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Lyudmila Korobenko

Publications and source records attributed to Lyudmila Korobenko.

At least 19 recordsLinked to original sources
arXiv2026

On the Necessity of Logarithmic Estimates for Hypoellipticity

This paper is focused on necessary conditions for hypoellipticity of an operator $L$ of the form $L=L_1(x)+g(x)L_2(y)$, where the operator $L_1$ is either elliptic or parabolic, $L_2$ is degenerately elliptic and $g(x)$ may itself vanish adding further degeneracy. First, we establish a logarithmic criterion: if the operator $L$ above is hypoelliptic and $L_1$ has a family of spectral solutions we define in the paper, then the remaining part $L_2$ must gain a power of a logarithm of a derivative. Such a property can be thought of as a restriction on degeneracy of the operator $L_2$. We then use this criterion to examine degenerate elliptic and parabolic operators closing gaps between sufficiency and necessity that have been open since 1980s in three and higher dimensions.

math.AP↗
arXiv2025

Degenerate elliptic equations with $Φ$-admissible weights

We develop regularity theory for degenerate elliptic equations with the degeneracy controlled by a weight. More precisely, we show local boundedness and continuity of weak solutions under the assumption of a weighted Orlicz-Sobolev and Poincaré inequalities. The proof relies on a modified DeGiorgi iteration scheme, developed in arXiv:1608.01630 and arXiv:1703.00774. The Orlicz-Sobolev inequality we assume here is much weaker than the classical $(2σ,2)$ Sobolev inequality with $σ>1$ which is typically used in the DeGiorgi or Moser iteration.

math.AP↗
arXiv2024

Carnot-Carathéodory metrics associated to degenerate elliptic operators in three dimensions

This note is a companion paper to arXiv:1608.01630 [math.CA]. Here we generalize some of the geometric results of arXiv:1608.01630 [math.CA] to the case of a $3\times 3$ matrix function $A(x)\approx \mathrm{diag}\{1,f(x_1), g(x_1)\}$. More precisely, we make explicit calculations of the geodesics in the Carnot-Carathéodory space associated to $A$, and provide estimates on the Lebesgue measures of metric balls centered at the origin in that space.

math.AP↗
arXiv2024

The Moser method and boundedness of solutions to infinitely degenerate elliptic equations

We show that if $\mathbb{R}^{n}$ is equipped with certain non-doubling metric and an Orlicz-Sobolev inequality holds for a special family of Young functions $Φ$, then weak solutions to quasilinear infinitely degenerate elliptic divergence equations of the form $$\mathrm{div}\mathcal{A}\left( x,u\right) \nabla u=ϕ_{0}-\mathrm{div}_{A} \vecϕ_{1}$$ are locally bounded. Furthermore, we establish a maximum principle for solutions whenever a global Orlicz-Soblev estimate is available. We obtain these results via the implementation of a Moser iteration method, what constitutes the first instance of such technique applied to infinite degenerate equations. These results partially extend previously known estimates for solutions of these equations but for which the right hand side did not have a drift term. We also obtain bounds for small negative powers of nonnegative solutions; these will be applied to obtain continuity of solutions in a subsequent paper.

math.AP↗
arXiv2023

Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators

This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form $L_1(x)+g(x)L_2(y)$, where $L_1(x)$ is one-dimensional and $g(x)$ may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our results do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.

math.AP↗
arXiv2023

Weighted 1-dimensional Orlicz-Poincaré inequalities

In this paper we establish necessary and sufficient conditions for weighted Orlicz-Poincaré inequalities in dimension one. Our theorems generalize the main results of Chua and Wheeden, who established necessary and sufficient conditions for weighted $(q,p)$ Poincaré inequalities. We give an example of a weight satisfying sufficient conditions for a $(Φ, p)$ Orlicz-Poincaré inequality where the gauge norm with respect to $Φ$ is a bump on the Lebesgue $L^p$ norm. This weight, on the other hand, does not satisfy a $(q,p)$ Poincaré inequality for any $q > p$.

math.FA↗
arXiv2022

Sum of squares I: scalar functions

This is the first in a series of three papers dealing with sums of squares and hypoellipticity in the infinite regime. We give a sharp sufficient condition on a smooth nonnegative function f on n-dimensional Euclidean space so that it can be written as a finite sum of squares of C^2,delta functions. Special consideration is given to analyzing the case when f vanishes only at the origin, answering a question of Bony et al.

math.FA↗
arXiv2021

Sums of squares III: hypoellipticity in the infinitely degenerate regime

This is the third in a series of papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We establish a C^2,delta generalization of M. Christ's sum of squares theorem, and use a bootstrap argument with the sum of squares theorem for matrix functions in the second paper of this series, in order to prove a hypoellipticity theorem generalizing work in the infinitely degenerate regime to include nondiagonal operators and more general degeneracies.

