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Erik Duse

Publications and source records attributed to Erik Duse.

12 recordsLinked to original sources
arXiv2024

Coerciveness and Morrey Inequalities for Elliptic Operators with Natural Boundary Conditions via Weitzenb\"ock Identities

We prove a Weitzenb\"ock identity for general pairs of constant coefficient homogeneous first order partial differential operators, and deduce from it sufficient algebraic conditions for coerciveness and Morrey estimates under the natural 1/2 boundary conditions. Our proof of the $W^{1,2}$ elliptic estimate relies on the Aronszajn-Necas-Smith coercive estimate. For generalized strongly pseudoconvex domains, we improve the Morrey estimate to a weighted $W^{1,2}$ square function estimate, using a generalized Cauchy-Pompieu reproducing formula and the T1 theorem for singular integrals. We use Van Schaftingen's notion of cocanceling to study the generalized Levi forms appearing.

math.AP↗
arXiv2024

On Fuglede's flux extensions and the point wise definition of linear partial differential operators

In this work we provide a survey of Fuglede's flux extensions of first order partial differential operators, a concept largely forgotten today. A long the way we also survey the classical weak and strong extensions of PDE operators and the works of Friedrichs and H\"ormander. We give several applications of this theory showing its usefulness, as well as connecting it to more recent developments in connection to various sharp versions of the divergence theorem. In particular, we use it to prove a generalization of Morera's theorem valid for general first order operators. Using this theory we also prove a new local limit formula for the maximal extension of a first order operator. We initiate a study of this limit and connect it to the wave cone of the operator, a concept that first arose in the theory of compensated compactness. Hopefully, this will contribute to a rival of Fuglede's beautiful ideas.

math.AP↗
arXiv2023

Lozenge Tilings of a Hexagon and q-Racah Ensembles

We study the limiting behavior of random lozenge tilings of the hexagon with a q-Racah weight as the size of the hexagon grows large. Based on the asymptotic behavior of the recurrence coefficients of the q-Racah polynomials, we give a new proof for the fact that that the height function for a random tiling concentrates near a deterministic limit shape and that the global fluctuations are described by the Gaussian Free Field. These results were recently proved using (dynamic) loop equation techniques. In this paper we extend the recurrence coefficient approach that was developed for (dynamic) orthogonal polynomial ensembles to the setting of q-orthogonal polynomials. An interesting feature is that the complex structure is easily found from the limiting behavior of the (explicitly known) recurrence coefficients. A particular motivation for studying this model is that the variational characterization of the limiting height function has an inhomogeneous term. The study of the regularity properties of the minimizer for general variation problems with such inhomogeuous terms is a challenging open problem. We show that, in a general setup, the variational problem gives rise to a natural complex structure that is associated to the same Beltrami equation as in the homogeneous situation. We also derive a relation between the complex structure and the complex slope. In case of the q-Racah weighting of lozenge tilings of the hexagon, our representation of the limit shape and their fluctuations in terms of the recurrence coefficients allows us to verify this relation explicitly.

math.PR↗
arXiv2023

Generic Ill-posedness of the Energy-Momentum Equations and Differential Inclusions

We show that the energy-momentum equations arising from inner variations whose Lagrangian satisfies a generic symmetry condition are generically ill-posed. This is done by proving that there exists a subclass of Lipschitz solutions that are also solutions to a differential inclusion. In particular these solutions can be nowhere C1. We prove that these solutions are not stationary points if the Lagrangian W is C1 and strictly rank-one convex. In view of the Lipschitz regularity result of Iwaniec, Kovalev and Onninen for solution of the energy-momentum equation in dimension 2 we give a sufficient condition for the non-existence of a partial C1-regularity result even under the condition that the mappings satisfy a positive Jacobian determinant condition. Finally we consider a number of well-known functionals studied in nonlinear elasticity and geometric function theory and show that these do not satisfy this obstruction to partial regularity.

math.AP↗
arXiv2021

Second Order Linear Elliptic Equations and Hodge-Dirac Operators

In this paper we show how a second order scalar uniformly elliptic equation on divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary condition. This first order system involves Hodge-Dirac operators and can be seen as a natural generalization of the Beltrami equation in the plane and we develop a theory for this equation, extending results from the plane to higher dimension. The reduction to a first order system applies both to linear as well as quasilinear second order equations and we believe this to be of independent interest. Using the first order system, we give a new representation formula of the solution of the Dirichlet problem both on simply and finitely connected domains. This representation formula involves only singular integral operators of convolution type and Neumann series there of, for which classical Calderón-Zygmund theory is applicable. Moreover, no use is made of any fundamental solution or Green's function beside fundamental solutions of constant coefficient operators. Remarkably, this representation formula applies also for solutions of the fully non-linear first order system. We hope that the representation formula could be used for numerically solving the equations. Using these tools we give a new short proof of Meyers' higher integrability theorem. Furthermore, we show that the solutions of the first order system are Hölder continuous with the same Hölder coefficient as the solutions of the second order equations. Finally, factorization identities and representation formulas for the higher dimensional Beurling-Ahlfors operator are proven.

math.AP↗
arXiv2020

A generalized Montel theorem for a class of first order elliptic equations with measurable coefficients

In this paper we prove a generalization of Montel's theorem for a class of first order elliptic equations with measurable coefficients involving Hodge-Dirac operators. We then apply this result to sequences of solutions of second order uniformly elliptic equations with measurable coefficients on divergence form and show that this results in a precompactness result for such sequences.

