Elliptic Case Research Articles

Parabolic Regularity and Dirichlet boundary value problems

We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form Lu=div(A∇u)−ut=0 in Lip(1,1∕2) time-varying cylinders, where the coefficient matrix A=aij(X,t) is uniformly elliptic and bounded.We show that if the Regularity problem (R)p for the equation Lu=0 is solvable for some 1<p<∞ then the Dirichlet problem (D∗)p′ for the adjoint equation L∗v=0 is also solvable, where p′=p∕(p−1). This result is an analogue of the result established in the elliptic case by Kenig and Pipher (1993). In the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by Hofmann and Lewis (1996) and Nyström (2006) for scalar parabolic systems. In comparison, our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains.

Computing the endomorphism ring of an ordinary abelian surface over a finite field

We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this algorithm only requires the heuristic assumptions required by the algorithm of Biasse and Fieker [2] which computes the class group of an order in a number field in subexponential time. Thus we avoid the multiple heuristic assumptions on isogeny graphs and polarized class groups which were previously required. The output of the algorithm is an ideal in the maximal totally real subfield of the endomorphism algebra, generalizing the elliptic curve case.

Quantum Lax Pairs via Dunkl and Cherednik Operators

We establish a direct link between Dunkl operators and quantum Lax matrices $\mathcal L$ for the Calogero--Moser systems associated to an arbitrary Weyl group $W$ (or an arbitrary finite reflection group in the rational case). This interpretation also provides a companion matrix $\mathcal A$ so that $\mathcal L, \mathcal A$ form a quantum Lax pair. Moreover, such an $\mathcal A$ can be associated to any of the higher commuting quantum Hamiltonians of the system, so we obtain a family of quantum Lax pairs. These Lax pairs can be of various sizes, matching the sizes of orbits in the reflection representation of $W$, and in the elliptic case they contain a spectral parameter. This way we reproduce universal classical Lax pairs by D'Hoker-Phong and Bordner-Corrigan-Sasaki, and complement them with quantum Lax pairs in all cases (including the elliptic case, where they were not previously known). The same method, with the Dunkl operators replaced by the Cherednik operators, produces quantum Lax pairs for the generalised Ruijsenaars systems for arbitrary root systems. As one of the main applications, we calculate a Lax matrix for the elliptic $BC_n$ case with nine coupling constants (van Diejen system), thus providing an answer to a long-standing open problem.

Continuity of the optimal stopping boundary for two-dimensional diffusions

We first show that a smooth fit between the value function and the gain function at the optimal stopping boundary for a two-dimensional diffusion process implies the absence of boundary’s discontinuities of the first kind (the right-hand and left-hand limits exist but differ). We then show that the smooth fit itself is satisfied over the flat portion of the optimal stopping boundary arising from any of its hypothesised jumps. Combining the two facts we obtain that the optimal stopping boundary is continuous whenever it has no discontinuities of the second kind. The derived fact holds both in the parabolic and elliptic case under the sole hypothesis of Holder continuous coefficients, thus improving upon all known results in the parabolic case, and establishing the fact for the first time in the elliptic case. The method of proof relies upon regularity results for the second-order parabolic/elliptic PDEs and makes use of the local time-space calculus techniques.

Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems

In this note, we study the mean curvature flow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is $ u_{\nu} = -\phi(x)(1+|Du|^2)^\frac{1-q}{2} $ for any parameter $ q&gt;0 $. Using the maximum principle, we prove the gradient estimates for the solutions of such a class of boundary value problems. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. In addition, we study the related additive eigenvalue problem for general boundary value problems and describe the asymptotic behavior of the solution at infinity time. The originality of the paper lies in the range $ 0&lt;q&lt;1 $, since there are no any related results before. For parabolic case, we generalize the result of Ma-Wang-Wei [25] to any $ q&gt;0 $. And in elliptic case, we generalize the results in [32] to any $ q\ge 0 $ and to any bounded smooth domain.

