Search Results

My stackexchange post was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes! Let $M$ be a compact Riemannian manifold and $\Delta$ be the …

I am looking for existence results on inhomogeneous linear heat equations. Concretely, I have the equation $$ \frac{\partial}{\partial t} u(t, x) = \Delta_t u(t, x), ~~~~~u(0, x) = u_0(x)$$ where $\De …

Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that $$L(H_k * F) = (LH_k)*F - F,$$ so if you define $F = \sum_{l=1}^\infty (LH_k)^{*l}$ and $p= H_k + H_k * F$, then $$ L p = LH_ …

Let $M$ be a compact Riemannian manifold and let $V \in C^\infty(M)$. Consider the operator $\Delta + V$ and let $p_t(x, y)$ be the corresponding heat kernel. If $\Delta + V$ is a positive operator su …

Is there an explicit formula in the literature for the heat kernel of the Hodge Laplacian on differential forms? I found some on functions, but not on forms of higher degree. What at least about 1- …

I am looking for references (with proof) for the following statement: Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be compac …

As far as I know, the term "Mehler Kernel" is used for the integral kernel of the heat equation corresponding the the harmonic oscillator, $$ \partial_tu + \Delta u + x^2 u = 0.$$ The equation you are …

Consider the heat equation $$ (\partial_t + \Delta + V)u = 0$$ on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential. Consider the semigroup gene …

This is crosspost from math.stackexchange https://math.stackexchange.com/questions/311213/heat-kernel-asymptotics-on-manifold-with-boundary where the question did not yield any answer On a closed Rie …

Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous. Question: Is there still an asymptotic expansion of the heat kernel of the form $$ p_t(x, y) \sim (4 …

I believe the answer to my question is well-known, but as I do not know too much about harmonic map flows, here it goes: Let $M$ be a complete Riemannian manifold. For a curves $c: [0,1] \longrightar …