We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conser...We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conservation equations for (probability) density functions give rise to velocity-field divergence equations the solutions of which generate—by way of superposition—the totality of solutions of various well-known classical and quantum-mechanical wave equations.展开更多
The problem of finding a L^∞-bounded two-dimensional vector field whose divergence is given in L2 is discussed from the numerical viewpoint. A systematic way to find such a vector field is to introduce a non-smooth v...The problem of finding a L^∞-bounded two-dimensional vector field whose divergence is given in L2 is discussed from the numerical viewpoint. A systematic way to find such a vector field is to introduce a non-smooth variational problem involving a L^∞-norm. To solve this problem from calculus of variations, we use a method relying on a well- chosen augmented Lagrangian functional and on a mixed finite element approximation. An Uzawa algorithm allows to decouple the differential operators from the nonlinearities introduced by the L^∞-norm, and leads to the solution of a sequence of Stokes-like systems and of an infinite family of local nonlinear problems. A simpler method, based on a L2- regularization is also considered. Numerical experiments are performed, making use of appropriate numerical integration techniques when non-smooth data are considered; they allow to compare the merits of the two approaches discussed in this article and to show the ability of the related methods at capturing L^∞bounded solutions.展开更多
In this paper,we consider the divergence degenerate elliptic equation with bounded coefficients constructed by Hörmander's vector fields.We prove a De Giorgi type result,i.e,the local Holder continuity for th...In this paper,we consider the divergence degenerate elliptic equation with bounded coefficients constructed by Hörmander's vector fields.We prove a De Giorgi type result,i.e,the local Holder continuity for the weak solutions to the equation by providing a De Giorgi type lemma and extending the Moser iteration to the setting here.As a consequence,the Harnack inequality of weak solutions is also given.展开更多
In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity...In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.展开更多
文摘We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conservation equations for (probability) density functions give rise to velocity-field divergence equations the solutions of which generate—by way of superposition—the totality of solutions of various well-known classical and quantum-mechanical wave equations.
文摘The problem of finding a L^∞-bounded two-dimensional vector field whose divergence is given in L2 is discussed from the numerical viewpoint. A systematic way to find such a vector field is to introduce a non-smooth variational problem involving a L^∞-norm. To solve this problem from calculus of variations, we use a method relying on a well- chosen augmented Lagrangian functional and on a mixed finite element approximation. An Uzawa algorithm allows to decouple the differential operators from the nonlinearities introduced by the L^∞-norm, and leads to the solution of a sequence of Stokes-like systems and of an infinite family of local nonlinear problems. A simpler method, based on a L2- regularization is also considered. Numerical experiments are performed, making use of appropriate numerical integration techniques when non-smooth data are considered; they allow to compare the merits of the two approaches discussed in this article and to show the ability of the related methods at capturing L^∞bounded solutions.
基金This work is sponsored by the China Scholarship Council with Grant Number 20200636-0116.
文摘In this paper,we consider the divergence degenerate elliptic equation with bounded coefficients constructed by Hörmander's vector fields.We prove a De Giorgi type result,i.e,the local Holder continuity for the weak solutions to the equation by providing a De Giorgi type lemma and extending the Moser iteration to the setting here.As a consequence,the Harnack inequality of weak solutions is also given.
基金This work is supported by the National Natural Science Foundation of China(No.1177-1354).
文摘In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.
