Continuity of Solutions of Parabolic and Elliptic Equations

@article{Nash1958ContinuityOS,
  title={Continuity of Solutions of Parabolic and Elliptic Equations},
  author={John F. Nash},
  journal={American Journal of Mathematics},
  year={1958},
  volume={80},
  pages={931}
}
  • J. Nash
  • Published 1 October 1958
  • Mathematics
  • American Journal of Mathematics
View via Publisher
karlin.mff.cuni.cz
1,088 Citations
Highly Influential Citations
132
Background Citations
288
Methods Citations
182
Results Citations
34

1,088 Citations

Citation Type

Citation Type

Nash Estimates and Upper Bounds for Non-homogeneous Kolmogorov Equations
We prove a Gaussian upper bound for the fundamental solutions of a class of ultra-parabolic equations in divergence form. The bound is independent on the smoothness of the coefficients and…
On-Diagonal Estimates on Schrödinger Semigroup Kernels and Reduced Heat Kernels
Abstract:We prove various estimates for the kernels of semigroups generated by Schrödinger operators with magnetic field and potential of polynomial growth. We also investigate the reduced heat…
Heat kernel of fractional Laplacian with Hardy drift via desingularizing weights
We establish sharp two‐sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical‐order singularity, by transferring it to appropriate weighted space with…
J an 2 01 3 Green ’ s function for second order parabolic systems with Neumann boundary condition ̊
We study the Neumann Green’s function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain Q “ Ω ˆ p ́8,8q, where Ω Ă Rn is an…
Two problems in Partial Differential Equations (in Portuguese)
In this work, we examine two important problems in the theory of nonlinear PDEs. In Part I, we propose and solve a more general and complete version of the celebrated Leray's problem for the…
Improved energy methods for nonlocal diffusion problems
We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations: Lu (x) := \int_{\mathbb{R}^N} K(x,y) (u(y) - u(x)) dy, where $L$ acts on a real…
Contractivity properties of a quantum diffusion semigroup
We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality.…
Boundary behavior of Green function for a parabolic equation in bounded domain
Abstract The purpose of the paper is to describe the boundary behavior of the Green function of the parabolic equation Δu + Vu + 2((∇ϕ)/ϕ)·∇u - ut = 0 in D×ℝ+, where D is a bounded C1,1 domain in ℝn…
Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations
Motivated by applications to vortex rings, we study the Cauchy problem for the three-dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the…
Variations on uncertainty principles for integral operators
The aim of this paper is to prove new uncertainty principles for an integral operator with a bounded kernel. To do so we prove a Nash-type inequality and a Carlson-type inequality for this…
...
...

References

SHOWING 1-6 OF 6 REFERENCES
Über die Wärmeleitungsgleichung mit nichtkonstanten Koeffizienten im räumlichen Falle
The fundamental solution of the parabolic equation, II
The imbedding problem for Riemannian manifolds
Sur le mouvement d'un liquide visqueux emplissant l'espace
Estimates and uniqueness of solutions of elliptic equations,
  • Commultications on Pure and Applied Mathematics, vol
  • 1956
On quasi-conformal mapping,
  • Journal d'analyse Mathgmatique, Jerusalem, vol
  • 1954