Parametrix method for a parabolic equation on a Riemannian manifold

The First Boundary Value Problem for a Parabolic Equation on a Manifold

“…It was also proved in [9] that φ(x, y) is a bounded function, which implies the inequality m 0 (t, x, y) ≤ cq(t, x, y), and the estimate l 0 (t, x, y) ≤ c 0 te c0t q(t, x, y) was obtained.…”

“…(1), dS y is the volume element of the submanifold S , and ν y is the unit outward normal of S in M at y. The conditions imposed on the manifold M are stated via the curvature tensor R(x) [9,10]:…”

“…A procedure, similar to the parametrix method, for constructing the fundamental solution of the Laplace-Beltrami equation on such a manifold ∂u ∂t = 1 2 ∆u was suggested in [9]; namely, the solution was represented by the sum p 0 (t, x, y) = m 0 (t, x, y) + l 0 (t, x, y) = m 0 (t, x, y) + where M (t, x, y) is the discrepancy in the Laplace-Beltrami equation for the initial approximation m(t, x, y). The factor exp{−φ(x, y)/2} in m(t, x, y) removes the singularity with respect to t in the discrepancy.…”

On the behavior of a simple-layer potential for a parabolic equation on a Riemannian manifold

“…(1), μ( , ) t x is the density of the potential, and dS y is the volume element of the submanifold S. The conditions imposed on the manifold M are formulated in terms of the curvature tensor R x ( ) [6] as follows:…”

“…Unfortunately, most works only give estimates for a solution but do not propose a scheme for its construction. A scheme for the construction of a fundamental solution of a parabolic equation on Cartan -Hadamard-type manifolds by the parametrix method was developed by Yosida [5] and Bondarenko [6]. For the convergence of an iteration procedure, several additional conditions must be imposed on the sectional curvature.…”