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Journal articles on the topic 'Second order elliptic equation'

Author: Grafiati

Published: 5 June 2025

Last updated: 2 August 2025

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1

Boughazi, Hichem. "Second-Order Elliptic Equation with Singularities." International Journal of Differential Equations 2020 (May 20, 2020): 1–16. http://dx.doi.org/10.1155/2020/4589864.

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On the compact Riemannian manifold of dimension n≥5, we study the existence and regularity of nontrivial solutions for nonlinear second-order elliptic equation with singularities. At the end, we give a geometric application of the above singular equation.

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Farg, Ahmed Saeed, A. M. Abd Elbary, and Tarek A. Khalil. "Applied method of characteristics on 2^nd order linear P.D.E." Journal of Physics: Conference Series 2304, no. 1 (2022): 012003. http://dx.doi.org/10.1088/1742-6596/2304/1/012003.

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Abstract PDEs are very important in dynamics, aerodynamics, elasticity, heat transfer, waves, electromagnetic theory, transmission lines, quantum mechanics, weather forecasting, prediction of crime places, disasters, how universe behave ……. Etc., second order linear PDEs can be classified according to the characteristic equation into 3 types hyperbolic, parabolic and elliptic; Hyperbolic equations have two distinct families of (real) characteristic curves, parabolic equations have a single family of characteristic curves, and the elliptic equations have none. All the three types of equations c

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3

Shan, Jin. "Singularly perturbed nonlinear second order elliptic equation." Applied Mathematics and Mechanics 8, no. 6 (1987): 547–59. http://dx.doi.org/10.1007/bf02017404.

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4

Azaiz, Seid, Hichem Boughazi, and Kamel Tahri. "On the singular second-order elliptic equation." Journal of Mathematical Analysis and Applications 489, no. 1 (2020): 124077. http://dx.doi.org/10.1016/j.jmaa.2020.124077.

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5

Caffarelli, Luis. "Elliptic second order equations." Rendiconti del Seminario Matematico e Fisico di Milano 58, no. 1 (1988): 253–84. http://dx.doi.org/10.1007/bf02925245.

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6

Tokibetov, Zh A., N. E. Bashar, and А. К. Pirmanova. "THE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS." BULLETIN Series of Physics & Mathematical Sciences 72, no. 4 (2020): 68–72. http://dx.doi.org/10.51889/2020-4.1728-7901.10.

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For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Diric

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7

Potts, Renfrey B. "Weierstrass elliptic difference equations." Bulletin of the Australian Mathematical Society 35, no. 1 (1987): 43–48. http://dx.doi.org/10.1017/s0004972700013022.

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The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

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8

Burazin, Krešimir, Jelena Jankov, and Marko Vrdoljak. "HOMOGENIZATION OF ELASTIC PLATE EQUATION∗." Mathematical Modelling and Analysis 23, no. 2 (2018): 190–204. http://dx.doi.org/10.3846/mma.2018.012.

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We are interested in general homogenization theory for fourth-order elliptic equation describing the Kirchhoﬀ model for pure bending of a thin solid symmetric plate under a transverse load. Such theory is well-developed for second-order elliptic problems, while some results for general elliptic equations were established by Zhikov, Kozlov, Oleinik and Ngoan (1979). We push forward an approach of Antoni´c and Balenovi´c (1999, 2000) by proving a number of properties of H-convergence for stationary plate equation.

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9

Ashyralyev, Allaberen, and Okan Gercek. "On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-Parabolic Equations." Abstract and Applied Analysis 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/230190.

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We are interested in studying a second order of accuracy implicit difference scheme for the solution of the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of this difference scheme is established. In an application, coercivity estimates in Hölder norms for approximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolic differential equations are obtained.

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10

SAMARSKII, ALEKSANDR A., and PETER N. VABISHCHEVICH. "REGULARIZED DIFFERENCE SCHEMES FOR EVOLUTIONARY SECOND ORDER EQUATIONS." Mathematical Models and Methods in Applied Sciences 02, no. 03 (1992): 295–315. http://dx.doi.org/10.1142/s0218202592000193.

