The Experts below are selected from a list of 213 Experts worldwide ranked by ideXlab platform

Durvudkhan Suragan - One of the best experts on this subject based on the ideXlab platform.

  • Layer Potentials kac s problem and refined hardy inequality on homogeneous carnot groups
    Advances in Mathematics, 2017
    Co-Authors: Michael Ruzhansky, Durvudkhan Suragan
    Abstract:

    Abstract We propose the analogues of boundary Layer Potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the Single Layer Potential and establish the Plemelj type jump relations for the double Layer Potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton Potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.

  • Layer Potentials green s formulae kac s problem and refined hardy inequality on homogeneous carnot groups
    arXiv: Analysis of PDEs, 2015
    Co-Authors: Michael Ruzhansky, Durvudkhan Suragan
    Abstract:

    We propose the analogues of boundary Layer Potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the Single Layer Potential and establish the Plemelj type jump relations for the double Layer Potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton Potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's "principle of not feeling the boundary". We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.

Michael Ruzhansky - One of the best experts on this subject based on the ideXlab platform.

  • Layer Potentials kac s problem and refined hardy inequality on homogeneous carnot groups
    Advances in Mathematics, 2017
    Co-Authors: Michael Ruzhansky, Durvudkhan Suragan
    Abstract:

    Abstract We propose the analogues of boundary Layer Potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the Single Layer Potential and establish the Plemelj type jump relations for the double Layer Potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton Potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.

  • Layer Potentials green s formulae kac s problem and refined hardy inequality on homogeneous carnot groups
    arXiv: Analysis of PDEs, 2015
    Co-Authors: Michael Ruzhansky, Durvudkhan Suragan
    Abstract:

    We propose the analogues of boundary Layer Potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the Single Layer Potential and establish the Plemelj type jump relations for the double Layer Potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton Potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's "principle of not feeling the boundary". We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.

Jukka Kemppainen - One of the best experts on this subject based on the ideXlab platform.

  • existence and uniqueness of the solution for a time fractional diffusion equation with robin boundary condition
    Abstract and Applied Analysis, 2011
    Co-Authors: Jukka Kemppainen
    Abstract:

    Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the Single-Layer Potential, the volume Potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data is continuous, the initial data is continuously differentiable, and the source term is Holder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.

Deyue Zhang - One of the best experts on this subject based on the ideXlab platform.

  • an integral equations method combined minimum norm solution for 3d elastostatics cauchy problem
    Computer Methods in Applied Mechanics and Engineering, 2014
    Co-Authors: Deyue Zhang
    Abstract:

    Abstract In this paper, we establish new density results for the equilibrium equations. Based on the denseness result of the elastic Potential functions, the Cauchy problem for the equilibrium equations is investigated. For this ill-posed problem, we construct a regularizing solution using the Single-Layer Potential function. The well-posedness of the regularizing solution as well as the convergence property is rigorously analyzed. The advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented. The method combines minimum norm solution with Morozov discrepancy principle to solve an inverse problem. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. The numerical convergence, accuracy, and stability of the method with respect to the discretisation about the integral equations on pseudo-boundary and the distance between the pseudo-boundary and the real boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are also analysed with some examples.

Laurent Kayser - One of the best experts on this subject based on the ideXlab platform.

  • gaussian lower bound for the neumann green function of a general parabolic operator
    Positivity, 2015
    Co-Authors: Mourad Choulli, Laurent Kayser
    Abstract:

    Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a Single-Layer Potential, we establish a Gaussian lower bound for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for Gaussian two-sided bounds for the fundamental solution.

  • gaussian lower bound for the neumann green function of ageneral parabolic operator
    arXiv: Analysis of PDEs, 2013
    Co-Authors: Mourad Choulli, Laurent Kayser
    Abstract:

    Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a Single-Layer Potential, we establish gaussian two-sided bounds for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for the gaussian two-sided bounds for the fundamental solution. We also indicate how our method can be adapted to get a gaussian lower bound for the Neumann heat kernel of a compact Riemannian manifold with boundary having non negative Ricci curvature.