Degeneracy Set Research Articles

Weighted Poincaré Inequalities and Degenerate Elliptic and Parabolic Problems: An Approach via the Distance Function

We obtain weighted Poincaré inequalities in bounded domains, where the weight is given by a symmetric nonnegative definite matrix, which can degenerate on submanifolds. Furthermore, we investigate uniqueness and nonuniqueness of solutions to degenerate elliptic and parabolic problems, where the diffusion matrix can degenerate on subsets of the boundary of the domain. Both the results are obtained by means of the distance function from the degeneracy set, which is used to construct suitable local sub– and supersolutions.

Complex-analytic intermediate hyperbolicity, and finiteness properties

Motivated by the finiteness of the set of automorphisms Aut([Formula: see text]) of a projective manifold of general type [Formula: see text], and by Kobayashi–Ochiai’s conjecture that a projective manifold dim([Formula: see text])-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of Aut([Formula: see text]) for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold [Formula: see text] which is (dim([Formula: see text])-1)-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-(dim([Formula: see text])-1)-analytically hyperbolic, strongly measure hyperbolic projective manifold [Formula: see text], under an additional hypothesis on the size of the degeneracy set. Some of the properties used during the proofs lead us to introduce a notion of intermediate Picard hyperbolicity, which we last discuss.

Poisson structures of near-symplectic manifolds and their cohomology

We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure π is defined on the tubular neighborhood of the singular locus Zω of the 2-form ω; it is of maximal rank 4, and it vanishes on a degeneracy set containing Zω. We compute its smooth Poisson cohomology, which depends on the modular vector field, and it is finite dimensional. We conclude with a discussion on the relation between the Poisson structure π and the overtwisted contact structure associated with a near-symplectic 4-manifold.

Minimizers of convex functionals with small degeneracy set

We study the question whether Lipschitz minimizers of \(\int F(\nabla u)\,dx\) in \(\mathbb {R}^n\) are \(C^1\) when F is strictly convex. Building on work of De Silva–Savin, we confirm the \(C^1\) regularity when \(D^2F\) is positive and bounded away from finitely many points that lie in a 2-plane. We then construct a counterexample in \(\mathbb {R}^4\), where F is strictly convex but \(D^2F\) degenerates on the Clifford torus. Finally we highlight a connection between the case \(n = 3\) and a result of Alexandrov in classical differential geometry, and we make a conjecture about this case.

Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation

This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate and fourth order non degenerate parabolic equations, in this work the a priori estimates of the error between the discrete solution and the weak solution to which it converges are obtained for a degenerate fourth order parabolic equation. The theoretical error estimates obtained in the present case state that the norms of the approximation errors, calculated on the support of the solution in the proper functional spaces, are bounded by power laws of the discretization parameters with exponent 1/2, while in the case of the classical Cahn-Hilliard equation with constant mobility the exponent is 1. The estimates are finally succesfully validated by simulation results in one and two space dimensions.

Poincaré inequalities for Sobolev spaces with matrix-valued weights and applications to degenerate partial differential equations

For bounded domains Ω, we prove that the Lp-norm of a regular function with compact support is controlled by weighted Lp-norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set, where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp-based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic, degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational: the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem.

A Cahn‐Hilliard–type equation with application to tumor growth dynamics

We consider a Cahn‐Hilliard–type equation with degenerate mobility and single‐well potential of Lennard‐Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn‐Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.

A quasiconformal composition problem for the $Q$-spaces

Given a quasiconformal mapping f:{\mathbb R}^n\to{\mathbb R}^n with n\ge2 , we show that (un-)boundedness of the composition operator {\mathbf C}_f on the spaces Q_{\alpha}({\mathbb R}^n) depends on the index \alpha and the degeneracy set of the Jacobian J_f . We establish sharp results in terms of the index \alpha and the local/global self-similar Minkowski dimension of the degeneracy set of J_f . This gives a solution to [3, Problem 8.4] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel–Lizorkin and Besov spaces. Consequently, Tukia–Väisälä's quasiconformal extension f:{\mathbb R}^n\to{\mathbb R}^n of an arbitrary quasisymmetric mapping g:{\mathbb R}^{n-p}\to {\mathbb R}^{n-p} is shown to preserve Q_{\alpha} ({\mathbb R}^n) for any (\alpha,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\} . Moreover, Q_{\alpha}({\mathbb R}^n) is shown to be invariant under inversions for all 0<\alpha<1 .

Exceptional Contours and Band Structure Design in Parity-Time Symmetric Photonic Crystals.

