The first parabolic version of the Harnack inequality is due to Hadamard ([Ha]) and Pini ([Pi]). Their result is the following: Let $u$ be a non-negative solution of the heat equation in a cylindric domain $\Omega_T$. Let $(x_0,t_0) \in \Omega_T$ and assume that the cylinder $(x_0,t_0) + Q_{2\rho} \subset \Omega_T$ where $Q_\rho \equiv B_\rho \times (-\rho^2,0)$. Then there exists a constant $\gamma$, depending only upon $N$, such that $$u(x_0,t_0) \geq \gamma \sup_{B_\rho(x_0)} u(x, t_0 -\rho^2) $$ The proof is based on local representations by means of heat potentials. A breakthrough in the theory is due to Moser, who in his celebrated paper [MO1] proved that the Harnack estimates continues to hold for nonnegative weak solution of linear parabolic equations with $L^\infty$ coefficients. The result of Moser can be extended (see [AS] and [TR]) to nonnegative weak solutions of quasilinear parabolic equation. The proof of Moser's result is based on suitable integral estimates for powers and logarithm of the solution $u$; the general structure follows the same one Moser used in his earlier work on Harnack's inequality in the elliptic case ([MO]). It is worth saying that the most difficult part of Moser's proof is an adaptation to the parabolic case of the exponential decay of the distribution function of a function with bounded mean oscillation. Going from the elliptic to the parabolic situation the difficulty lies in the special role played by the time variable. Indeed Moser's proof is hard to follow and the need for a possible simplification was immediately felt. Moser himself published a new proof of Harnack inequality in 1971 ([MO2]) with the expressed purpose to avoid the use of his parabolic John - Nirenberg Lemma ([JN]); through estimates on the logarithm of the solution via a measure lemma based on a result of Bombieri ([BO], [BG]). Quite surprisingly Moser's method does not work in the case of the $p$-Laplacean and this is not simply a matter of technique. As the $p$-Laplacean is invariant by the scaling $x\rightarrow hx$ and $t \rightarrow h^p t$, one would guess that Harnack estimates would hold in the cylinder $[(x_0, t_0) + B_\rho \times (\rho^p,0)]$, but this is not true as one can easily check considering the explicit solution introduced by Barenblatt in [BA]. DiBenedetto overcame this difficulty realizing that a specific Harnack estimate holds with an intrinsic time scale exactly of the order $u(x_0,t_0)^{2-p}$. More precisely the following result is proved in [DB], and [DBK]: Let $u$ be a nonnegative weak solution of the $p$-Laplacean and let $p> \frac{2N}{N+1}$. Fix $(x_0,t_0) \in \Omega_T$ and assume that $u(x_0,t_0) >0$. There exists constants $\gamma>1$ and $C>0$, depending only upon $N$ and $p$, such that $$u(x_0,t_0) \leq \gamma \inf_{B_\rho(x_0)}u(\cdot, t_0 +\theta)$$ where $$\theta \equiv \frac {C\rho^p}{[u(x_0,t_0)]^{p-2}}$$ provided that the cylinder $(x_0,t_0) + B_{4\rho}\times (-4\theta,4\theta)$ is contained in $\Omega_T$. The proof given by DiBenedetto is based on explicit representation of solutions and cannot be extended to the general case. In this paper we filled this gap and we extended this result to a general class of degenerate parabolic equations. The proof is based on DeGiorgi techniques ([DG]) and on a measure-theoretical lemma ([DBGV]). This new approach is purely geometrical and does not rely on BMO properties or on Bombieri’s lemma. The paper originated several paper concerning a new approach to regularity for degenerate parabolic equations (see for instance [GSV]) and asymptotic behavior of the solutions (see for instance [RVV]). REFERENCES [AS ] D.G. Aronson - J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal. 25 (1967), pp. 81-123. [BA] G.I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mech. 16 (1952), pp. 67-78. [BO] E. Bombieri, Theory of minimal surfaces and a counterexample to