Correction to: Asymptotic analysis of linearly elastic shells in normal compliance contact: convergence for the elliptic membrane case

Correction to: Asymptotic analysis of linearly elastic shells in normal compliance contact: convergence for the elliptic membrane case

Three-dimensional interaction of a shock with a thin non-circular channel of low-density gas

Numerical simulation of three-dimensional interaction of a shock with thin circular or non-circular – elliptic or rectangular – low-density gas channel is presented. Euler’s equations for inviscid gas are solved using 5th-order finite-difference WENO scheme. Large-scaled shock wave precursor structure is described in detail. It is shown that in three-dimensional flow internal shear layer instabilities develop faster than in axisymmetric case. High-pressure jet cumulation effect is found to amplify moderately for elliptic and rectangular cases. An influence of channel cross-section shape on the precursor growth rate is studied. It is found that the duration of linear precursor growth phase is greatly increased for stretched channel cross-section shapes.

Berry\u2013Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

We study the random conductance model on the lattice {mathbb {Z}}^d, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension dge 3 quantitative central limit theorems for the random walk in form of a Berry–Esseen estimate with speed t^{-frac{1}{5}+varepsilon } for dge 4 and t^{-frac{1}{10}+varepsilon } for d=3. Additionally, in the uniformly elliptic case in low dimensions d=2,3 we improve the rate in a quantitative Berry–Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for dge 3 we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.

Spatial Effects of Interaction of a Shock with a Lateral Low-Density Gas Channel

Three-dimensional interaction of a shock with lateral low-density gas channel of round, elliptic or rectangular cross-section is numerically studied using Euler’s equations. The structure of formed shock wave precursor is described in detail. Internal shear layer instabilities in three-dimensional flow are shown to develop faster than in axisymmetric case. Moderate amplification of high-pressure jet cumulation effect is noted for elliptic and rectangular channel cases. Dependence of precursor growth rate on cross-section shape is studied. It is found that stretching of cross-section shape significantly increases the duration of linear precursor growth phase.

Asymptotic analysis of linearly elastic shells in normal compliance contact: convergence for the elliptic membrane case

We consider a family of linearly elastic shells all sharing the same middle surface and in normal compliance contact with a deformable foundation on the lower face. By using formal asymptotic analysis, we first characterize a variety of possible limit two-dimensional problems, ranging from elliptic membranes to flexural shells, depending on the geometry of the middle surface, the boundary conditions, the order of the applied forces, and the order of the normal compliance response. Then we focus in the elliptic membrane case, for which we provide a rigorous convergence result.

Properties of blow-up solutions and their initial data for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper is concerned with blow-up solutions to the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type{ut=∇⋅(∇um−uq−1∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0 under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂RN (N≥3), m≥1, q≥2. As the basis on this study, it was recently shown that there exist radial initial data such that the corresponding solutions blow up in the case q>m+2N ([5]). In the parabolic–elliptic case Sugiyama [27] established behavior of blow-up solutions; however, behavior in the parabolic–parabolic case has not been studied. The purpose of this paper is to give many finite-time blow-up solutions and behavior of blow-up solutions in a neighborhood of blow-up time in the parabolic–parabolic case.

Spectral analysis of the hypoelliptic Robin problem

This paper is devoted to a functional analytic approach to the study of the hypoelliptic Robin problem for a second-order, uniformly elliptic differential operator with a complex parameter $$\lambda $$ , under the probabilistic condition that either the absorption phenomenon or the reflection phenomenon occurs at each point of the boundary. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous Robin problem when $$\vert \lambda \vert $$ tends to $$\infty $$ . More precisely, we prove the spectral properties of the closed realization of the uniformly elliptic differential operator, similar to the elliptic (non-degenerate) case. However, in the degenerate case we cannot use Green’s formula to characterize the adjoint operator of the closed realization. Hence, we shift our attention to its resolvent. In the proof, we make use of the Boutet de Monvel calculus in order to study the resolvent and its adjoint in the framework of $$L^{2}$$ Sobolev spaces.