We investigate the properties of two-dimensional parity-time symmetric periodic systems whose non-Hermitian periodicity is an integer multiple of the underlying Hermitian system's periodicity. This creates a natural set of degeneracies that can undergo thresholdless PT transitions. We derive a k·p perturbation theory suited to the continuous eigenvalues of such systems in terms of the modes of the underlying Hermitian system. In photonic crystals, such thresholdless PT transitions are shown to yield significant control over the band structure of the system, and can result in all-angle supercollimation, a PT-superprism effect, and unidirectional behavior.

Can we measure individual black-hole spins from gravitational-wave observations?

Measurements of black-hole spins from gravitational-wave observations of black-hole binaries with ground-based detectors are expected to be hampered by partial degeneracies in the gravitational-wave phasing: between the two component spins, and between the spins and the binary's mass ratio, at least for signals that are dominated by the binary's inspiral. Through the merger and ringdown, however, a different set of degeneracies apply. This suggests the possibility that, if the inspiral, merger and ringdown are all within the sensitive frequency band of a detector, we may be able to break these degeneracies and more accurately measure both spins. In this work we investigate our ability to measure individual spins for non-precessing binaries, for a range of configurations and signal strengths, and conclude that in general the spin of the larger black hole will be measurable (at best) with observations from Advanced LIGO and Virgo. This implies that in many applications waveform models parameterized by only one \emph{effective spin} will be sufficient. Our work does not consider precessing binaries or sub-dominant harmonics, although we provide some arguments why we expect that these will not qualitatively change our conclusions.

Unique continuation and approximate controllability for a degenerate parabolic equation

This article studies unique continuation for weakly degenerate parabolic equations in one-space dimension. A new Carleman estimate of local type is obtained to deduce that all solutions that vanish on the degeneracy set, together with their conormal derivative, are identically equal to zero. An approximate controllability result for weakly degenerate parabolic equations under Dirichlet boundary condition is deduced.

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

In this paper we deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for the linear equation. A good example of such an operator is the Grushin operator on ℝ^{d+k} , to which we devote particular attention. As an application of these tools in the degenerate elliptic setting, we prove a partial symmetry result for classical solutions of semilinear problems on bounded, symmetric and suitably convex domains, which is a generalization of the result of Gidas–Ni–Nirenberg [12, 13] and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space, which is a generalization of the result of Gidas–Spruck [14] and Chen–Li [6]. The method we use to obtain these results is the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator.

A lattice approach to the conformal [formula omitted] supercoset sigma model. Part II: The boundary spectrum

We consider the partition function of the boundary OSp ( 2 S + 2 | 2 S ) coset sigma model on an annulus, based on the lattice regularization introduced in the companion paper. Using results for the action of OSp ( 2 S + 2 | 2 S ) and B L ( 2 ) on the corresponding spin chain, as well as mini-superspace and small g σ 2 calculations, we conjecture the full spectrum and set of degeneracies on the entire critical line. Potential relationship with the OSp ( 2 S + 2 | 2 S ) Gross–Neveu model is also discussed.

TheK-property of four billiard balls

A further step is achieved toward establishing the celebrated Boltzmann-Sinai ergodic hypothesis: for systems of four hard balls on the ν-torus (ν>2) it is shown that, on the submanifold of the phase specified by the trivial conservation laws, the system is aK-flow. All parts of our previous demonstration providing the analogous result for three hard balls are simplified and strengthened. The main novelties are: (i) A refinement of the geometric-algebraic methods used earlier helps us to bound the codimension of the arising implicitly given set of degeneracies even if we can not calculate their exact dimension that was possible for three-billiards. As a matter of fact, it is this part of our arguments, where further understanding and new ideas are necessary before attacking the general ergodic problem; (ii) In the “pasting” part of the proof, which is a sophisticated version of Hopf's classical device, the arguments are so general that it is hoped they work in the general case, too. This is achieved for four balls, in particular, by a version of the Transversal Fundamental Theorem which, on one hand, is simpler and more suitable for applications than the previous one and, on the other hand, as we have discovered earlier, is the main tool to prove global ergodicity of semi-dispersing billiards; (iii) The verification of the Chernov-Sinai ansatz is essentially simplified and the new idea of the proof also promises to work in the general case.

On the Morphology of Multi‐Valued Solutions of a Simple Shallow Arch Problem

AbstractThe exact, global, strong and complete solution of a shallow arch problem is constructed equivalently by means of the boundary value problem, by means of a variational problem in terms of displacements, or by means of a variational problem in terms of dual static field quantities, respectively. It is shown that in contrast to all known results, duality and equivalence of the variational problems are preserved on the degeneracy set, too. The paradox appearing in the so‐called complementary variational approach, where the same solution point should be stable in a displacement approach and simultaneously unstable in a genuinely dual static approach is resolved.