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The questions of approximate solution of unstable problems for evolutionary second order equations are discussed in this paper. The classical Cauchy problem for elliptic type equation is a significant example of such problem. Incorrectness of this problem (the Hadamard example) is due to instability of the solution towards small perturbations of the initial conditions. The extension problem of the solutions of well-posed elliptic problems beyond the calculation region boundary is also discussed. The stability of corresponding difference schemes is investigated by basing on general theory of ρ-

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11

Gromak, V. I. "On Bäcklund transformations to stationary equations in hierarchy of the second Painlevé equation." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 60, no. 3 (2024): 195–202. http://dx.doi.org/10.29235/1561-2430-2024-60-3-195-202.

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The analytical properties of solutions to stationary equations of the second and fourth orders in the hierarchy of the second Painlevé equation are considered. For the second-order equation, it is shown that the Bäcklund transformation in the general case determines the formula of the addition theorem for the Weierstrass elliptic function. For the fourth and sixth-order equations, Bäcklund transformations and special classes of solutions are constructed. It has been established that, for a certain relationship between the parameters, the set of solutions to the first term of the stationary hie

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12

Gebran, Hicham G., and Charles A. Stuart. "FREDHOLM AND PROPERNESS PROPERTIES OF QUASILINEAR ELLIPTIC SYSTEMS OF SECOND ORDER." Proceedings of the Edinburgh Mathematical Society 48, no. 1 (2005): 91–124. http://dx.doi.org/10.1017/s0013091504000550.

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AbstractFor a large class of subsets $\varOmega\subset\mathbb{R}^{N}$ (including unbounded domains), we discuss the Fredholm and properness properties of second-order quasilinear elliptic operators viewed as mappings from $W^{2,p}(\varOmega;\mathbb{R}^{m})$ to $L^{p}(\varOmega;\mathbb{R}^{m})$ with $N\ltp\lt\infty$ and $m\geq1$. These operators arise in the study of elliptic systems of $m$ equations on $\varOmega$. A study in the case of a single equation ($m=1$) on $\mathbb{R}^{N}$ was carried out by Rabier and Stuart.AMS 2000 Mathematics subject classification: Primary 35J45; 35J60. Secondar

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13

Borsuk, Mikhail. "Boundary value problems for singular p- and p(x)- Laplacian equations in a domain with conical point on the boundary." Ukrainian Mathematical Bulletin 17, no. 4 (2020): 455–83. http://dx.doi.org/10.37069/1810-3200-2020-17-4-1.

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This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

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14

Denche, Mohamed. "Nonlocal boundary value problem for second order abstract elliptic differential equation." Abstract and Applied Analysis 4, no. 3 (1999): 153–68. http://dx.doi.org/10.1155/s1085337599000135.

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We establish conditions that guarantee Fredholm solvability in the Banach spaceLpof nonlocal boundary value problems for elliptic abstract differential equations of the second order in an interval. Moreover, in the spaceL2we prove in addition the coercive solvability, and the completeness of root functions (eigenfunctions and associated functions). The obtained results are then applied to the study of a nonlocal boundary value problem for Laplace equation in a cylindrical domain.

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15

Janušauskas, A. I. "On classification of elliptic second order partial equation systems." Lithuanian Mathematical Journal 37, no. 2 (1997): 134–45. http://dx.doi.org/10.1007/bf02465886.

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16

Ferreira, José A., Elias Gudiño, and Paula de Oliveira. "A Second Order Approximation for Quasilinear Non-Fickian Diffusion Models." Computational Methods in Applied Mathematics 13, no. 4 (2013): 471–93. http://dx.doi.org/10.1515/cmam-2013-0017.