Characteristics and mechanism of mixing enhancement for noncircular synthetic jets at low Reynolds number

In this paper, the evolution of the flow structures and the mixing characteristics in low-aspect-ratio (AR) noncircular incompressible synthetic jets are investigated. Two elongated configurations, elliptic and rectangular orifices with AR = 3 and 5 are used to produce synthetic jets at a Reynolds number of 166. The velocity fields in different spatial planes are measured by time-resolved two-dimensional particle image velocimetry (2D-PIV), and stereoscopic particle image velocimetry (S-PIV), respectively. The results show that the evolution of the vortical structures and the flow physics are influenced by the orifice configuration and AR. The first axis-switching location of the elliptic cases is closer to the jet orifice than that of the rectangular cases, while the AR = 3 cases undergo more upstream location and more times of axis switching than the AR = 5 cases. In addition to axis switching, noncircular synthetic jets are observed to develop streamwise vortices. The rectangular cases develop stronger streamwise vortices than the elliptic cases. The extended angle of streamwise vortices increases with the increasing AR, and is larger for the rectangular cases than the elliptic cases. In particular, the AR = 3 rectangular case has larger values of centerline streamwise velocity fluctuation, Reynolds stresses, mass flow rate and momentum flux than the other cases in the second axis-switching region, suggesting significantly enhanced mixing and momentum transportation characteristics. The mechanism could be attributed to its more evident vortex deformation and stronger streamwise vortices.

Iterated elliptic and hypergeometric integrals for Feynman diagrams

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop quantum chromodynamics corrections to the ρ-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of 2F1 Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi’s ϑi functions and Dedekind’s η-function. The corresponding representations can be traced back to polynomials out of Lambert–Eisenstein series, having representations also as elliptic polylogarithms, a q-factorial 1/ηk(τ), logarithms, and polylogarithms of q and their q-integrals. Due to the specific form of the physical variable x(q) for different processes, different representations do usually appear. Numerical results are also presented.

Green’s formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators

This article treats boundary value problems for the fractional Laplacian , a > 0, and more generally for classical pseudodifferential operators (ψdo’s) P of order 2a with even symbol, applied to functions on a smooth subset Ω of . There are several meaningful local boundary conditions, such as the Dirichlet and Neumann conditions , k = 0, 1, where . We show a new Green’s formula where B is a first-order ψdo on depending on the first two terms in the symbol of P. Moreover, we show in the elliptic case how the Poisson-like solution operator KD for the nonhomogeneous Dirichlet problem is constructed from P+ in the factorization obtained in earlier work. The Dirichlet-to-Neumann operator is derived from this as a first-order ψdo on , with an explicit formula for the symbol. This leads to a characterization of those operators P for which the Neumann problem is Fredholm solvable.

Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems

We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The L-M method transforms the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the L-M method is rigorously established for the nonlinear elliptic and parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the L-M iteration. Then the surrogate functional approach is adopted to solve the strongly ill-conditioned convex minimizations, resulting in an explicit solution of the minimisation at each L-M iteration for both the elliptic and parabolic cases. Numerical experiments are provided to demonstrate the accuracy, efficiency and quadratic convergence of the methods.

The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan–Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is also shown, along with comparison principles, both in the parabolic and in the elliptic case. Our results complete previous analyses of the porous medium equation flow on negatively curved Riemannian manifolds, which were carried out first for the hyperbolic space and then for general Cartan–Hadamard manifolds with a negative curvature having at most quadratic growth. We point out that no similar analysis seems to exist for the linear heat flow. We also translate such results into some weighted porous medium equations in the Euclidean space having special weights.

The ε-form of the differential equations for Feynman integrals in the elliptic case

Feynman integrals are easily solved if their system of differential equations is in ε-form. In this letter we show by the explicit example of the kite integral family that an ε-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The ε-form is obtained by a (non-algebraic) change of basis for the master integrals.

Euler-type Relative Equilibria and their Stability in Spaces of Constant Curvature

AbstractWe consider three point positivemasses moving onS2andH2. An Eulerian-relative equilibrium is a relative equilibrium where the three masses are on the same geodesic. In this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of S2, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classiûcation of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibriumis spectrally stable or unstable. OnH2, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible, we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible; in this case the Eulerian-relative equilibria are unstable.

The Green's function and a maximum principle for a Caputo two-point boundary value problem with a convection term

A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order α∈(1,2) and where a convection term is also present is considered. Its boundary conditions are of Robin type and include Dirichlet boundary conditions as a special case. An explicit formula for the associated Green's function is obtained in terms of two-parameter Mittag-Leffler functions. Some new properties of these Mittag-Leffler functions are derived; from these, one can deduce necessary and sufficient conditions on the boundary conditions that ensure non-negativity of the Green's function and hence a maximum principle for the boundary value problem. In particular, unlike the classical elliptic case α=2, Dirichlet boundary conditions will not in general yield a maximum principle.