The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

A MIXED ELECTRIC BOUNDARY VALUE PROBLEM FOR AN ANTI-PLANE PIEZOELECTRIC CRACK

The analytical continuation method is adopted to solve a mixed electric boundary value problem for a piezoelectric medium under anti-plane deformation.The crack face is partly conductive and partly impermeable.The results show that the stress intensity factor is identical with the mode Ⅲ stress intensity factor independent of the conducting length.But the electric field and the electric displacement are dependent on the electric boundary conditions on the crack faces and are singular not only at the crack tips but also at the junctures between the impermeable part and conducting portions.

GLOBAL EXISTENCE OF WEAKLY DISCONTINUOUS SOLUTIONS TO A KIND OF MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.

AN ANALYTICAL METHOD FOR A MIXED BOUNDARY VALUE PROBLEM OF CIRCULAR PLATES UNDER ARBITRARY LATERAL LOADS

In this paper by using the concept of mixed boundary funetions, an analytical method is proposed for a mixed boundary value problem of circular plates. The trial functions are constructed by using the series of particular solutions of the biharmonic equations in the polar coordinate system. Three examples are presented to show the stability and high convergence rate of the method.

PERIODIC BOUNDARY VALUE PROBLEMS FOR NONLINEAR IMPULSIVEINTEGRODIFFERENTIAL EQUATIONS OF MIXED TYPE IN BANACH SPACES

In this paper, the author obtains an existence theorem of minimal and maximal solutions for the periodic boundary value problems of nonlinear impulsive integrodifferential equations of mixed type in Banach space by means of the monotone iterative technique and cone theory based on a comparison result.

A UNIFORMLY DIFFERENCE SCHEME OF SINGULAR PERTURBATION PROBLEM FOR A SEMILINEAR ORDINARY DIFFERENTIAL EQUATION WITH MIXED BOUNDARY VALUE CONDITION

In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.

SEMI-GLOBAL C^1 SOLUTION TO THE MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

By means of an equivalent invariant form of boundary conditions, the authors get the exis- tence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions.

TWO-SCALE FEM FOR ELLIPTIC MIXED BOUNDARY VALUE PROBLEMS WITH SMALL PERIODIC COEFFICIENTS

Presents a study which proposed a dual approximate expression of the exact solution for mixed boundary value problems of second order elliptic partial differential equation with small periodic coefficients. Dual approximate error estimates; Main results.

Petrov-Galerkin Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems on Polygons

The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.

Guest Review:Potential theory method for 3D crack and contact problems of multi-field coupled media: A survey

This paper presents an overview of the recent progress of potential theory method in the analysis of mixed boundary value problems mainly stemming from three-dimensional crack or contact problems of multi-field coupled media. This method was used to derive a series of exact three dimensional solutions which should be of great theoretical significance because most of them usually cannot be derived by other methods such as the transform method and the trial-and-error method. Further, many solutions are obtained in terms of elementary functions that enable us to treat more complicated problems easily. It is pointed out here that the method is usually only applicable to media characterizing transverse isotropy, from which, however, the results for the isotropic case can be readily obtained.

Existence of Local Solutions to a Class of Doubly Degenerate Parabolic Equations

This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate pLaplace equation （p 〉 2）ut-div（b（x,t,u）｜↓△u｜^p-2↓△u）=f（x,u,t）The initial-boundary value problem in a bounded domain of R^N is considered under mixed boundary conditions. The existence of local-in-time weak solutions is obtained.

SPECTRAL METHOD FOR MIXED INHOMOGENEOUS BOUNDARY VALUE PROBLEMS IN THREE DIMENSIONS

In this paper, we investigate spectral method for mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approxima- tion in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique, with which we could handle mixed inho- mogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms is proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.

THREE SOLUTIONS FOR A CLASS OF GRADIENT MIXED BOUNDARY VALUE SYSTEMS DEPENDING ON TWO PARAMETERS

In this paper we discuss the existence of at least three solutions for a class of gradient mixed boundary value systems. The approach is fully based on a recent three critical points theorem of B. Ricceri [A three critical points theorem revisited, Nonlinear Anal., 70：9（2009）,3084-3089].

Superlinear Elliptic Equation with Mixed Boundary Value in Annular Domain

In this paper,we study superlinear elliptic equations with mixed boundary value conditions in annular domains.It is assumed that the nonlinearities depend on the derivative terms.Some results about existence of solutions are established by using the Nehari manifold technique,as well as iterative technique.

THE LINEAR SAMPLING METHOD FOR RECONSTRUCTING A PENETRABLE CAVITY WITH UNKNOWN EXTERNAL OBSTACLES

We consider the interior inverse scattering problem for recovering the shape of a penetrable partially coated cavity with external obstacles from the knowledge of measured scattered waves due to point sources.In the first part,we obtain the well-posedness of the direct scattering problem by the variational method.In the second part,we establish the mathematical basis of the linear sampling method to recover both the shape of the cavity,and the shape of the external obstacle,however the exterior transmission eigenvalue problem also plays a key role in the discussion of this paper.

Asymptotic Stability of Equilibrium State to the Mixed Initial-Boundary Value Problem for Quasilinear Hyperbolic Systems

Under the internal dissipative condition, the Cauchy problem for inhomogeneous quasilinear hyperbolic systems with small initial data admits a unique global C1 solution, which exponentially decays to zero as t →＋∞, while if the coefficient matrix 19 of boundary conditions satisfies the boundary dissipative condition, the mixed initialboundary value problem with small initial data for quasilinear hyperbolic systems with nonlinear terms of at least second order admits a unique global C1 solution, which also exponentially decays to zero as t →＋∞. In this paper, under more general conditions, the authors investigate the combined effect of the internal dissipative condition and the boundary dissipative condition, and prove the global existence and exponential decay of the C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with small initial data. This stability result is applied to a kind of models, and an example is given to show the possible exponential instability if the corresponding conditions are not satisfied.

GLOBAL WEAKLY DISCONTINUOUS SOLUTIONS TO A KIND OF MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR INHOMOGENEOUS QUASILINEAR HYPERBOLIC SYSTEMS

In this paper we study the mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems in the domain D -- {（t, x） I t 〉 O, x 〉 0}. Under the assumption that the source term satisfies the matching condition, a sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.

PROBABILISTIC APPROACH TO SEMILINEAR AND GENERALIZED MIXED BOUNDARY VALUE PROBLEMS

A probabilistic approach is developed to solve semilinear and generalized mixed boundaryvalue problems involving Schrodinger operators. The results obtained in this paper generalize thecorresponding results of [1] and partly generalize the result of [2] as well.

LIFE-SPAN OF CLASSICAL SOLUTIONS OF INITIAL-BOUNDARY VALUE PROBLEM FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS

In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in a half-unbounded domain {（t, x）｜t ≥ 0, x ≥ 0}. Under the assumption that the positive eigenvalues are not all weakly linearly degenerate,we obtain the blow-up phenomenon of the first order derivatives of C^1 solution with small and decaying initial data.We also give precise estimate of the life-span of C^1 solution.

Kirchhoff-Type Problem with Mixed Boundary Condition in a Variable Exponent Sobolev

.In this paper,we consider a mixed boundary value problem for the stationary Kirchhoff-type equation containing p(-)-Laplacian.More precisely,we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary.We show the existence of at least one,two or infinitely many non-trivial weak solutions according to hypotheses on given functions.