporary destabilization process inherent in the large-amplitude equilibration. As the waves grow they reduce the baroclinic growth by reducing the vertical shear of the mean flow, and reduce the barotropic decay by reducing the mean potential vorticity gradient at the inner sides of the upper-layer critical levels. Temporary destabilization occurs when the reduction in barotropic decay is larger than the reduction in baroclinic growth, leading to a larger total growth rate. Ekman friction and radiative damping are found to play a major role in sustaining the vertical shear of the mean flow and enabling the baroclinic growth to continue. By controlling the mean flow potential vorticity gradient near the critical level, the model evolution can be changed from one type of equilibration to the other.

A Simple Status Quo that Ensures Participation (with Application to Efficient Bargaining)

We consider Bayesian incentive-compatible mechanisms with independent types and either private values or interdependent values that satisfy a form of congruence. We show that in these settings, interim participation constraints are satisfied when the status quo is the randomized allocation that has the same distribution as the equilibrium allocation in the mechanism. Moreover, when utilities are convex in the allocation, we can instead satisfy participation constraints with the deterministic status quo equal to the expected equilibrium allocation in the mechanism. For quasilinear settings, these observations imply the possibility of efficient bargaining when the status quo has the expected efficient decision provided that the total surplus is convex in the decision.

Modulating Pulse Solutions to Quadratic Quasilinear Wave Equations over Exponentially Long Length Scales

This paper presents an existence proof for modulating pulse solutions to a wide class of quadratic quasilinear Klein-Gordon equations of the form $$\partial_t^2 u = \partial_x^2 u - u + f_1(u, \partial_x u, \partial_t u)\partial_x^2 u + f_2(u, \partial_x u, \partial_t u).$$ Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave-train; they are also referred to as ‘moving breathers’ since they are time-periodic in a moving frame of reference. The problem is formulated as an infinite-dimensional dynamical system with three stable, three unstable and infinitely many neutral directions. By transforming part of the equation into a normal form with an exponentially small remainder term and using a generalisation of local invariant-manifold theory to the quasilinear setting, we prove the existence of small-amplitude modulating pulses on domains in space whose length is exponentially large compared to the magnitude of the pulse.

Modulating pulse solutions for quasilinear wave equations

This paper presents an existence proof for symmetric modulating pulse solutions of a quasilinear wave equation. Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave train; they are also referred to as ‘moving breathers’ since they are time periodic in a moving frame of reference. The problem is formulated as an infinite-dimensional dynamical system with two stable, two unstable and infinitely many neutral directions. Using a partial normal form and a generalisation of local invariant-manifold theory to the quasilinear setting we prove the existence of modulating pulses on arbitrarily large, but finite domains in space and time.