Equations Of Elliptic Type Research Articles

Strict Solution for a Second Order Differential Equation in Holder Spaces

In this paper we give some results on abstract second order differential elliptic equations of mixed type. The study is performed in Holder continuous Banach spaces. Our main purpose is the study of necessary and sufficient conditions on the data for obtaining existence, uniqueness and maximal regularity properties of the strict solution. The techniques used are based on analytic semigroups theory.

Backward SDEs and infinite horizon stochastic optimal control

We study an optimal control problem on infinite horizon for a controlled stochastic differential equation driven by Brownian motion, with a discounted reward functional. The equation may have memory or delay effects in the coefficients, both with respect to state and control, and the noise can be degenerate. We prove that the value, i.e. the supremum of the reward functional over all admissible controls, can be represented by the solution of an associated backward stochastic differential equation (BSDE) driven by the Brownian motion and an auxiliary independent Poisson process and having a sign constraint on jumps. In the Markovian case when the coefficients depend only on the present values of the state and the control, we prove that the BSDE can be used to construct the solution, in the sense of viscosity theory, to the corresponding Hamilton-Jacobi-Bellman partial differential equation of elliptic type on the whole space, so that it provides us with a Feynman-Kac representation in this fully nonlinear context. The method of proof consists in showing that the value of the original problem is the same as the value of an auxiliary optimal control problem (called randomized), where the control process is replaced by a fixed pure jump process and maximization is taken over a class of absolutely continuous changes of measures which affect the stochastic intensity of the jump process but leave the law of the driving Brownian motion unchanged.

Формирование моделей радиолокационных изображений в виде стохастических дифференциальных уравнений для распознавания космических объектов

The resource problems of the traditional use of detailed radar images for reliable recognition of space objects are shown. The urgent task of forming a new type of model of radar images of space objects to determine the signs of their recognition is posed. Corresponding mathematical models of such images based on stochastic differential equations of elliptic type are presented. The adequacy of the developed models to the real radar images of a space object was assessed. It is established that for the description of radar images of space objects the most suitable is a modified model in the form of a mixed derivative of an elliptical model. To test the hypothesis about the possibility of using the radar image model when constructing descriptive recognition signs, an experiment was conducted to distinguish four different types of space objects. The experimental results showed the possibility of using a mixed derivative of the elliptical model to determine signs of recognition of space objects.

On a boundary value problem for a mixed type equation of the second kind

Stationary processes of different physical nature (oscillations, thermal conductivity, diffusion, electrostatics, etc.) are described by equations of elliptic type. In particular, some models, such as hydro and gas dynamics, consider elliptic equations. In this paper, we study a nonlocal boundary value problem with the Poincaré condition for an equation of the elliptic-hyperbolic type of the second kind, i.e. for an equation where the line of degeneration is a characteristic. Refs.: 8 titles.Keywords: nonlocal boundary value problem, Poincaré conditions, equations of elliptic - hyperbolic type, equations of the second kind.

Geometric grid network and third-order compact scheme for solving nonlinear variable coefficients 3D elliptic PDEs

By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordon equations, elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values. The results corroborate the reliability and efficiency of the scheme.

Treatment of long-range interactions arising in the Enskog–Vlasov description of dense fluids

The kinetic theory of rarefied gases and numerical schemes based on the Boltzmann equation, have evolved to the cornerstone of non-equilibrium gas dynamics. However, their counterparts in the dense regime remain rather exotic for practical non-continuum scenarios. This problem is partly due to the fact that long-range interactions arising from the attractive tail of molecular potentials, lead to a computationally demanding Vlasov integral. This study focuses on numerical remedies for efficient stochastic particle simulations based on the Enskog–Vlasov kinetic equation. In particular, we devise a Poisson type elliptic equation which governs the underlying long-range interactions. The idea comes through fitting a Green function to the molecular potential, and hence deriving an elliptic equation for the associated fundamental solution. Through this transformation of the Vlasov integral, efficient Poisson type solvers can be readily employed in order to compute the mean field forces. Besides the technical aspects of different numerical schemes for treatment of the Vlasov integral, simulation results for evaporation of a liquid slab into the vacuum are presented. It is shown that the proposed formulation leads to accurate predictions with a reasonable computational cost.

Solvability of nonlinear elliptic type equation with two unrelated non standard growths

In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths in a bounded domain $\Omega \subset \mathbb{R}^{n}$. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.

Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

Abstract We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a nonregular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn’s topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved by us, and extend previous results established in the powerlike case.