math.AP↗
arXiv2020

Dimer Models and Conformal Structures

In this work we study the variational problem associated to dimer models, a class of models from integrable probability and statistical mechanics in dimension two which have been the focus of intense research efforts over the last decades. These models give rise to an infinite family of non-differentiable functionals on Lipschitz functions with gradient constraint, determined by solutions of the Dirichlet problem on compact convex polygons for a class of Monge-Amp\`ere equations. We settle a number or outstanding open questions for this infinite class functionals. In particular we prove a complete classification of the regularity of minimizers, also known as height functions, for all dimer models for a natural class of polygonal (simply or multiply connected) domains much studied in numerical simulations and elsewhere. Our classification in particular implies that the Pokrovsky-Talapov law holds for all dimer models at a generic point on the frozen boundary and in addition shows a very strong local rigidity of dimer models which can be interpreted as a geometric universality result. Furthermore, we give a complete classification of the regularity of the associated free boundary, also known in the literature as frozen boundary or arctic curves and prove that they are all algebraic curves. The lack of differentiability of the functionals is intimately connected to the boundary behaviour of the solutions to the Monge-Amp\`ere equations and we prove a complete classification for these, of independent interest.

math.AP↗
arXiv2018

Universal edge fluctuations of discrete interlaced particle systems

We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models, and are closely related to the eigenvalue minor processes of a broad class of random Hermitian matrices. They have a determinantal structure, with a known correlation kernel. We rescale the systems by $\frac1n$, and examine the asymptotic behaviour, as $n \to \infty$, under weak asymptotic assumptions for the (rescaled) particles on row $n$: The empirical distribution of these converges weakly to a probability measure with compact support, and they otherwise satisfy mild regulatory restrictions. We prove that the correlation kernel of particles in the neighbourhood of `typical edge points' convergences to the extended Airy kernel. To do this, we first find an appropriate scaling for the fluctuations of the particles. We give an explicit parameterisation of the asymptotic edge, define an analogous non-asymptotic edge curve (or finite $n$-deterministic equivalent), and choose our scaling such that that the particles fluctuate around this with fluctuations of order $O(n^{-\frac13})$ and $O(n^{-\frac23})$ in the tangent and normal directions respectively. While the final results are quite natural, the technicalities involved in studying such a broad class of models under such weak asymptotic assumptions are unavoidable and extensive.

math.PR↗
arXiv2015

The Cusp-Airy Process

At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point.

math.PR↗
arXiv2015

Asymptotic Geometry of Discrete Interlaced Patterns: Part II

We study the boundary of the liquid region $\mathcal{L}$ in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line $(x,1)$, $x\in\mathbb{R}\equiv \partial \mathbb{H}$. We assume that the initial particle configuration converges weakly to a limiting density $ϕ(x)$, $0\le ϕ\leq 1$. The liquid region is given by a homeomorphism $W_{\mathcal{L}}: \mathcal{L}\to \mathbb{H}$, the upper half plane, and we consider the extension of $W_{\mathcal{L}}^{-1}$ to $\overline{\mathbb{H}}$. Part of $\partial \mathcal{L}$ is given by a curve, the edge $\mathcal{E}$, parametrized by intervals in $\partial \mathbb{H}$, and this corresponds to points where $ϕ$ is identical to $0$ or $1$. If $0<ϕ<1$, the non-trivial support, there are two cases. Either $W_{\mathcal{L}}^{-1}(w)$ has the limit $(x,1)$ as $w\to x$ non-tangentially and we have a \emph{regular point}, or we have what we call a singular point. In this case $W_{\mathcal{L}}^{-1}$ does not extend continuously to $\overline{\mathbb{H}}$. Singular points give rise to parts of $\partial \mathcal{L}$ not given by $\mathcal{E}$ and which can border a frozen region, or be "inside" the liquid region. This shows that in general the boundary of $\partial \mathcal{L}$ can be very complicated. We expect that on the singular parts of $\partial \mathcal{L}$ we do not get a universal point process like the Airy or the extended Sine kernel point processes. Furthermore, $\mathcal{E}$ and the singular parts of $\partial \mathcal{L}$ are shocks of the complex Burgers equation.

math-ph↗
arXiv2015

Asymptotic geometry of discrete interlaced patterns: Part I

A discrete Gelfand-Tsetlin pattern is a configuration of particles in Z^2. The particles are arranged in a finite number of consecutive rows, numbered from the bottom. There is one particle on the first row, two particles on the second row, three particles on the third row, etc, and particles on adjacent rows satisfy an interlacing constraint. We consider the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform probability measure on the set of all tilings of certain polygons with lozenges. We prove a determinantal structure, and calculate the correlation kernel. We consider the asymptotic behaviour of the system as the size increases under the assumption that the empirical distribution of the deterministic particles on the top row converges weakly. We consider the asymptotic `shape' of such systems. We provide parameterisations of the asymptotic boundaries and investigate the local geometric properties of the resulting curves. We show that the boundary can be partitioned into natural sections which are determined by the behaviour of the roots of a function related to the correlation kernel. This paper should be regarded as a companion piece to the upcoming paper, [4], in which we resolve some of the remaining issues. Both of these papers serve as background material for the upcoming papers, [5] and [6], in which we examine the edge asymptotic behaviour.

math.PR↗