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Asymptotic behaviour of fundamental solutions and potential theory of parabolic operators with variable coefficients. (English) Zbl 0638.35003
Let $$L=div(A(x,t)\nabla _ x)-D_ t$$ be a parabolic operator in $${\mathbb{R}}^{n+1}$$ with $$C^{\infty}$$ coefficient matrix $$A(x,t)=(a_{ij}(x,t))$$. In this paper we prove the following asymptotic expansion for the fundamental solution $$\Gamma$$ of L: As $$t\to 0$$ $$+$$ and for x close to y $(1)\quad \Gamma (x,t;y,0)\sim (4\pi t)^{-n/2} \exp (-d^ 2(x,y,t)/4t)\sum ^{\infty}_{j=0}t\quad ju\quad _ j(x,y,t).$ In (1) d(x,y,t) denotes the Riemannian distance between x and y in the metric generated on $${\mathbb{R}}^ n$$at time t by $$A^{-1}(\cdot,t)$$. Expansions similar to (1) hold for the derivatives of $$\Gamma$$. The above result extends to time-dependent operators a classical result of Minashisundaram and Pleijel.
Using (1) we address several questions in parabolic potential theory. In previous work of Fabes and Garofalo and of Garofalo and Lanconelli it was shown that solutions of $$Lu=0$$ in $${\mathbb{R}}^{n+1}$$ and, more in general, any $$C^{\infty}$$ function on $${\mathbb{R}}^{n+1}$$ can be suitably represented as weighted averages on the level sets of the fundamental solution $$\Gamma$$ of L. In this paper, we prove some new mean value formulas which extend the previous cited results. The advantage of these new formulas consists in the fact that not only the kernels appearing in them are bounded, but they also possess an arbitrarily high degree of regularity. The proof of these results crucially relies on (1). Using these new formulas we are able to give an elementary proof of Harnack inequality for nonnegative solutions of $$Lu=0$$ which is modelled on the classical proof for harmonic functions, and therefore does not use the parabolic BMO machinery of Moser.
The last two sections of the paper are devoted to study the connection between the averaging operators introduced and L-superparabolic functions on $${\mathbb{R}}^{n+1}$$. The main result states that every L-superparabolic function can be monotonically approximated by a sequence of L- superparabolic functions with an arbitrarily high degree of smoothness. This result plays a crucial role in our previous work on Wiener’s criterion for the regularity of boundary points in Dirichlet problem.
Reviewer: N.Garofalo

##### MSC:
 35A08 Fundamental solutions to PDEs 35K10 Second-order parabolic equations 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 35C20 Asymptotic expansions of solutions to PDEs
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