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Статті в журналах з теми "Degenerate elliptic operators"

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Автор: Grafiati

Опубліковано: 7 липня 2024

Оновлено: 7 липня 2024

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1

Shakhmurov, Veli B. "Degenerate Differential Operators with Parameters." Abstract and Applied Analysis 2007 (2007): 1–27. http://dx.doi.org/10.1155/2007/51410.

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The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valuedLp−spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

2

Duc, Duong Minh. "A class of strongly degenerate elliptic operators." Bulletin of the Australian Mathematical Society 39, no. 2 (April 1989): 177–200. http://dx.doi.org/10.1017/s0004972700002665.

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Using a weighted Poincaré inequality, we study (ω1,…,ωn)-elliptic operators. This method is applied to solve singular elliptic equations with boundary conditions in W1,2. We also obtain a result about the regularity of solutions of singular elliptic equations. An application to (ω1,…,ωn)-parabolic equations is given.

3

Igisinov, S. Zh, L. D. Zhumaliyeva, A. O. Suleimbekova, and Ye N. Bayandiyev. "Estimates of singular numbers (s-numbers) for a class of degenerate elliptic operators." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 107, no. 3 (September 30, 2022): 51–58. http://dx.doi.org/10.31489/2022m3/51-58.

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In this paper we study a class of degenerate elliptic equations with an arbitrary power degeneracy on the line. Based on the research carried out in the course of the work, the authors propose methods to overcome various difficulties associated with the behavior of functions from the definition domain for a differential operator with piecewise continuous coefficients in a bounded domain, which affect the spectral characteristics of boundary value problems for degenerate elliptic equations. It is shown the conditions imposed on the coefficients at the lowest terms of the equation, which ensure the existence and uniqueness of the solution. The existence, uniqueness, and smoothness of a solution are proved, and estimates are found for singular numbers (s-numbers) and eigenvalues of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration.

4

Robinson, Derek W., and Adam Sikora. "L1-uniqueness of degenerate elliptic operators." Studia Mathematica 203, no. 1 (2011): 79–103. http://dx.doi.org/10.4064/sm203-1-5.

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5

Morimoto, Yoshinori. "Non-hypoellipticity for degenerate elliptic operators." Publications of the Research Institute for Mathematical Sciences 22, no. 1 (1986): 25–30. http://dx.doi.org/10.2977/prims/1195178369.

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6

Hua, Chen, and Chen Hongge. "Eigenvalue problem of degenerate elliptic operators." SCIENTIA SINICA Mathematica 51, no. 6 (March 8, 2021): 833. http://dx.doi.org/10.1360/ssm-2020-0219.

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7

Robinson, Derek W., and Adam Sikora. "Degenerate elliptic operators in one dimension." Journal of Evolution Equations 10, no. 4 (April 23, 2010): 731–59. http://dx.doi.org/10.1007/s00028-010-0068-9.

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8

Ouhabaz, El Maati, and Derek W. Robinson. "Uniqueness properties of degenerate elliptic operators." Journal of Evolution Equations 12, no. 3 (May 11, 2012): 647–73. http://dx.doi.org/10.1007/s00028-012-0148-0.

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9

Levendorskiĭ, S. Z. "ON TYPES OF DEGENERATE ELLIPTIC OPERATORS." Mathematics of the USSR-Sbornik 66, no. 2 (February 28, 1990): 523–40. http://dx.doi.org/10.1070/sm1990v066n02abeh001183.

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10

Morimoto, Yoshinori. "Estimates for degenerate Schrödinger operators and hypoellipticity for infinitely degenerate elliptic operators." Journal of Mathematics of Kyoto University 32, no. 2 (1992): 333–72. http://dx.doi.org/10.1215/kjm/1250519539.

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11

Chen, Li, José María Martell, and Cruz Prisuelos-Arribas. "Conical square functions for degenerate elliptic operators." Advances in Calculus of Variations 13, no. 1 (January 1, 2020): 75–113. http://dx.doi.org/10.1515/acv-2016-0062.

