Equations Of Elliptic Type Research Articles

Asymptotics of a solution to an optimal boundary control problem

A problem of optimal boundary control of solutions to an elliptic-type equation with a small coefficient at the highest derivative and integral constraints on the control is considered. Asymptotic estimates for solutions to a problem that approximates the original problem are obtained.

Global solution curves for a class of elliptic systems

We study curves of positive solutions for a system of elliptic equations of Hamiltonian type on a unit ball. We give conditions for all positive solutions to lie on global solution curves, allowing us to use the analysis similar to the case of one equation, as developed in P. Korman, Y. Li and T. Ouyang [An exact multiplicity result for a class of semilinear equations, Commun. PDE 22 (1997), pp. 661–684.] (see also T. Ouyang and J. Shi [Exact multiplicity of positive solutions for a class of semilinear problems, II, J. Diff. Eqns. 158(1) (1999), pp. 94–151].). As an application, we obtain some non-degeneracy and uniqueness results. For the one-dimensional case we also prove the positivity for the linearized problem, resulting in more detailed results.

Gradient estimates for viscosity solutions of singular fully nonlinear elliptic equations

We establish Lipschitz regularity for solutions to a family of non-isotropic fully nonlinear partial differential equations of elliptic type. In general such a regularity is optimal. No sign constraint is imposed on the solution, thus limiting free boundaries may have two-phases. Our estimates are then employed in combination with fine regularizing techniques to prove existence of viscosity solutions to singular nonlinear PDEs.

Gradient continuity estimates

We consider degenerate elliptic equations of p-Laplacean type $$-{\rm{div}}\, (\gamma(x)|Du|^{p-2}Du)=\mu\,,$$ and give a sufficient condition for the continuity of Du in terms of a natural non-linear Wolff potential of the right-hand side measure μ. As a corollary we identify borderline condition for the continuity of Du in terms of the data: namely μ belongs to the Lorentz space L(n, 1/(p − 1)), and γ(x) is a Dini continuous elliptic coefficient. This last result, together with pointwise gradient bounds via non-linear potentials, extends to the non homogeneous p-Laplacean system, thereby giving a positive answer in the vectorial case to a conjecture of Verbitsky. Continuity conditions related to the density of μ, or to the decay rate of its Ln-norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.

Study of boundary value and transmission problems governed by a class of variable operators verifying the Labbas-Terreni non commutativity assumption

The aim of this work is the study of some transmission problems, which are written as an abstract second order differential equation of elliptic type with variable operator coefficients, in the framework of Holder spaces. Here, we do not assume the differentiability of the resolvent operators. However, we suppose that the family of variable operators verifies the Labbas-Terreni assumption inspired by the sum theory and similar to the Acquistapace-Terreni one. We use Dunford calculus, interpolation spaces and semigroup theory in order to obtain existence, uniqueness and maximal regularity results for the solution of the problem.

A second-order and two-scale computation method for the quasi-periodic structures of composite materials

In this paper, we shall present the multi-scale analysis of boundary value problems for second-order elliptic type equation for the quasi-periodic structures of composite materials, and give the mathematical proof of the convergence estimate for displacement, which is the basis of finite element computation. Finally, numerical results show that the method presented in this paper is effective and reliable.

Entire Large Solutions of Quasilinear Elliptic Equations of Mixed Type

In this paper, the existence and nonexistence of nonnegative entire large solutions for the quasilinear elliptic equation are established, where , and are nondecreasing and vanish at the origin. The locally H lder continuous functions and are nonnegative. We extend results previously obtained for special cases of and g.

On the eigenvalue set structure for higher-order equations of elliptic type with discontinuous nonlinearities

We study the existence of nonzero solutions of the Dirichlet problem for a higherorder equation of elliptic type with a discontinuous nonlinearity. By the variational method, we prove the existence of a ray of positive eigenvalues and prove an upper bound for the bifurcation parameter in the problem.

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

Sign‐Changing Solutions to Equations of Elliptic Type

1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 Mathematisches Institut, University Giessen, Arndtstr. 2, 35392 Giessen, Germany 3 Graduate School of Environment and Information Science, Yokohama National University, Tokiwadai, Yokohama 240-8501, Japan 4 Mathematics Department, University of California, Irvine, CA 92697-3875, USA 5 Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-4205, USA

Transmission problem for an abstract fourth-order differential equation of elliptic type in UMD spaces

In this work, we present a new result ofexistence, uniqueness and maximal regularity for the restriction solutions of the bilaplacian transmission problem set in the juxtaposition of two rectangular bodies. The study is performed in the space $L^{p}( (-1, 0) \cup(0, \delta) ;X),$ $1 < p < \infty,$ where $\delta $ is a small parameter which is destined to tend to zero and $X$ is a UMD Banach space. The geometry of the bodies allows us to find an explicit representation of the solutions in virtue of the operational Dunford calculus. We then use essentially the famous Dore-Venni theorem among others for the analysis of the solutions.