Global Sobolev regularity for general elliptic equations of p-Laplacian type

We derive global gradient estimates for $W^{1,p}_0(\Omega)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and H\"older continuous in $u.$ In the case when $p\geq n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonlinearity in $u$ from Lipschitz continuity to H\"older one, but we also find a very lower level of geometric assumptions on the boundary of the domain to ensure global character of the obtained gradient estimates.

A meshless method for solving the nonlinear inverse Cauchy problem of elliptic type equation in a doubly-connected domain

In the paper a nonlinear inverse Cauchy problem of nonlinear elliptic type partial differential equation in an arbitrary doubly-connected plane domain is solved using a novel meshless numerical method. The unknown Dirichlet data on an inner boundary are recovered by over-specifying the Cauchy data on an outer boundary. A homogenization function is derived to annihilate the Cauchy data on the outer boundary, and then a homogenization technique generates a transformed equation in terms of a transformed variable, whose outer Cauchy boundary conditions are homogeneous. When the numerical solution is expanded by a sequence of boundary functions, which automatically satisfy the homogeneous Cauchy boundary conditions on the outer boundary, we can solve the transformed equation by the domain type meshless collocation method. For the nonlinear inverse Cauchy problems we require to iteratively solve the linear systems with the right-hand sides varying per iteration step. The accuracy and robustness of the homogenization boundary function method (HBFM) are examined through seven numerical examples, where we compare the exact Dirichlet data on the inner boundary to the ones recovered by the HBFM under a large noisy disturbance.

Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains

Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains

On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system

We study stability and convergence of hp-streamline diffusion (SD) finite element, and Nitsche's schemes for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system and Maxwell's equations, respectively. For the hp scheme for the Vlasov-Maxwell system, assuming that the exact solution is in the Sobolev space HS+1(Omega), we derive global a priori error bound of order O(h/p)(s+1/2), where h(= max(K) h(K)) is the mesh parameter and p(= max(K) p(K)) is the spectral order. This estimate is based on the local version with h(K) = diam K being the diameter of the phase-space-time element K and pR-is the spectral order (the degree of approximating finite element polynomial) for K. As for the Nitsche's scheme, by a simple calculus of the field equations, first we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of O(h(2) +k(2)), where h is the spatial mesh size and k is the time step. Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justification of the results, in lower dimensions, is presented and is also the subject of a forthcoming computational work [22].

On Boundary-Value Problems for Second-Order Elliptic and Parabolic Systems

For nonstandard boundary-value problems for systems of equations of elliptic and parabolic types with vector boundary conditions, the well-posedness is proved. Typical examples are provided.

On a problem for an elliptic type equation of the second kind with a conormal and integral condition

On a problem for an elliptic type equation of the second kind with a conormal and integral condition

On the Definitions of Boundary Values of Generalized Solutions to an Elliptic-Type Equation

An elliptic-type equation with variable coefficients is considered. An overview is given of the definitions of boundary values of generalized solutions to this equation. Conditions for the existence of boundary values as well as conditions for the existence and uniqueness of solutions to the corresponding Dirichlet problem are analyzed.

An end point Orlicz type estimate for nonlinear elliptic equations

We study nonlinear elliptic equations of p-Laplacian type in divergence form to establish a natural Calderón–Zygmund type theory of an Orlicz space type, where the Lebesgue space is the special case with Φ(t)=tqp: Φ(|F|p)∈L1(Q2R)⟹Φ(|Du|p)∈L1(QR).In the previous results, the estimates obtained were strictly above the natural exponents, and the function such as Φ(t)=tlog(1+t) was ruled out for the candidate of Φ. But with our approach, Φ can be selected up to the end point case of the estimates, and the functions Φ(t)=t and Φ(t)=tlog(1+t) can be additionally selected for Φ.

On Approximation of Coefficient Inverse Problems for Differential Equations in Functional Spaces

This paper is devoted to the theory of approximation of coefficient inverse problems for differential equations of parabolic, elliptic, and hyperbolic types in functional spaces. We present general statements of problems and their approximations and review results obtained earlier in the literature.

Elliptic problems involving the p-Laplacian in ℝN with locally Lipschitz functional

ABSTRACTWe study the existence of infinitely many weak solutions for the elliptic type equation via a general nonsmooth critical point theorem.

A Note on a New Method of Investigation of Solutions of Boundary Value Problems for the Elliptic Type Equations of the First Order

The paper deals with investigation of the solutions of the boundary value problem for the Cauchy-Riemann equation on the curvelinear strip in the real plane. Here the boundary conditions are in the special type and are fundamental with respect to the direction solutions are used. An analytic form of the solution of considered boundary value problem is obtained.

Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential

We study a generalized Kirchhoff type equation with trapping potential. The existence and blow-up behavior of solutions with normalized L2-norm for this problem are discussed.