Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary. We provide some examples satisfying such a local boundedness property, and thus have Gaussian bounds for their Green's functions.     

A linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition

The problem of shock reflection by a wedge, which the flow is dominated by the unsteady potential flow equation, is a important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. 57 (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint (2003)], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle, which between incident shock and wedge, is small, we can see that weak regular reflection as the disturbance of normal reflection as in [Shuxing Chen, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21 (78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with above viewpoint, we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, delicate a priori estimate and compact arguments, we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

带有加权Hardy-Sobolev临界指数的非齐次Neumann边界奇异的多解问题

主要研究了非齐次Neumann边界奇异的问题,利用Ekeland变分原理、山路引理和一些分析技巧,证明了正解的存在性.     

A linear approximation for the regular reflection of a weak shock at a wedge

The problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. LVII (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint, 2003], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle between incident shock and wedge is small, we can see the weak regular reflection as the disturbance of normal reflection as in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118], we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, a delicate a priori estimate and compact arguments, we show that the solution of the linearized problem is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

A free boundary problem of elliptic equation arising in supersonic flow past a conical body

We study free boundary value problems of elliptic equation caused by a supersonic flow past a non-symmetric conical body. The flow is described by the potential flow equation. In the self-similar coordinate system the problem can be reduced to a boundary value problem of second order nonlinear elliptic equation with a free boundary. Applying the partial hodograph transformation and the method of nonlinear alternative iteration we proved the existence of solution to this boundary value problem. Consequently, we also proved the conclusion that for the problem of supersonic flow past a conical body, if the conical body is slightly different from a circular cone with its vertex angle less than a given value determined by the parameters of the coming flow, then there exists a weak entropy solution with an attached conical shock.     

A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary     

Transmission problems for Maxwell's equations with weakly Lipschitz interfaces

We prove sufficient conditions on material constants, frequency and Lipschitz regularity of interface for well posedness of a generalized Maxwell transmission problem in finite energy norms. This is done by embedding Maxwell's equations in an elliptic Dirac equation, by constructing the natural trace space for the transmission problem and using Hodge decompositions for operators d and δ on weakly Lipschitz domains to prove stability. We also obtain results for boundary value problems and transmission problems for the Hodge–Dirac equation and prove spectral estimates for boundary singular integral operators related to double layer potentials. Copyright © 2005 John Wiley & Sons, Ltd.     

Comparison of Green's functions for a family of boundary value problems for fractional difference equations

ABSTRACT

In this paper, we obtain sign conditions and comparison theorems for Green's functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Green's function converges to a uniquely defined Green's function of a singular boundary value problem.     

A singular multi-dimensional piston problem in compressible flow

This paper concerns the multi-dimensional piston problem, which is a special initial boundary value problem of multi-dimensional unsteady potential flow equation. The problem is defined in a domain bounded by two conical surfaces, one of them is shock, whose location is also to be determined. By introducing self-similar coordinates, the problem can be reduced to a free boundary value problem of an elliptic equation. The existence of the problem is proved by using partial hodograph transformation and nonlinear alternating iteration. The result also shows the stability of the structure of shock front in symmetric case under small perturbation.     

Selections of shape functions for dimensional reduction to Helmholtz's equation

The boundary value problem of Helmholtz's equation on a n + 1 dimensional thin slab is approximated by appropriate systems of the n‐dimensional boundary value problem. The very detailed estimates for modeling error in the H¹‐norm demonstrate convergence when the thickness of the slab approaches 0 as well as when the size of the systems approaches infinity. Shape functions through the thickness are first selected by finitely many eigenfunctions, and the tail is then selected to consist of polynomials. The presence of two types of functions gives rise to a certain choice in the selection of a particular set of shape functions. Numerical results provide a good illustration of the effect of different choices for specific problems. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 169–190, 1999     

具有脉冲的分数阶Bagley-Torvik 模型边值问题

将具有脉冲的分数阶Bagley-Torvik微分方程边值问题巧妙地转化为积分方程,定义加权Banach空间及全连续算子,运用不动点定理获得该边值问题解的存在性定理.举例说明了定理的应用.最后提出有趣的研究问题.     