the Bernstein conjecture in high dimension, Mimeographed Notes of Lectures held at Courant Institut, New York University, 1970. [BG] E. Bombieri - E. Giusti, Harnack’s inequality for elliptic differential equations on minimal surfaces, Inventiones Math. 15 (1972), pp. 24-46. [DG] E. DeGiorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat., Ser. 3, 3 (1957), pp. 25-43. [DB] E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Rat. Mech. Anal. 100 (1988), pp. 129-147. [DBGV] E. DiBenedetto; U. Gianazza,; V. Vespri,; Local clustering of the non-zero set of functions in $W^ {1,1}(E)$. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006), no. 3, 223--225. [DBK] E. DiBenedetto - Y.C. Kwong, Intrinsic Harnack estimates and extinction profile for certain singular parabolic equations, Transactions of the A.M.S. 330 (1992), pp. 783-811. [GSV] U. Gianazza, M. Surnachev,; V. Vespri,; On a new proof of Hölder continuity of solution of p-Laplace type parabolic equations. Adv, Calc. Var 3 (2010), 263-278. [HA] J. Hadamard, Extension a l'equation de la chaleur d'un theoreme de A. Harnack, Rend. Circ. Mat. di Palermo, Ser. 2(3), (1954), 337-346. [JN] F. John, L. Nirenberg; On functions of bounded mean oscillation, Comm. Pure Appl. Math 14 (1961) 415-426. [MO] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14, (1961), 577-591. [MO1] J. Moser, A Harnack Inequality for Parabolic Differential Equations, Comm. Pure Appl. Math. 17, (1964), 101-134. [MO2] J. Moser - On a Pointwise Estimate for Parabolic Differential Equations - Comm. Pure Appl. Math. 24, (1971), 727-740. [PI] B. Pini, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Mat. Univ. Padova, 23,(1954), 422-434. [RVV] F.  Ragnedda, Francesco, S. Vernier-Piro, V. Vespri; Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations. Math. Annalen 348 (2010) 779-795. [TR] N.S. Trudinger, Pointwise Estimates and Quasi-Linear Parabolic Equations, Comm. Pure Appl. Math. 21, (1968), 205-226.

Harnack Estimates for Quasi-Linear Degenerate Parabolic Differential Equations

GIANAZZA, UGO PIETRO;
2008-01-01

Abstract

The first parabolic version of the Harnack inequality is due to Hadamard ([Ha]) and Pini ([Pi]). Their result is the following: Let $u$ be a non-negative solution of the heat equation in a cylindric domain $\Omega_T$. Let $(x_0,t_0) \in \Omega_T$ and assume that the cylinder $(x_0,t_0) + Q_{2\rho} \subset \Omega_T$ where $Q_\rho \equiv B_\rho \times (-\rho^2,0)$. Then there exists a constant $\gamma$, depending only upon $N$, such that $$u(x_0,t_0) \geq \gamma \sup_{B_\rho(x_0)} u(x, t_0 -\rho^2) $$ The proof is based on local representations by means of heat potentials. A breakthrough in the theory is due to Moser, who in his celebrated paper [MO1] proved that the Harnack estimates continues to hold for nonnegative weak solution of linear parabolic equations with $L^\infty$ coefficients. The result of Moser can be extended (see [AS] and [TR]) to nonnegative weak solutions of quasilinear parabolic equation. The proof of Moser's result is based on suitable integral estimates for powers and logarithm of the solution $u$; the general structure follows the same one Moser used in his earlier work on Harnack's inequality in the elliptic case ([MO]). It is worth saying that the most difficult part of Moser's proof is an adaptation to the parabolic case of the exponential decay of the distribution function of a function with bounded mean oscillation. Going from the elliptic to the parabolic situation the difficulty lies in the special role played by the time variable. Indeed Moser's proof is hard to follow and the need for a possible simplification was immediately felt. Moser himself published a new proof of Harnack inequality in 1971 ([MO2]) with the expressed purpose to avoid the use of his parabolic John - Nirenberg Lemma ([JN]); through estimates on the logarithm of the solution via a measure lemma based on a result of Bombieri ([BO], [BG]). Quite surprisingly Moser's method does not work in the case of the $p$-Laplacean and this is not simply a matter of technique. As the $p$-Laplacean is invariant by the scaling $x\rightarrow hx$ and $t \rightarrow h^p t$, one would guess that Harnack estimates would hold in the cylinder $[(x_0, t_0) + B_\rho \times (\rho^p,0)]$, but this is not true as one can easily check considering the explicit solution introduced by Barenblatt in [BA]. DiBenedetto overcame this difficulty realizing that a specific Harnack estimate holds with an intrinsic time scale exactly of the order $u(x_0,t_0)^{2-p}$. More precisely the following result is proved in [DB], and [DBK]: Let $u$ be a nonnegative weak solution of the $p$-Laplacean and let $p> \frac{2N}{N+1}$. Fix $(x_0,t_0) \in \Omega_T$ and assume that $u(x_0,t_0) >0$. There exists constants $\gamma>1$ and $C>0$, depending only upon $N$ and $p$, such that $$u(x_0,t_0) \leq \gamma \inf_{B_\rho(x_0)}u(\cdot, t_0 +\theta)$$ where $$\theta \equiv \frac {C\rho^p}{[u(x_0,t_0)]^{p-2}}$$ provided that the cylinder $(x_0,t_0) + B_{4\rho}\times (-4\theta,4\theta)$ is contained in $\Omega_T$. The proof given by DiBenedetto is based on explicit representation of solutions and cannot be extended to the general case. In this paper we filled this gap and we extended this result to a general class of degenerate parabolic equations. The proof is based on DeGiorgi techniques ([DG]) and on a measure-theoretical lemma ([DBGV]). This new approach is purely geometrical and does not rely on BMO properties or on Bombieri’s lemma. The paper originated several paper concerning a new approach to regularity for degenerate parabolic equations (see for instance [GSV]) and asymptotic behavior of the solutions (see for instance [RVV]). REFERENCES [AS ] D.G. Aronson - J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal. 25 (1967), pp. 81-123. [BA] G.I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mech. 16 (1952), pp. 67-78. [BO] E. Bombieri, Theory of minimal surfaces and a counterexample to the Bernstein conjecture in high dimension, Mimeographed Notes of Lectures held at Courant Institut, New York University, 1970. [BG] E. Bombieri - E. Giusti, Harnack’s inequality for elliptic differential equations on minimal surfaces, Inventiones Math. 15 (1972), pp. 24-46. [DG] E. DeGiorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat., Ser. 3, 3 (1957), pp. 25-43. [DB] E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Rat. Mech. Anal. 100 (1988), pp. 129-147. [DBGV] E. DiBenedetto; U. Gianazza,; V. Vespri,; Local clustering of the non-zero set of functions in $W^ {1,1}(E)$. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006), no. 3, 223--225. [DBK] E. DiBenedetto - Y.C. Kwong, Intrinsic Harnack estimates and extinction profile for certain singular parabolic equations, Transactions of the A.M.S. 330 (1992), pp. 783-811. [GSV] U. Gianazza, M. Surnachev,; V. Vespri,; On a new proof of Hölder continuity of solution of p-Laplace type parabolic equations. Adv, Calc. Var 3 (2010), 263-278. [HA] J. Hadamard, Extension a l'equation de la chaleur d'un theoreme de A. Harnack, Rend. Circ. Mat. di Palermo, Ser. 2(3), (1954), 337-346. [JN] F. John, L. Nirenberg; On functions of bounded mean oscillation, Comm. Pure Appl. Math 14 (1961) 415-426. [MO] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14, (1961), 577-591. [MO1] J. Moser, A Harnack Inequality for Parabolic Differential Equations, Comm. Pure Appl. Math. 17, (1964), 101-134. [MO2] J. Moser - On a Pointwise Estimate for Parabolic Differential Equations - Comm. Pure Appl. Math. 24, (1971), 727-740. [PI] B. Pini, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Mat. Univ. Padova, 23,(1954), 422-434. [RVV] F.  Ragnedda, Francesco, S. Vernier-Piro, V. Vespri; Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations. