On the entropy solution to an elliptic problem in anisotropic Sobolev–Orlicz spaces

For a certain class of anisotropic elliptic equations with the right-hand side from L ₁ in an arbitrary unbounded domains, the Dirichlet problem with an inhomogeneous boundary condition is considered. The existence and uniqueness of the entropy solution in anisotropic Sobolev–Orlicz spaces are proven.     

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

We consider a class of anisotropic elliptic equations of second order with variable exponents of non-linearity where a special Radon measure is used as the right-hand side. We establish uniqueness of entropy and renormalized solutions of the Dirichlet problem in anisotropic Sobolev spaces with variable exponents of non-linearity for arbitrary domains and certain other their properties. In addition, we prove the equivalence of entropy and renormalized solutions of the problem under consideration.     

Fast Algorithm for Best Anisotropic Walsh Bases and Relatives

We introduce a new family of Nlog Nbest basis search algorithms for functions of more than one variable. These algorithms search the collection of anisotropic wavelet packet and cosine packet bases and output a minimum entropy basis for a given function. These algorithms are constructed after treating the model problem of computing best Walsh packet bases. Several intermediate algorithms for conducting mixed isotropic/anisotropic best basis searches in the function's various coordinate directions are also presented.     

Stochastic integral representations,stochastic derivatives and minimal variance hedging

The stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale. In relation to this, a certain stochastic derivative is defined. It is shown that this derivative determines the integrand in the stochastic integral which serves as the best L 2 - approximation to the random variable considered. For a general Lévy process as integrator some specification of the suggested stochastic derivative is given. In the case of the Wiener process the considered specification reduces to the well-known Clark-Haussmann-Ocone formula. This result provides a general solution to the problem of minimal variance hedging in incomplete markets.     

Multi-dimensional scalar balance laws with discontinuous flux

We consider scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect to both the space variable x and the unknown quantity u. We formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem. The problem is formulated in the framework of multi-valued mappings. The notion of entropy measure-valued solutions is used to prove the so-called contraction principle and comparison principle.     

Existence and Uniqueness of Entropy Solution of Scalar Conservation Laws with a Flux Function Involving Discontinuous Coefficients

ABSTRACT

In this paper, the question of existence and uniqueness for entropy solutions of scalar conservation laws with a flux function which is discontinuous with respect to the space variable is investigated. We show that no extra assumption of convexity or genuine non-linearity with respect to the state variable of the flux function is required for the problem to be well-posed and prove it. The proof uses a kinetic formulation of the conservation law.     

Singular and degenerate anisotropic parabolic equations with a nonlinear source

The Dirichlet problem in arbitrary domain for degenerate and singular anisotropic parabolic equations with a nonlinear source term is considered. We state conditions that guarantee the existence and uniqueness of a global weak solution to the problem. A similar result is proved for the parabolic p-Laplace equation.     

Asymptotic behavior of solutions to anisotropic conservation laws in two-dimensional space

In this paper, we investigate the asymptotic behavior of solutions for anisotropic conservation laws in two-dimensional space, provided with step-like initial conditions that approach the constant states u_± (u₋<u₊) as x→±∞, respectively. It shows that there is a global classical solution that converges toward the rarefaction wave, ie, the unique entropy solution of the Riemann problem for the nonviscous Burgers' equation in one-dimensional space.     

Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients

One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients.

Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L ² space for solution to the inverse problem under consideration.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Prewavelet analysis of the heat equation

Summary. We consider the heat equation in a smooth domain of R with Dirichlet and Neumann boundary conditions. It is solved by using its integral formulation with double-layer potentials, where the unknown , the jump of the solution through the boundary, belongs to an anisotropic Sobolev space. We approximate by the Galerkin method and use a prewavelet basis on , which characterizes the anisotropic space. The use of prewavelets allows to compress the stiffness matrix from to when N is the size of the matrix, and the condition number of the compressed matrix is uniformly bounded as the initial one in the prewavelet basis. Finally we show that the compressed scheme converges as fast as the Galerkin one, even for the Dirichlet problem which does not admit a coercive variational formulation. Received April 16, 1999 / Published online August 2, 2000     

Dirichlet problem for complex model partial differential equations

ABSTRACT

Complex model partial differential equations of arbitrary order are considered. The uniqueness of the Dirichlet problem is studied. It is proved that the Dirichlet problem for higher order complex partial differential equations with one complex variable has infinitely many solutions.     

