Fundamental solutions of degenerate or singular elliptic equations

Exitence of fundamental solutions is established for a class of degenerate or singular, second-order, linear elliptic partial differential equations. This class contains, for example, Tricomi's equation in the upper half-plane which arises in the study of aerodynamics; the equation of Weinstein's generalized axially symmetric potential theory which arises in the study of fluid dynamics and elasticity; and Schrödinger's equation with a singular potential which arises in quantum mechanics.     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.     

Stationary solutions of SPDEs and infinite horizon BDSDEs

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.     

Fundamental groups of singular quasi-projective varieties

We express, under appropriate conditions, the fundamental group of a singular complex quasi-projective variety as a quotient of the fundamental group of a general hyperplane section, using a generic pencil. The subgroup by which the quotient is taken is described with the help of the monodromies around the exceptional hyperplanes of the pencil. This is a new generalization of the classical Zariski-van Kampen theorem on curves.     

Log-Harnack inequality for mild solutions of SPDEs with multiplicative noise

Due to technical reasons, existing results concerning Harnack type inequalities for SPDEs with multiplicative noise apply only to the case where the coefficient in the noise term is a Hilbert–Schmidt perturbation of a constant bounded operator. In this paper we obtained gradient estimates, log-Harnack inequality for mild solutions of general SPDEs with multiplicative noise whose coefficient is even allowed to be unbounded which cannot be Hilbert–Schmidt. Applications to stochastic reaction–diffusion equations driven by space–time white noise are presented.     

Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the -valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the -valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.     

Joint continuity of the solutions to a class of nonlinear SPDEs

For a one-dimensional superprocess in random environment, a nonlinear SPDE was derived by Dawson et?al. (Ann Inst Henri Poincaré Probab Stat 36(2):167–180, 2000) for its density process. The time-space joint continuity of the density process was left as an open problem. In this paper we give an affirmative answer to this problem.     

Peaked singular wave solutions associated with singular curves

We present new types of singular wave solutions with peaks in this paper. When a heteroclinic orbit connecting two saddle points intersects with the singular curve on the topological phase plane for a generalized KdV equation, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed. It is shown for the first time that there coexist different types of singular waves corresponding to one heteroclinic orbit.     

On existence of singular solutions

In the paper sufficient conditions are given under which the differential equation y⁽ⁿ⁾=f(t,y,…,y⁽ⁿ⁻²⁾)g(y⁽ⁿ⁻¹⁾) has a singular solution y :[T,τ)→R, τ<∞ fulfilling

     

Mild solutions for a class of fractional SPDEs and their sample paths

We introduce a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset ${D\subset\mathbb{R}^{d}}We introduce a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset and driven by an infinite-dimensional fractional noise. We prove the existence of such a solution, establish its relation with the variational solution introduced by Nualart and Vuillermot (J Funct Anal 232:390–454, 2006) and the H?lder continuity of its sample paths when we consider it as an L ²(D)-valued stochastic process. When h is an affine function, we also prove uniqueness. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness. M. Sanz-Solé was supported by the grant MTM 2006-01351 from the Dirección General de Investigación, Ministerio de Educación y Ciencia, Spain.     

Fundamental solutions and eigenfunction expansions for Schrödinger operators I. Fundamental solutions

Sans résumé     

Fundamental solutions of some degenerate operators

In this paper we modify the usual series computation for a parametrix of an elliptic operator and find a method that is applicable to some simple degenerate operators. In fact, we are able to construct fundamental solutions for operators such as D_x + ixD_y and D_x² + x²D_y² + cD_y which explain the lack of local solvability in a natural way.     

Computing the singular behavior of solutions of the cauchy singular integral equations

Stenger's formula for numerical computation of principal value integrals isused to determine the singular behavior of solutions of homogeneous Cauchy singular integral equations near the end-points of the domain of integration.     

On relaxed control and singular solutions

A relation is shown to exist between relaxed problems and singular control. It is shown that a problem that does not possess an optimal solution, but satisfies some general assumptions, has a singular relaxed solution.     

Nonlinear wave equations and singular solutions

We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface, which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order. The method of Fuchsian reduction is employed.

     

Hyperbolic-like solutions for singular Hamiltonian systems

We study the existence of unbounded solutions of singular Hamiltonian systems: where is a potential with a singularity. For a class of singular potentials with a strong force , we show the existence of at least one hyperbolic-like solutions. More precisely, for given and , we find a solution q(t) of (*) satisfying Received October 1998