A new generalization of the PSWFs with applications to spectral approximations on quasi-uniform grids

We define a new family of generalized prolate spheroidal wave functions (GPSWFs), which extends the prolate spheroidal wave functions of order zero (PSWFs or Slepian functions; Slepian and Pollak, 1961 [45]) to real order α > − 1 , and also generalizes the Gegenbauer polynomials to an orthogonal system with an intrinsic tuning parameter c > 0 . We show that the GPSWFs, defined as the eigenfunctions of a Sturm–Liouville problem, are also the eigenfunctions of an integral operator. We present a number of analytic and asymptotic formulae for the GPSWFs and the associated eigenvalues, and introduce efficient algorithms for their evaluations. Moreover, we derive a set of optimal results on the GPSWF approximations featured with explicit dependence on the parameter c. As an important application, we implement and analyze the GPSWF spectral methods for elliptic-type equations. We illustrate that the presence of c provides flexibility to design high-order approximations on quasi-uniform grids, and endows the GPSWFs with some favorable advantages over their polynomial counterparts.

On Hölder regularity for elliptic equations of non-divergence type in the plane

This paper is concerned with strong solutions of uniformly elliptic equations of non-divergence type in the plane. First, we use the notion of quasiregular gradient mappings to improve Morrey’s theorem on the Holder continuity of gradients of solutions. Then we show that the Gilbarg-Serrin equation does not produce the optimal Holder exponent in the considered class of equations. Finally, we propose a conjecture for the best possible exponent and prove it under an additional restriction. Mathematics Subject Classification (2000): 35B65 (primary); 30C62, 35J15 (secondary).

The method of calculation of the geophysical fields in piecewise homogeneous media

Recurrent algorithms for solving 3D direct problems on the distribution of geophysical fields in the piecewise-homogeneous media are built, based on methods of integral transforms, integral representations, and integral equations. The algorithms allow the recalculation of Green’s functions of one medium into Green’s function of another medium. The computational procedure is applicable to problems with equations of elliptic, parabolic, hyperbolic, and mixed types. The algorithms can be split and implemented on computing clusters and multiprocessor complexes. Inverse problems and algorithms of searching for boundaries of piecewise-homogeneous media as extremals of the Tikhonov regulating functional, approximated by the spline functions, are formulated.

Analysis of spectral approximations using prolate spheroidal wave functions

In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of non-periodic functions in Sobolev spaces. These results serve as an indispensable tool for the analysis of PSWF spectral methods. A PSWF spectral-Galerkin method is proposed and analyzed for elliptic-type equations. Illustrative numerical results consistent with the theoretical analysis are also presented.

On solutions of elliptic-type differential equations in a Banach space on a semiaxis

We describe all classical solutions of an abstract m-harmonic equation on (0,∞) and study their properties on this interval and in the neighborhood of the singular point 0.

Jacobian elliptic solitons in inhomogeneous alpha-helical proteins

In this paper, we study the effects of mass inhomogeneities on the energy transfer by solitons in alpha-helical proteins. We demonstrate that such a system is governed by a standard elliptic type equation. Total energy of the soliton and its effective mass are calculated. We determine critical solitons' velocities at the borders of the inhomogeneous zone. In the case of light-mass-induced impurities, we obtain that oscillatory motions are possible provided cn-type solitons are considered. In the case of heavy-mass-induced impurities, oscillatory motion can be introduced by dn-type solitons. Pinning frequencies of oscillations are estimated.

Multigrid Methods for PDE Optimization

Research on multigrid methods for optimization problems is reviewed. Optimization problems considered include shape design, parameter optimization, and optimal control problems governed by partial differential equations of elliptic, parabolic, and hyperbolic type.

Uniqueness of boundary blow-up solutions on unbounded domain of [formula omitted

In this paper, we consider the existence of positive solutions of the degenerate logistic type elliptic equation − Δ u = a ( x ) u − b ( x ) f ( u ) , x ∈ R N ∖ D u | ∂ D = ∞ , where N ≥ 2 , D ⊂ R N is a bounded smooth domain and a ( x ) , b ( x ) are continuous functions on R N with b ( x ) ≥ 0 , b ( x ) ≢ 0 , especially b ( x ) = 0 on ∂ D and f ( u ) ∈ C ( 0 , ∞ ) . We show that under rather general conditions on a ( x ) and b ( x ) for large | x | and f ( u ) behaves like u q where q > 1 . Without the behavior of b ( x ) near the boundary ∂ D , it will be shown that there exists a unique positive solution. Our results improve the corresponding ones in [W. Dong, Pang, Uniqueness of boundary blow-up solutions on exterior domain of R N , J. Math. Anal. Appl. 330 (2007), 654–664] and [Y. Du, L. Ma, Logistic type equations on R N by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., (2) 64 (2001), 107–124].

Quadratic-Argument Approach to Nonlinear Schrödinger Equation and Coupled Ones

The two-dimensional cubic nonlinear Schrödinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schrödinger equations are used to describe the interaction of electromagnetic waves with different polarizations in nonlinear optics. Mathematically, they are fundamental nonlinear partial differential equations of elliptic type. In this paper, we solve the above equations by imposing a quadratic condition on the related argument functions and using their symmetry transformations. More complete families of exact solutions of such type are obtained. Many of them are the periodic, quasi-periodic, aperiodic and singular solutions that may have practical significance.