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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

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
Full Text: DOI EuDML
[1] [Ba] Bauer H.: Harmonische R?ume und ihre Potentialtheorie (Lecture Notes in Mathematics, Vol. 22). Berlin Heidelberg New York: Springer 1966
[2] [BGM] Berger, M., Gauduchon, P. Mazet, E.: Le Spectre d’une Vari?t? Riemanniene (Lecture Notes in Mathematics, Vol. 194). Berlin Heidelberg New York: Springer 1971
[3] [EG] Evans, L.C., Gariepy, R.F.: Wiener’s criterion for the heat equation. Arch. Rat. Mech. Anal.78, 293-314 (1982) · Zbl 0508.35038 · doi:10.1007/BF00249583
[4] [FG] Fabes, E.B., Garofalo, N.: Mean value properties of solutions to parabolic equations with variable coefficients. J. Math. Anal. Appl.121, 305-316 (1987) · Zbl 0627.35041 · doi:10.1016/0022-247X(87)90249-6
[5] [Fe] Federer, H.: Geometric measure theory (Die Grundlehren der, mathematischen Wissenschaften, Vol. 153). Berlin Heidelberg New York: Springer 1969
[6] [F] Folland, G.B: Introduction to partial differential equations. Princeton Univ. Press 1976 · Zbl 0325.35001
[7] [Fr] Friedman, A.: Partial differential equations of parabolic type. New York. Prentice-Hall 1964 · Zbl 0144.34903
[8] [Fu] Fulks, W.: A mean value theorem for the heat equation. Proc. Am. Math. Soc.17, 6-11 (1966) · Zbl 0152.10503 · doi:10.1090/S0002-9939-1966-0192200-3
[9] [GL] Garofalo, N., Lanconelli, E.: Wiener’s criterion for parabolic equations with variable coefficients and its consequences, Trans. Am. Math. Soc.307 (to appear) (1988) · Zbl 0698.31007
[10] [H] Helms, L.L.: Introduction to potential theory. New York: Wiley-Interscience 1969 · Zbl 0188.17203
[11] [K] Kannai, Y.: Off diagonal short time asymptotics for fundamental solutions of diffusion equations. Commun. Partial Differ. Equations2, 781-830 (1977) · Zbl 0381.35039 · doi:10.1080/03605307708820048
[12] [Ku] Kupcov, L.P.: The mean property and the maximum principle for the parabolic equations of second order. Dokl. Akad. Nauk SSSR242, no. 3 (1978); English transl.: Soviet Math. Dokl.19, 1140-1144 (1978)
[13] [L1] Littman, W.: A strong maximum principle for weaklyL-subharmonic functions. J. Math. Mech.8, 761-770 (1959) · Zbl 0090.08201
[14] [L2] Littman, W.: Generalized subharmonic functions: Monotonic approximations and an improved maximum principle. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.17, 207-222 (1963) · Zbl 0123.29104
[15] [M] Moser, J.: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math.17, 101-134 (1964) · Zbl 0149.06902 · doi:10.1002/cpa.3160170106
[16] [P1] Pini, B.: Sulle equazioni a derivate parziali lineari del secondo ordine in due variabili di tipo parabolico. Ann. Mat. Pura e Appl.32, 179-204 (1951) · Zbl 0044.31602 · doi:10.1007/BF02417958
[17] [P2] Pini, B.: Magioranti e minoranti delle soluzioni delle equazioni paraboliche, Ann. Mat. Pura Appl., IV. Ser.37, 249-264 (1954) · Zbl 0057.32703 · doi:10.1007/BF02415101
[18] [P3] Pini, B.: Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico. Rend. Semin. Mat. Univ. Padova23, 422-434 (1954) · Zbl 0057.32801
[19] [W] Watson, N.A.: A theory of temperatures in several variables. Proc. Lond. Math. Soc.26, (3) 385-417 (1973) · Zbl 0253.35045 · doi:10.1112/plms/s3-26.3.385
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