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Abstract. In this paper initial boundary value problems, defined using quasilinear diffusion equations of Volterra type, are considered. These equations arise for instance to describe diffusion processes in viscoelastic media whose behavior is represented by a Voigt–Kelvin model or a Maxwell model. A finite difference discretization defined on a general non-uniform grid with second order convergence order in space is proposed. The analysis does not follow the usual splitting of the global error using the solution of an elliptic equation induced by the integro-differential equation. The new app

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17

Fiedler, F. "Oscillation criteria of Nehari-type for Sturm–Liouville operators and elliptic differential operators of second order and the lower spectrum." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 109, no. 1-2 (1988): 127–44. http://dx.doi.org/10.1017/s030821050002672x.

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SynopsisSufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

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18

Sabelfeld, Karl K., and Dmitrii Smirnov. "A global random walk on grid algorithm for second order elliptic equations." Monte Carlo Methods and Applications 27, no. 3 (2021): 211–25. http://dx.doi.org/10.1515/mcma-2021-2092.

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Abstract We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Gre

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19

Ivanov, A. V. "Nonlinear nonuniformly elliptic second-order equations." Journal of Soviet Mathematics 32, no. 5 (1986): 448–69. http://dx.doi.org/10.1007/bf01372196.

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20

Merkov, A. B. "SECOND-ORDER ELLIPTIC EQUATIONS ON GRAPHS." Mathematics of the USSR-Sbornik 55, no. 2 (1986): 493–509. http://dx.doi.org/10.1070/sm1986v055n02abeh003017.

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21

Xu, Zhiting, and Hongyan Xing. "Integral operator and oscillation of second order elliptic equations." Tamkang Journal of Mathematics 36, no. 2 (2005): 93–101. http://dx.doi.org/10.5556/j.tkjm.36.2005.121.

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By using integral operator, some oscillation criteria for second order elliptic differential equation$$ \sum^d _{i,j=1} D_i[A_{ij}(x)D_jy]+ q(x)f(y)=0, \;x \in \Omega\qquad \eqno{(E)} $$are established. The results obtained here can be regarded as the extension of the well-known Kamenev theorem to Eq.$(E)$.

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22

Tsai, Ching-Piao, Hong-Bin Chen, and John R. C. Hsu. "Second-Order Time-Dependent Mild-Slope Equation for Wave Transformation." Mathematical Problems in Engineering 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/341385.

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This study is to propose a wave model with both wave dispersivity and nonlinearity for the wave field without water depth restriction. A narrow-banded sea state centred around a certain dominant wave frequency is considered for applications in coastal engineering. A system of fully nonlinear governing equations is first derived by depth integration of the incompressible Navier-Stokes equation in conservative form. A set of second-order nonlinear time-dependent mild-slope equations is then developed by a perturbation scheme. The present nonlinear equations can be simplified to the linear time-d

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23

Dumanyan, V. Zh. "On solvability of Dirichlet problem for second order elliptic equation." Journal of Contemporary Mathematical Analysis 46, no. 2 (2011): 77–88. http://dx.doi.org/10.3103/s1068362311020038.

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24

Gushchin, A. K. "ON THE DIRICHLET PROBLEM FOR A SECOND-ORDER ELLIPTIC EQUATION." Mathematics of the USSR-Sbornik 65, no. 1 (1990): 19–66. http://dx.doi.org/10.1070/sm1990v065n01abeh002075.

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25

Wang, Yuhan, Peiyao Wang, Rongpei Zhang, and Jia Liu. "Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method." Mathematics 12, no. 12 (2024): 1892. http://dx.doi.org/10.3390/math12121892.

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This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hyb

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26

Amanov, Rabil A., and Farman I. Mamedov. "On Wiener's Criterion for an Elliptic Equation with Nonuniform Degeneration." gmj 14, no. 4 (2007): 607–26. http://dx.doi.org/10.1515/gmj.2007.607.

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Abstract For some class of nonuniformly degenerated elliptic equations of second order, a necessary and sufficient condition for boundary points to be regular is found. This condition is an analogue of Wiener's criterion for the Laplace equation.

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27

Freitas, Pedro, and Guido Sweers. "Positivity results for a nonlocal elliptic equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 4 (1998): 697–715. http://dx.doi.org/10.1017/s0308210500021727.

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In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the l

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28

Sabelfeld, Karl K., Dmitry Smirnov, Ivan Dimov, and Venelin Todorov. "A global random walk on grid algorithm for second order elliptic equations." Monte Carlo Methods and Applications 27, no. 4 (2021): 325–39. http://dx.doi.org/10.1515/mcma-2021-2097.

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Abstract In this paper we develop stochastic simulation methods for solving large systems of linear equations, and focus on two issues: (1) construction of global random walk algorithms (GRW), in particular, for solving systems of elliptic equations on a grid, and (2) development of local stochastic algorithms based on transforms to balanced transition matrix. The GRW method calculates the solution in any desired family of prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula. The use in local random walk methods of balanced trans

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29

Shilkov, A. V. "On Solving Second-Order Linear Elliptic Equations." Mathematical Models and Computer Simulations 12, no. 4 (2020): 597–612. http://dx.doi.org/10.1134/s2070048220040171.

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30

Allegretto, W. "Second order elliptic equations with degenerate weight." Proceedings of the American Mathematical Society 107, no. 4 (1989): 989. http://dx.doi.org/10.1090/s0002-9939-1989-0977929-4.

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31

Delanoë, Philippe. "Perturbing fully nonlinear second order elliptic equations." Topological Methods in Nonlinear Analysis 20, no. 1 (2002): 63. http://dx.doi.org/10.12775/tmna.2002.025.

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32

Vabishchevich, Petr N. "NUMERICAL SOLVING UNSTEADY SPACE-FRACTIONAL PROBLEMS WITH THE SQUARE ROOT OF AN ELLIPTIC OPERATOR." Mathematical Modelling and Analysis 21, no. 2 (2016): 220–38. http://dx.doi.org/10.3846/13926292.2016.1147000.

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An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two- level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Ada

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33

Babich, E. R., and I. P. Martynov. "On the absence of logarithmic singularities in the solutions of Lamé-type equations." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 57, no. 4 (2021): 428–34. http://dx.doi.org/10.29235/1561-2430-2021-57-4-428-434.

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The object of this research is linear differential equations of the second order with regular singularities. We extend the concept of a regular singularity to linear partial differential equations. The general solution of a linear differential equation with a regular singularity is a linear combination of two linearly independent solutions, one of which in the general case contains a logarithmic singularity. The well-known Lamé equation, where the Weierstrass elliptic function is one of the coefficients, has only meromorphic solutions. We consider such linear differential equations of the seco

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34

Karppinen, Arttu, and Saara Sarsa. "Local second order regularity of solutions to elliptic Orlicz–Laplace equation." Nonlinear Analysis 253 (April 2025): 113737. https://doi.org/10.1016/j.na.2024.113737.

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35

Gushchin, A. K., and V. P. Mikhaĭlov. "ON SOLVABILITY OF NONLOCAL PROBLEMS FOR A SECOND-ORDER ELLIPTIC EQUATION." Russian Academy of Sciences. Sbornik Mathematics 81, no. 1 (1995): 101–36. http://dx.doi.org/10.1070/sm1995v081n01abeh003617.

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36

Aibeche, Aissa, Nasreddine Amroune, and Stephane Maingot. "General non local boundary value problem for second order elliptic equation." Mathematische Nachrichten 291, no. 10 (2018): 1470–85. http://dx.doi.org/10.1002/mana.201700032.

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37

Alurrfi, Khaled A. E., Ayad M. Shahoot, Mohamed O. M. Elmrid, Ali M. Almsiri, and Abdullah M. H. Arwiniya. "THE (G′/G) -EXPANSION METHOD FOR SOLVING A NONLINEAR PDE DESCRIBING THE NONLINEAR LOW-PASS ELECTRICAL LINES." EPH - International Journal of Applied Science 1, no. 4 (2015): 30–40. http://dx.doi.org/10.53555/eijas.v1i4.23.

