Hölder estimates of solutions to a degenerate diffusion equation

This paper is concerned with the Hölder estimates of weak solutions of the Cauchy problem for the general degenerate parabolic equations

with the initial data , where the diffusion function can be a constant on a nonzero measure set, such as the equations of two-phase Stefan type. Some explicit Hölder exponents of the composition function with respect to the space variables are obtained by using the maximum principle.

     

On the best Hölder exponent for two dimensional elliptic equations in divergence form

We obtain an estimate for the Hölder continuity exponent for weak solutions to the following elliptic equation in divergence form:

where is a bounded open subset of and, for every , is a symmetric matrix with bounded measurable coefficients. Such an estimate ``interpolates' between the well-known estimate of Piccinini and Spagnolo in the isotropic case , where is a bounded measurable function, and our previous result in the unit determinant case . Furthermore, we show that our estimate is sharp. Indeed, for every we construct coefficient matrices such that is isotropic and has unit determinant, and such that our estimate for reduces to an equality, for every .

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains

We consider the operator

acting on functions in . We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on and . In contrast to previous work, the need only be nonnegative on the boundary rather than strictly positive, at the expense of the and being Hölder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhan's perturbation argument, but the underlying function space is now a weighted Hölder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.

     

Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces

We introduce a new method for proving the estimate

where solves the equation . The method can be applied to the Laplacian on . It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.

     

Boundary Hölder and estimates for local solutions of the tangential Cauchy-Riemann equation

We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood of a point when is a generic -concave manifold of real codimension in , where . Our method is to first derive a homotopy formula for in when is the intersection of with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any -closed form on without shrinking. We obtain Hölder and estimates up to the boundary for the solution operator. RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage d'un point d'une sous-variété générique -concave de codimension quelconque de . Nous construisons une formule d'homotopie pour le sur , lorsque est l'intersection de et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme -fermée sur . Nous en déduisons des estimations et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur .

     

Hydrodynamic approach to constructing solutions of the nonlinear Schrödinger equation in the critical case

Proceeding from the hydrodynamic approach, we construct exact solutions to the nonlinear Schrödinger equation with special properties. The solutions describe collapse, in finite time, and scattering, over infinite time, of wave packets. They generalize known blow-up solutions based on the ``ground state'.

     

Regularity of weak solutions to the Monge-Ampère equation

We study the properties of generalized solutions to the Monge-Ampère equation , where the Borel measure satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When , this condition, which we call , admits the possibility of vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when , which implies that is doubling. The main difference between the case and the case when is bounded between two positive constants is the need to use a variant of the Aleksandrov maximum principle (due to Jerison) and some tools from convex geometry, in particular the Hausdorff metric.

     

High‐energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and application

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short‐range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high‐energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy E. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in L²(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy E, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some E, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge‐conjugation and time‐reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd.     

A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications

The system of partial differential equations

Existence and regularity of weak solutions to a model for coarsening in molecular beam epitaxy

Taking into account the occurrence of a zero of the surface diffusion current and the requirement of the Ehrlich–Schwoebel effect, Siegert et al. formulated a model of Langevin type that describes the growth of pyramid‐like structures on a surface under conditions of molecular beam epitaxy and that the slope of these pyramids is selected by the crystalline symmetries of the growing film. In this article, the existence and uniqueness of weak solution to an initial boundary value problem for this model is proved, in the case that the noise is neglected. The regularity of the weak solution to models, with/without slope selection, is also investigated. Copyright © 2012 John Wiley & Sons, Ltd.     

Loss of regularity for the solutions to hyperbolic equations with non‐regular coefficients—an application to Kirchhoff equation

We consider the Cauchy problem for second‐order strictly hyperbolic equations with time‐depending non‐regular coefficients. There is a possibility that singular coefficients make a regularity loss for the solution. The main purpose of this paper is to derive an optimal singularity for the coefficient that the Cauchy problem is C^∞ well‐posed. Moreover, we will apply such a result to the estimate of the existence time of the solution for Kirchhoff equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Moments for a multivariate linear model with an application to the growth curve model

An extended growth curve model is considered which, among other things, is useful when linear restrictions exist on the mean in the ordinary growth curve model. The maximum likelihood estimators consist of complicated stochastic expressions. It is shown how, by the aid of fairly elementary calculations, the dispersion matrix for the estimator of the mean and the expectation of the estimated dispersion matrix are obtained. Results for Wishart, inverted Wishart, and inverse beta variables are utilized. Additionally, some asymptotic results are presented.     

Approximation of the Navier–Stokes system with variable viscosity by a system of Cauchy–Kowaleska type

In this paper, we study the existence of weak solutions when n?4 of the mixed problem for the Navier–Stokes equations defined in a bounded domain Q using approximation by a system of Cauchy–Kowaleska type. Periodical solutions are also analyzed. Copyright © 2008 John Wiley & Sons, Ltd.     

Constructing an atlas for the diffeomorphism group of a compact manifold with boundary,with application to the analysis of image registrations

This paper considers the problem of defining a parameterization (chart) on the group of diffeomorphisms with compact support, motivated primarily by a problem in image registration, where diffeomorphic warps are used to align images. Constructing a chart on the diffeomorphism group will enable the quantitative analysis of these warps to discover the normal and abnormal variation of structures in a population.     

A new proof of the existence of weak solutions to a model for phase evolution driven by material forces

We prove the existence of weak solutions to a one‐dimensional initial‐boundary value problem for a model system of partial differential equations, which consists of a sub‐system of linear elasticity and a nonlinear non‐uniformly parabolic equation of second order. To simplify the existence proof of weak solutions in the 2006 paper of Alber and Zhu, we replace the function in that work by . The model is formulated by using a sharp interface model for phase transformations that are driven by material forces. Copyright © 2017 John Wiley & Sons, Ltd.     

Existence of maximal and minimal solutions to a boundary value problem for the equation of the bending of an elastic beam

By introducing a partial order and using the Mönch fixed point theorem, we establish the existence of maximal and minimal solutions in Banach spaces to a boundary value problem for the equation of the bending of an elastic beam.