УЗАГАЛЬНЕНИЙ РОЗВ'ЯЗОК ЗАДАЧІ ДІРІХЛЕ ДЛЯ МОДЕЛЬНОГО АНІЗОТРОПНОГО ВАГОВОГО РІВНЯННЯ

For a class of second-order anisotropic elliptic equations with variable nonlinearity indices and summable right-hand sides, we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable exponents.

We establish a number of results on the integrability of the entropy and the weak solutions of the Dirichlet problem for nonlinear elliptic equations of second order depending on the integrability properties of the right-hand sides of these equations.

The mixed problem for a nonlinear degenerate elliptic equation of second order

In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give its applications to the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.

We report a new 9-point compact discretization of order two in y- and order four in x-directions, based on cubic spline approximation, for the solution of two-dimensional nonlinear elliptic partial differential equations of the form $$ A\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial x^{2} }} + B\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial y^{2} }} = f\left( {x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}} \right),\left( {x,y} \right) \in\Omega $$ defined in the domain \( \Omega = \left\{ {\left( {x,y} \right):0 0 \) and \( B\left( {x,y} \right) > 0 \) in \( \Omega \). The corresponding Dirichlet boundary conditions are prescribed by $$ u\left( {x,y} \right) = \psi \left( {x,y} \right),\quad \left( {x,y} \right) \in \partial\Omega $$ The main spline relations are presented and incorporated into solution procedures for elliptic partial differential equations. Available numerical methods based on cubic spline approximations for the numerical solution of nonlinear elliptic equations are of second-order accurate. Although 9-point finite difference approximations of order four accurate for the solution of nonlinear elliptic differential equations are discussed in the past, but these methods require five evaluations of the function f. In this piece of work, using the same number of grid points and three evaluations of the function f, we have derived a new stable cubic spline method of order 2 in y- and order 4 in x-directions for the solution of nonlinear elliptic equation. However, for a fixed parameter (Δy/Δx 2), the proposed method behaves like a fourth order method. The accuracy of the proposed method is exhibited from the computed results. The proposed method is applicable to Poisson’s equation and two-dimensional Navier-Stokes’ equations of motion in polar coordinates, which is main highlight of the work. The convergence analysis of the proposed cubic spline approximation for the nonlinear elliptic equation is discussed and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.

Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems

The maximal regularity properties of a fractional anisotropic elliptic equation are investigated. The equation involves an abstract linear operator A in Fourier type Banach space E and convolution terms. We find the sufficient conditions that guarantee the separability of this problems in E-valued weighted Lp spaces. We prove that an anisotropic elliptic operator generated by this problem is sectorial and generates an analytic semigroup. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation and the boundary value problem for anisotropic elliptic convolution equation are obtained in mixed Lp norms.

Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations

Nonlinear elliptic equations with BMO coefficients in Reifenberg domains

We study the behavior of eigenvalues and eigenfunctions of the Dirichlet problem for nonlinear elliptic second-order equations in domains with fine-grain boundary.

In this paper, we consider the problem of finding positive axially symmetric solutions to boundary value problems for nonlinear elliptic differential equations using the method of two-sided approximations. The first boundary value problem, or Dirichlet problem, is solved. The nonlinearity involved is of an anti-monotonic type, characterized by a power-law dependence with an exponent ranging from –1 to 0. By transitioning to a polar coordinate system and exploiting the axial symmetry of the solution, the original boundary value problem for an elliptic equation is reduced to a boundary value problem for an ordinary differential equation on a finite interval. The solution depends solely on the radial coordinate, eliminating the dependence on the angular variable. In this case, the pole of the polar coordinate system becomes a singular point, where it is necessary to impose a boundedness condition on the solution. For the boundary value problem, the Green’s function is constructed, after which the problem is reduced to a Hammerstein integral equation. This integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on a finite interval, partially ordered by the cone of non-negative functions on that interval. The corresponding operator is studied with respect to properties such as anti-monotonicity (antitonicity), positivity, boundedness, and pseudo-concavity. The next stage involves determining the initial approximation as the endpoints of a strongly invariant conical segment for the antitone operator, in a way that ensures the highest possible convergence rate of the iterative process. Two iterative sequences of two-sided approximations are then constructed. The first sequence is non-decreasing with respect to the cone, while the second is non-increasing. At each iteration, the arithmetic mean of the upper and lower approximations is chosen as the current estimate. The iterative process continues until the error estimate for the solution meets the prescribed accuracy. The theoretical results obtained in this work were verified through computational experiments. The dependence of the solution and the convergence rate of the iterative process on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs.

In response to the growing use of reaction diffusion problems in many fields, this monograph gives a systematic treatment of a class of nonlinear parabolic and elliptic differential equations and their applications these problems. It is an important reference for mathematicians and engineers, as well as a practical text for graduate students.

On the singular second-order elliptic equation

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.

In this paper, we study boundary value problems for parameter-dependent elliptic differential-operator equations with variable coefficients in smooth domains. Uniform regularity properties and Fredholmness of this problem are obtained in vector-valued $$ {\text{L}}_{p} $$ -spaces. We prove that the corresponding differential operator is positive and is a generator of an analytic semigroup. Then, via maximal regularity properties of the linear problem, the existence and uniqueness of the solution to the nonlinear elliptic problem is obtained. As an application, we establish maximal regularity properties of the Cauchy problem for abstract parabolic equations, Wentzell–Robin-type mixed problems for parabolic equations, and anisotropic elliptic equations with small parameters.