A Time-Fractional Step Method for Conservation Law Related Obstacle Problems

We are interested in approximating the solution of a first-order quasi-linear equation associated with a forced unilateral obstacle condition. With this view, we make use of the time-splitting method developed classically to compute discontinuous solutions of nonhomogeneous scalar conservation laws. Here, one proves that this fractional step method converges in L¹ to the weak entropy solution of the considered obstacle problem. In the case of the Cauchy problem, an L¹-error bound in is established.     

Entropic perturbation method for solving a system of linear inequalities

The problem of finding an $\text{x∈}\text{R}{}^{\mathrm{n}}$ such that Ax⩽b and x⩾0 arises in numerous contexts. We propose a new optimization method for solving this feasibility problem. After converting Ax⩽b into a system of equations by introducing a slack variable for each of the linear inequalities, the method imposes an entropy function over both the original and the slack variables as the objective function. The resulting entropy optimization problem is convex and has an unconstrained convex dual. If the system is consistent and has an interior solution, then a closed-form formula converts the dual optimal solution to the primal optimal solution, which is a feasible solution for the original system of linear inequalities. An algorithm based on the Newton method is proposed for solving the unconstrained dual problem. The proposed algorithm enjoys the global convergence property with a quadratic rate of local convergence. However, if the system is inconsistent, the unconstrained dual is shown to be unbounded. Moreover, the same algorithm can detect possible inconsistency of the system. Our numerical examples reveal the insensitivity of the number of iterations to both the size of the problem and the distance between the initial solution and the feasible region. The performance of the proposed algorithm is compared to that of the surrogate constraint algorithm recently developed by Yang and Murty. Our comparison indicates that the proposed method is particularly suitable when the number of constraints is larger than that of the variables and the initial solution is not close to the feasible region.     

The obstacle problem for nonlinear elliptic equations with variable growth and L1-data

The aim of this paper is twofold: to prove, for L ¹-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy-Stampacchia inequalities to the general framework of L ¹.     

The homogeneous dirichlet problem for quasilinear anisotropic degenerate parabolic-hyperbolic equation with L initial value

The aim of this paper is to prove the well-posedness (existence and uniqueness) of the L^p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L^p initial value. We use the device of doubling variables and some technical analysis to prove the uniqueness result. Moreover we can prove that the L^p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.     

On the coefficient inverse problem of heat conduction in a degenerating domain

ABSTRACT

In the paper, we consider a coefficient inverse problem for the heat equation in a degenerating angular domain. It has been shown that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with the integral condition. Moreover, the solution of the considered inverse problem is found in explicit form. In conclusion, statements of possible generalizations and the results of numerical calculations are given.     

The problem of the bending of a beam lying on an elastic base

The plane contact problem of the transmission of a normal force of specified strength onto an elastic anisotropic, wedge-shaped plate by an elastic beam of variable flexural stiffness is considered. The beam is coupled to one of the edges of the plate and its other edge is stress-free. The solution of the problem is obtained in closed form by reducing it to a Karleman boundary-value problem with shear for a strip. A conclusion is reached concerning the nature of the discontinuity of the normal contact stress at the vertex of the wedge.     

Dynamics of Bianchi Type I Solutions of the Einstein Equations with Anisotropic Matter

We analyze the global dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The matter model is not specified explicitly but only through a set of mild and physically motivated assumptions; thereby our analysis covers matter models as different from each other as, e.g., collisionless matter, elastic matter and magnetic fields. The main result we prove is the existence of an ‘anisotropy classification’ for the asymptotic behaviour of Bianchi type I cosmologies. The type of asymptotic behaviour of generic solutions is determined by one single parameter that describes certain properties of the anisotropic matter model under extreme conditions. The anisotropy classification comprises the following types. The convergent type A₊: Each solution converges to a Kasner solution as the singularity is approached and each Kasner solution is a possible past asymptotic state. The convergent types B₊ and C₊: Each solution converges to a Kasner solution as the singularity is approached; however, the set of Kasner solutions that are possible past asymptotic states is restricted. The oscillatory type D₊: Each solution oscillates between different Kasner solutions as the singularity is approached. Furthermore, we investigate non-generic asymptotic behaviour and the future asymptotic behaviour of solutions. Submitted: October 28, 2008.; Accepted: January 26, 2009.