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In this paper, we apply the (G′/G)-expansion method based on three auxiliary equations namely, the generalized Riccati equation, the Jacobi elliptic equation and the second order linear ordinary differential equation to find many new exact solutions of a nonlinear partial differential equation (PDE) describing the nonlinear low-pass electrical lines. The given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. Solitons wave solutions, periodic functions solutions, rational functions solutions and Jacobi elliptic

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38

Laipanova, A. M. "ON ANALOGUE OF THE TRIKOMI PROBLEM FOR A THIRD-ORDER EQUATION OF MIXED TYPE." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 13, no. 2 (2021): 17–23. http://dx.doi.org/10.14529/mmph210203.

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It is commonly known that equations of mixed type are partial differential equations that belong to different types in different parts of the domain under consideration. For example, the equation may belong to the elliptic type in one part of the domain and to the hyperbolic type in another one; these parts are separated by a transition line, at which the equation degenerates into parabolic or undefined. In 1923, the Italian mathematician F. Tricomi considered a boundary value problem for one equation of mixed elliptic-hyperbolic type (later named after him) in a domain bounded in the upper ha

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39

Clements, D. L., M. Haselgrove, and D. M. Barnett. "A note on the boundary integral equation method for the solutions of second order elliptic equations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 26, no. 4 (1985): 415–21. http://dx.doi.org/10.1017/s0334270000004628.

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AbstractThe boundary integral equation method is obtained by expressing a solution to a particular partial differential equation in terms of an integral taken round the boundary of the region under consideration. Various methods exist for the numerical solution of this integral equation and the purpose of this note is to outline an improvement to one of these procedures.

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40

Rozmej, Piotr, and Anna Karczewska. "New Exact Superposition Solutions to KdV2 Equation." Advances in Mathematical Physics 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/5095482.

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New exact solutions to the KdV2 equation (also known as the extended KdV equation) are constructed. The KdV2 equation is a second-order approximation of the set of Boussinesq’s equations for shallow water waves which in first-order approximation yields KdV. The exact solutionsA/2dn2[B(x-vt),m]±m cn[B(x-vt),m]dn[B(x-vt),m]+Din the form of periodic functions found in the paper complement other forms of exact solutions to KdV2 obtained earlier, that is, the solitonic ones and periodic ones given by singlecn2ordn2Jacobi elliptic functions.

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41

Gybkina, N. V., S. M. Lamtyugova, and M. V. Sidorov. "TWO-SIDED APPROXIMATIONS METHOD BASED ON THE GREEN’S FUNCTIONS USE FOR CONSTRUCTION OF A POSITIVE SOLUTION OF THE DIRICHLE PROBLEM FOR A SEMILINEAR ELLIPTIC EQUATION." Radio Electronics, Computer Science, Control, no. 3 (October 6, 2021): 26–41. http://dx.doi.org/10.15588/1607-3274-2021-3-3.

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Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation.  Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to stu

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42

Giri, Ratan, and Debajyoti Choudhuri. "A study of second order semilinear elliptic PDE involving measures." Filomat 33, no. 8 (2019): 2489–506. http://dx.doi.org/10.2298/fil1908489g.

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The objective of this article is to study the boundary value problem for the general semilinear elliptic equation of second order involving L1 functions or Radon measures with finite total variation. The study investigates the existence and uniqueness of ?very weak? solutions to the boundary value problem for a given L1 function. However, a ?very weak? solution need not exist when an L1 function is replaced with a measure due to which the corresponding reduced limits has been found for which the problem admits a solution in a ?very weak? sense.

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Mebrate, Benyam, and Giovanni Porru. "Convexity of solutions to elliptic PDE's." Electronic Journal of Differential Equations 2024, no. 01-?? (2024): 51. http://dx.doi.org/10.58997/ejde.2024.51.

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This article concerns the convexity or concavity of solutions to special second order elliptic partial differential equations in convex domains. We concentrate our investigation to boundary blow up solutions as well as to solutions of particular singular equations. Following a method due to Korevaar and Kennington, we find a new sufficient condition for proving convexity or concavity. This sufficient condition is useful when the semilinear component of the equation is the sum of two or more terms. For more information see https://ejde.math.txstate.edu/Volumes/2024/51/abstr.html

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KUMARI, RUPALI, and RASMITA KAR. "Weak solution for biharmonic equation with Navier boundary conditions." MATHEMATICA 66 (89), no. 1 (2024): 114–22. http://dx.doi.org/10.24193/mathcluj.2024.1.10.