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AbstractThe aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let {L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)}, where {w\in A_{2}} and A is an {n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the {L^{2}(w)}-Kato square root problem obtaining that {\sqrt{L_{w}}} is equivalent to the gradient on {L^{2}(w)}. The same authors in collaboration with the second named author of this paper studied the {L^{p}(w)}-boundedness of operators that are naturally associated with {L_{w}}, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in {L^{p}(v\,dw)} for {v\in A_{\infty}(w)}), and in particular a class of “degeneracy” weights w was found in such a way that the classical {L^{2}}-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on {L^{p}(w)} and on {L^{p}(v\,dw)}, with {v\in A_{\infty}(w)}, of the conical square functions that one can construct using the heat or Poisson semigroup associated with {L_{w}}. As a consequence of our methods, we find a class of degeneracy weights w for which {L^{2}}-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with {L_{w}}.

12

Cannarsa, P., G. Da Prato, and H. Frankowska. "Invariant measures associated to degenerate elliptic operators." Indiana University Mathematics Journal 59, no. 1 (2010): 53–78. http://dx.doi.org/10.1512/iumj.2010.59.3886.

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13

Himonas, A. Alexandrou. "On degenerate elliptic operators of infinite type." Mathematische Zeitschrift 220, no. 1 (December 1995): 449–60. http://dx.doi.org/10.1007/bf02572625.

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14

TER ELST, A. F. M., DEREK W. ROBINSON, and ADAM SIKORA. "FLOWS AND INVARIANCE FOR DEGENERATE ELLIPTIC OPERATORS." Journal of the Australian Mathematical Society 90, no. 3 (June 2011): 317–39. http://dx.doi.org/10.1017/s1446788711001315.

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AbstractLetSbe a sub-Markovian semigroup onL2(ℝd) generated by a self-adjoint, second-order, divergence-form, elliptic operatorHwithW1,∞(ℝd) coefficientsckl, and let Ω be an open subset of ℝd. We prove that ifeither C∞c(ℝd) is a core of the semigroup generator of the consistent semigroup onLp(ℝd) for somep∈[1,∞] or Ω has a locally Lipschitz boundary, thenSleavesL2(Ω) invariant if and only if it is invariant under the flows generated by the vector fields ∑dl=1ckl∂lfor allk. Further, for allp∈[1,2] we derive sufficient conditions on the coefficients for the core property to be satisfied. Then by combination of these results we obtain various examples of invariance in terms of boundary degeneracy both for Lipschitz domains and domains with fractal boundaries.

15

Horiuchi, Toshio. "Kato's Inequalities for Degenerate Quasilinear Elliptic Operators." Kyungpook mathematical journal 48, no. 1 (March 31, 2008): 15–24. http://dx.doi.org/10.5666/kmj.2008.48.1.015.

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16

Himonas, A. Alexandrou, and Gerson Petronilho. "On global hypoellipticity of degenerate elliptic operators." Mathematische Zeitschrift 230, no. 2 (February 1999): 241–57. http://dx.doi.org/10.1007/pl00004693.

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17

Taira, A. Favini, S. Romanelli, K. "Feller Semigroups Generated by Degenerate Elliptic Operators." Semigroup Forum 60, no. 2 (March 1, 2000): 296–309. http://dx.doi.org/10.1007/s002339910022.

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18

Altomare, Francesco, Mirella Cappelletti Montano, and Sabrina Diomede. "Degenerate elliptic operators, Feller semigroups and modified Bernstein-Schnabl operators." Mathematische Nachrichten 284, no. 5-6 (March 21, 2011): 587–607. http://dx.doi.org/10.1002/mana.200810196.

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19

Dmytryshyn, M. I., and O. V. Lopushansky. "Spectral approximations of strongly degenerate elliptic differential operators." Carpathian Mathematical Publications 11, no. 1 (June 30, 2019): 48–53. http://dx.doi.org/10.15330/cmp.11.1.48-53.

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We establish analytical estimates of spectral approximations errors for strongly degenerate elliptic differential operators in the Lebesgue space $L_q(\Omega)$ on a bounded domain $\Omega$. Elliptic operators have coefficients with strong degeneration near boundary. Their spectrum consists of isolated eigenvalues of finite multiplicity and the linear span of the associated eigenvectors is dense in $L_q(\Omega)$. The received results are based on an appropriate generalization of Bernstein-Jackson inequalities with explicitly calculated constants for quasi-normalized Besov-type approximation spaces which are associated with the given elliptic operator. The approximation spaces are determined by the functional $E\left(t,u\right)$, which characterizes the shortest distance from an arbitrary function ${u\in L_q(\Omega)}$ to the closed linear span of spectral subspaces of the given operator, corresponding to the eigenvalues such that not larger than fixed ${t>0}$. Such linear span of spectral subspaces coincides with the subspace of entire analytic functions of exponential type not larger than ${t>0}$. The approximation functional $E\left(t,u\right)$ in our cases plays a similar role as the modulus of smoothness in the functions theory.