math.FA↗
arXiv2021

Sums of squares II: matrix functions

This is the second in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We give sharp conditions on the entries of a positive semidefinite NxN matrix function F on n-dimensional Euclidean space, whose determinant vanishes only at the origin and such that F is comparable to its diagonal matrix, in order that F is a finite sum of squares of C^2,delta vector fields. We also consider slightly more general decompositions in which a single quasiconformal term need not be a sum of squares.

math.FA↗
arXiv2020

Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators

We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap between necessary and sufficient conditions for solvability of the Dirichlet problem.

math.AP↗
arXiv2020

Continuity of infinitely degenerate weak solutions via the trace method

In 1971 Fediĭ proved the remarkable theorem that the linear second order partial differential operator in the plane with coefficients 1 and f^2 is hypoelliptic provided that f is smooth, vanishes at the origin and is positive otherwise. Variants of this result, with hypoellipticity replaced by continuity of weak solutions, were recently given by the authors, together with Cristian Rios and Ruipeng Shen, to infinitely degenerate elliptic divergence form equations where the nonnegative matrix A(x,u) has bounded measurable coefficients with trace roughly 1 and determinant comparable to f, and where F=ln(1/f) is essentially doubling. However, in the plane, these variants assumed additional geometric constraints on f, something not required in Fediĭ's theorem. In this paper we in particular remove these additional geometric constraints in the plane for homogeneous equations with F essentially doubling.

math.CA↗
arXiv2019

Sharp local boundedness and maximum principle in the infinitely degenerate regime via DeGiorgi iteration

We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form equations, and the local boundedness turns out to be sharp in more than two dimensions, answering the `Moser gap' problem left open in arXiv:1506.09203v5. Finally we obtain a maximum principle for weak solutions under the same condition on the degeneracy.

math.CA↗
arXiv2019

Continuity of weak solutions to rough infinitely degenerate equations

We obtain a generalization of the DeGiorgi Lemma to the infinitely degenerate regime and apply it to obtain continuity of weak solutions to certain infinitely degenerate equations. This reproduces the continuity result obtained in arXiv:1506.09203 via Moser iteration, but only for homogeneous equations. However, the proofs are much less technical and more transparent.

math.AP↗
arXiv2019

Nonhedgeable risk and Credit Risk Pricing

We introduce a new model for pricing corporate bonds, which is a modification of the classical model of Merton. In this new model, we drop the liquidity assumption of the firm's asset value process, and assume that there is a liquidly traded asset in the market whose value is correlated with the firm's asset value, and all portfolios can be constructed using solely this asset and the money market account. We formulate the market price of the corporate bond as the product of the price of an optimal replicating portfolio and exp(- kappa x replication error), where kappa is a positive constant. The interpretation is that the representative investor accepts the price of the optimal replicating portfolio as a benchmark, however, requests compensation for the non-hedgeable risk. We show that if the replication error is measured relative to the firm's value, the resulting formula is arbitrage free with mild restrictions on the parameters.

q-fin.PR↗
arXiv2019

Orlicz Sobolev Inequalities and the Doubling Condition

In [12] it has been shown that $(p,q)$ Sobolev inequality with $p>q$ implies the doubling condition on the underlying measure. We show that even weaker Orlicz-Sobolev inequalities, where the gain on the left-hand side is smaller than any power bump, imply doubling. Moreover, we derive a condition on the quantity that should replace the radius on the righ-hand side (which we call `superradius'), that is necessary to ensure that the space can support the Orlicz-Sobolev inequality and simultaneously be non-doubling.

math.AP↗
arXiv2017

Hypoellipticity without loss of derivatives for Fedii's type operators

We prove that second order linear operators on $\mathbb{R}^{n+m}$ of the form $L(x,y,D_x,D_y) = L_1(x,D_x) + g(x) L_2(y,D_y)$, where $L_1$ and $L_2$ satisfy Morimoto's super-logarithmic estimates and $g$ is smooth, nonnegative, and vanishes only at the origin in $\mathbb{R}^n$ (but to any arbitrary order) are hypoelliptic without loss of derivarives. We also show examples in which our hypotheses are necessary for hypoellipticity.

math.AP↗
arXiv2016

Local boundedness, maximum principles, and continuity of solutions to infinitely degenerate elliptic equations

We develop subrepresentation inequalities for infinitely degenerate metrics, and obtain corresponding Poincare and Sobolev inequalities. We then derive conditions on the degenerate metric under which weak solutions to associated infinitely degenerate equations with rough coefficients are locally bounded, satisfy a maximum principle, or are continuous. As an application we obtain W-hypoellipticity of certain infinitely degenerate quasilinear equations with smooth coefficients having mild nonlinearities and degeneracies.

math.CA↗