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45

Aldshev, S. A., and A. K. Tanirbergen. "Correctness of the mixed problem for degenerate multidimensional elliptic-parabolic equations." Bulletin of the National Engineering Academy of the Republic of Kazakhstan 93, no. 3 (2024): 267–77. http://dx.doi.org/10.47533/2024.1606-146x.66.

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The initial-boundary value problem (Dirichlet problem) for general elliptic-parabolic equations of second order was first posed by G. Fichera. Further investigation of this problem was carried out in the monograph by O.A. Oleinik and E.V. Radkevich and the works by V.N. Vragov. In these works, the authors examined mixed problems for degenerate multidimensional elliptic equations. The articles by S.A. Aldashev focused on the correctness (in the sense of uniqueness of solvability) of the Dirichlet problem in a cylindrical domain for multidimensional elliptic-parabolic equations. A mixed problem

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BUROVSKIY, P. A., E. V. FERAPONTOV, and S. P. TSAREV. "SECOND-ORDER QUASILINEAR PDEs AND CONFORMAL STRUCTURES IN PROJECTIVE SPACE." International Journal of Mathematics 21, no. 06 (2010): 799–841. http://dx.doi.org/10.1142/s0129167x10006215.

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We investigate second-order quasilinear equations of the form fijuxixj = 0, where u is a function of n independent variables x1, …, xn, and the coefficients fij depend on the first-order derivatives p1 = ux1, …, pn = uxn only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n + 1, R), which acts by projective transformations on the space Pn with coordinates p1, …, pn. The coefficient matrix fij defines on Pn a conformal structure fij(p)dpidpj. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reduc

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Furusho, Yasuhiro, and Takaŝi Kusano. "On the Existence of Positive Decaying Entire Solutions for a Class of Sublinear Elliptic Equations." Canadian Journal of Mathematics 40, no. 5 (1988): 1156–73. http://dx.doi.org/10.4153/cjm-1988-048-1.

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In recent years there has been a growing interest in the existence and asymptotic behavior of entire solutions for second order nonlinear elliptic equations. By an entire solution we mean a solution of the elliptic equation under consideration which is guaranteed to exist in the whole Euclidean N-space RN, N ≧ 2. For standard results on the subject the reader is referred to the papers [2-7, 9-21].The study of entire solutions, which at an early stage was restricted to simple equations of the form Δu + f(x, u) = 0, x ∊ RN, Δ being the N-dimensional Laplacian, has now been extended and generaliz

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Vabishchevich, Nikolay, and Petr Vabishchevich. "VAGO METHOD FOR THE SOLUTION OF ELLIPTIC SECOND‐ORDER BOUNDARY VALUE PROBLEMS." Mathematical Modelling and Analysis 15, no. 4 (2010): 533–45. http://dx.doi.org/10.3846/1392-6292.2010.15.533-545.

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Mathematical physics problems are often formulated using differential operators of vector analysis, i.e. invariant operators of first order, namely, divergence, gradient and rotor (curl) operators. In approximation of such problems it is natural to employ similar operator formulations for grid problems. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for t

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Caffarelli, L., L. Nirenberg, and J. Spruck. "Correction to: The dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampere equation." Communications on Pure and Applied Mathematics 40, no. 5 (1987): 659–62. http://dx.doi.org/10.1002/cpa.3160400508.

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Honda, Naofumi, Ching-Lung Lin, Gen Nakamura, and Satoshi Sasayama. "Unique continuation property of solutions to general second order elliptic systems." Journal of Inverse and Ill-posed Problems 30, no. 1 (2021): 5–21. http://dx.doi.org/10.1515/jiip-2020-0073.

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Abstract This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carle

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