20

Amano, Kazuo. "The Dirichlet problem for degenerate elliptic 2-dimensional Monge-Ampère equation." Bulletin of the Australian Mathematical Society 37, no. 3 (June 1988): 389–410. http://dx.doi.org/10.1017/s0004972700027015.

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We study the following Dirichlet problem for the degenerate elliptic Monge-Ampère equation: Given , f ≥ 0 and , find a solution , t ≥ 2, satisfying in Ω and u = g on ∂Ω. Since f is nonnegative, we cannot apply any standard elliptic methods. In this paper, we use an iteration scheme of Nash-Moser type and a priori estimates for degenerate elliptic operators, and solve the Dirichlet problem for a certain class of f and g.

21

Albano, Paolo. "On the Eikonal equation for degenerate elliptic operators." Proceedings of the American Mathematical Society 140, no. 5 (May 1, 2012): 1739–47. http://dx.doi.org/10.1090/s0002-9939-2011-11132-8.

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22

Gianazza, Ugo, and Vincenzo Vespri. "Generation of analytic semigroups by degenerate elliptic operators." NoDEA : Nonlinear Differential Equations and Applications 4, no. 3 (July 1, 1997): 305–24. http://dx.doi.org/10.1007/s000300050017.

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23

Baldes, Alfred. "Degenerate elliptic operators diagonal systems and variational integrals." Manuscripta Mathematica 55, no. 3-4 (September 1986): 467–86. http://dx.doi.org/10.1007/bf01186659.

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24

AMANO, Kazuo. "The global hypoellipticity of degenerate elliptic-parabolic operators." Journal of the Mathematical Society of Japan 40, no. 2 (April 1988): 181–204. http://dx.doi.org/10.2969/jmsj/04020181.

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25

Morimoto, Yoshinori. "Erratum to: ``Non-hypoellipticity for degenerate elliptic operators''." Publications of the Research Institute for Mathematical Sciences 30, no. 4 (1994): 533–34. http://dx.doi.org/10.2977/prims/1195165789.

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26

Koike, Minoru. "A note on hypoellipticity of degenerate elliptic operators." Publications of the Research Institute for Mathematical Sciences 27, no. 6 (1991): 995–1000. http://dx.doi.org/10.2977/prims/1195169008.

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27

Delgado, Julio, and Alex M. Zamudio. "Invertibility for a class of degenerate elliptic operators." Journal of Pseudo-Differential Operators and Applications 1, no. 2 (March 10, 2010): 207–31. http://dx.doi.org/10.1007/s11868-010-0003-4.

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28

Robinson, Derek W., and Adam Sikora. "Analysis of degenerate elliptic operators of Grušin type." Mathematische Zeitschrift 260, no. 3 (December 7, 2007): 475–508. http://dx.doi.org/10.1007/s00209-007-0284-3.

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29

Beals, Richard, Peter Greiner, and Bernard Gaveau. "Green's Functions for Some Highly Degenerate Elliptic Operators." Journal of Functional Analysis 165, no. 2 (July 1999): 407–29. http://dx.doi.org/10.1006/jfan.1999.3421.

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30

Ferrari, Fausto, and Antonio Vitolo. "Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators." Advanced Nonlinear Studies 20, no. 1 (February 1, 2020): 213–41. http://dx.doi.org/10.1515/ans-2019-2069.

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AbstractWe consider the elliptic differential operator defined as the sum of the minimum and the maximum eigenvalue of the Hessian matrix, which can be viewed as a degenerate elliptic Isaacs operator, in dimension larger than two. Despite of nonlinearity, degeneracy, non-concavity and non-convexity, such an operator generally enjoys the qualitative properties of the Laplace operator, as for instance maximum and comparison principles, ABP and Harnack inequalities, Liouville theorems for subsolutions or supersolutions. Existence and uniqueness for the Dirichlet problem are also proved as well as local and global Hölder estimates for viscosity solutions. All results are discussed for a more general class of weighted partial trace operators.

31

Muratbekov, Mussakan, and Sabit Igissinov. "Estimates of Eigenvalues of a Semiperiodic Dirichlet Problem for a Class of Degenerate Elliptic Equations." Symmetry 14, no. 4 (March 28, 2022): 692. http://dx.doi.org/10.3390/sym14040692.

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In this paper, we consider a class of degenerate elliptic equations with arbitrary power degeneration. The issues about the existence, uniqueness, and smoothness of solutions of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration are studied. The two-sided estimates for singular numbers (s-numbers) are obtained. Note that estimates of singular numbers (s-numbers) show the rate of approximation of the found solutions by finite-dimensional subspaces. Here, we also obtain estimates for the eigenvalues. We note that, in this paper, apparently, two-sided estimates of singular numbers (s-numbers) for degenerate elliptic operators are obtained for the first time. At the end of the paper, a symmetric operator is considered, i.e., a self-adjoint case.

32

Le, Vy Khoi. "On some noncoercive variational inequalities containing degenerate elliptic operators." ANZIAM Journal 44, no. 3 (January 2003): 409–30. http://dx.doi.org/10.1017/s1446181100008117.

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AbstractWe are concerned with the solvability of variational inequalities that contain degenerate elliptic operators. By using a recession approach, we find conditions on the boundary conditions such that the inequality has at least one solution. Existence results of Landesman-Lazer type for a nonsmooth inequality and a resonance problem for a weighted p-Laplacian are discussed in detail.

33

Fazio, Giuseppe Di, Maria Stella Fanciullo, and Pietro Zamboni. "Harnack inequality for degenerate elliptic equations and sum operators." Communications on Pure and Applied Analysis 14, no. 6 (September 2015): 2363–76. http://dx.doi.org/10.3934/cpaa.2015.14.2363.

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34

Horiuchi, Toshio. "On the Neumann problems for certain degenerate elliptic operators." Proceedings of the Japan Academy, Series A, Mathematical Sciences 69, no. 9 (1993): 372–76. http://dx.doi.org/10.3792/pjaa.69.372.

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35

Ping, Gao. "The boundary harnack principle for some degenerate elliptic operators." Communications in Partial Differential Equations 18, no. 12 (January 1993): 2001–22. http://dx.doi.org/10.1080/03605309308821003.

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36

Gao, Ping. "The boundary harnack principle for some degenerate elliptic operators." Communications in Algebra 18, no. 12 (1993): 2001–22. http://dx.doi.org/10.1080/00927879308824121.

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37

Ling, Jun. "Unique continuation for a class of degenerate elliptic operators." Journal of Mathematical Analysis and Applications 168, no. 2 (August 1992): 511–17. http://dx.doi.org/10.1016/0022-247x(92)90176-e.

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38

Hakulinen, Ville. "Passive Advection and the Degenerate Elliptic Operators M n." Communications in Mathematical Physics 235, no. 1 (April 1, 2003): 1–45. http://dx.doi.org/10.1007/s00220-002-0778-0.

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39

Negrini, Paolo, and Vittorio Scornazzani. "Wiener criterion for a class of degenerate elliptic operators." Journal of Differential Equations 66, no. 2 (February 1987): 151–64. http://dx.doi.org/10.1016/0022-0396(87)90029-5.

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40

Gadoev, Makhmadrakhim Gafurovich, and Sulaimon Abunasrovich Iskhokov. "Spectral properties of degenerate elliptic operators with matrix coefficients." Ufimskii Matematicheskii Zhurnal 5, no. 4 (2013): 37–48. http://dx.doi.org/10.13108/2013-5-4-37.

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41

Triebel, Hans. "Eigenvalue distributions of some non-isotropic degenerate elliptic operators." Revista Matemática Complutense 24, no. 2 (May 19, 2010): 343–55. http://dx.doi.org/10.1007/s13163-010-0042-7.

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42

Altomare, Francesco, and Vita Leonessa. "Continuous selections of Borel measures, positive operators and degenerate evolution problems." Journal of Numerical Analysis and Approximation Theory 36, no. 1 (February 1, 2007): 9–23. http://dx.doi.org/10.33993/jnaat361-852.