Reconstruction in the Calderón Problem with Partial Data

We consider the problem of recovering the coefficient σ(x) of the elliptic equation ?·(σ?u) = 0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sjöstrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by solving an integral equation which involves only the given partial Cauchy data. The construction entails a new family of Green's functions for the Laplacian, and corresponding single layer potentials, which may be of independent interest.     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Maximum‐norm superapproximation of the gradient for the trilinear block finite element

For a model elliptic boundary value problem in three dimensions, we give the weak estimate of the first type for trilinear block elements and the estimate for W^1,1‐seminorm of the discrete derivative Green's function over rectangular partitions of the domain, from which we obtain maximum‐norm superapproximation of the gradient for the trilinear block finite element approximation. Furthermore, utilizing this superapproximation, we can also obtain maximum‐norm superconvergence of the gradient. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Quasilinear elliptic inclusions of Clarke's gradient type under local growth conditions

We consider a class of elliptic inclusions under Dirichlet boundary conditions involving multifunctions of Clarke's generalized gradient that satisfies only a one-sided local growth condition. An appropriate modification of the associated potential function along with truncation techniques will allow us to apply the theory of multivalued pseudomonotone operators for obtaining existence and comparison results under only one-sided local growth conditions on Clarke's generalized gradient.     

Solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations at resonance

In this paper, we study the solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations with fractional derivative under resonant conditions. Firstly, the kernel function is presented through the Laplace transform and the properties of the kernel function are obtained. And then, some new results on the solvability for the boundary value problem are established by using Mawhin''s coincidence degree theory. Finally, two examples are presented to illustrate the applicability of our main results.     

L p Estimates for Maxwell's Equations with Source Term

The goal of this paper is to establish interior and global L ^p -type estimates for the solutions of Maxwell's equations with source term in a domain filled with two different materials separated by a ² interface. The usual elliptic estimates cannot be applied directly, due to the singularity of the dielectric permittivity. A special curl-div decomposition is introduced for the electric field to reduce the problem to an elliptic equation in divergence form with jump coefficients. The potential analysis and the jump condition lead to the interior L ^p estimates which are superior to the straightforward Nash-Moser estimates. The reduction procedure is expected to be useful for future numerical simulation. Because of the natural physical requirements, the boundary condition is nonlocal and involves a first order pseudo-differential operator, the boundary estimate is established by delicate new maximum principles and Riesz convexity arguments. These estimates are then employed to solve a nonlinear optics problem that arises in the modeling of surface enhanced second-harmonic generation of nonlinear diffractive optics in periodic structures (gratings).     

Existence and regularity of solution of mixed boundary value problem of a Busemann-equation with nonlinear absorb term

The Busemann-equation is a classical equation coming from fluid dynamics. The well-posed problem and regularity of solution of Busemann-equation with nonlinear term are interesting and important. The Busemann-equation is elliptic in y>0 and is degenerate at the line y=0 in R². With a special nonlinear absorb term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain. By means of elliptic regularization technique, a delicate prior estimate and compact argument, we show that the solution of mixed boundary value problem of Busemann-equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary on some conditions. The result is better than the result of classical boundary degenerate elliptic equation.     

Régularité des solutions d'un problème mêlé Dirichlet–Signorini dans un domaine polygonal plan

We prove a rgularity result, in Hölder spaces, for solutions to an elliptic problem with mixed boundary condition, namely Dirichlet on a part of the boundary and Signorini on the remaining part, in a regular or polygonal, domain of IR² We give the behaviour of the solution near points where the boundary condition tyoe changes.

Nous montrons la Régularité jusqu'au bord, dans les espaces de Hölder des solutions d'un problème aux limites de type mêlé avec une condition de Dirichlet sur une partie de la frontière et des conditions de type Signorini sur le complémentaire, et ceci dans un ouvert plan ré gulier ou polygonal. Nous donnons en particulier le comportement de la solution au volisinge d'un point de la frontiére où la condition aux limites change de type.

Mots Clés: In´quations variationnells–conditions unilatélrales. Domains polygonaux.

Classification AMS 35 (J) 49 (A)     

Superconvergence of tetrahedral quadratic finite elements for a variable coefficient elliptic equation

For a variable coefficient elliptic boundary value problem in three dimensions, using the properties of the bubble function and the element cancelation technique, we derive the weak estimate of the first type for tetrahedral quadratic elements. In addition, the estimate for the W^1,1‐seminorm of the discrete derivative Green's function is also given. Finally, we show that the derivatives of the finite element solution u_h and the corresponding interpolant Π₂u are superclose in the pointwise sense of the L^∞‐norm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013