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In this paper we continue the study of a sequence of positive linear operators which we have introduced in [9] and which are associated with a continuous selection of Borel measures on the unit interval. We show that the iterates of these operators converge to a Markov semigroup whose generator is a degenerate second-order elliptic differential operator on the unit interval. Some qualitative properties of the semigroup, or equivalently, of the solutions of the corresponding degenerate evolution problems, are also investigated.

43

Brüning, Jochen, and Toshikazu Sunada. "On the spectrum of periodic elliptic operators." Nagoya Mathematical Journal 126 (June 1992): 159–71. http://dx.doi.org/10.1017/s0027763000004049.

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It was observed in [Su5] that the spectrum of a periodic Schrödinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only finitely many. The purpose of this paper is to show, by a slightly different method, that this is also true for general periodic elliptic self-adjoint operators.

44

Hafeez, Usman, Theo Lavier, Lucas Williams, and Lyudmila Korobenko. "Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators." Electronic Journal of Differential Equations 2021, no. 01-104 (September 23, 2021): 82. http://dx.doi.org/10.58997/ejde.2021.82.

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We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequalityin 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. For more information see https://ejde.math.txstate.edu/Volumes/2021/82/abstr.html

45

Ouhabaz, El Maati. "On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators." Semigroup Forum 57, no. 3 (November 1998): 305–14. http://dx.doi.org/10.1007/pl00005980.

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46

Battaglia, Erika, and Stefano Biagi. "Superharmonic functions associated with hypoelliptic non-Hörmander operators." Communications in Contemporary Mathematics 22, no. 04 (November 16, 2018): 1850071. http://dx.doi.org/10.1142/s0219199718500712.

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In this paper, we consider a class of degenerate-elliptic linear operators [Formula: see text] in quasi-divergence form and we study the associated cone of superharmonic functions. In particular, following an abstract Potential-Theoretic approach, we prove the local integrability of any [Formula: see text]-superharmonic function and we characterize the [Formula: see text]-superharmonicity of a function [Formula: see text] in terms of the sign of the distribution [Formula: see text]; we also establish some Riesz-type decomposition theorems and we prove a Poisson–Jensen formula. The operators involved are [Formula: see text]-hypoelliptic but they do not satisfy the Hörmander Rank Condition nor subelliptic estimates or Muckenhoupt-type degeneracy conditions.

47

GAWARECKI, L., V. MANDREKAR, and B. RAJEEV. "THE MONOTONICITY INEQUALITY FOR LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 12, no. 04 (December 2009): 575–91. http://dx.doi.org/10.1142/s0219025709003902.

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We prove the monotonicity inequality for differential operators A and L that occur as coefficients in linear stochastic partial differential equations associated with finite-dimensional Itô processes. We characterize the solutions of such equations. A probabilistic representation is obtained for solutions to a class of evolution equations associated with time dependent, possibly degenerate, second-order elliptic differential operators.

48

Hoshiro, Toshihiko. "Hypoellipticity for infinitely degenerate elliptic and parabolic operators II, operators of higher order." Journal of Mathematics of Kyoto University 29, no. 3 (1989): 497–513. http://dx.doi.org/10.1215/kjm/1250520223.

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49

Gao, Lili, Ming Huang, and Lu Yang. "Wong–Zakai approximations for non-autonomous stochastic parabolic equations with X-elliptic operators in higher regular spaces." Journal of Mathematical Physics 64, no. 4 (April 1, 2023): 042701. http://dx.doi.org/10.1063/5.0111876.

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Анотація:

In this paper, we consider the regularity of Wong–Zakai approximations of the non-autonomous stochastic degenerate parabolic equations with X-elliptic operators. We first establish the pullback random attractors for the random degenerate parabolic equations with a general diffusion. Then, we prove the convergence of solutions and the upper semi-continuity of random attractors of the Wong–Zakai approximation equations in L p( D N) ∩ H.

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Cruz-Uribe, David, José María Martell, and Cristian Rios. "On the Kato problem and extensions for degenerate elliptic operators." Analysis & PDE 11, no. 3 (January 1, 2018): 609–60. http://dx.doi.org/10.2140/apde.2018.11.609.

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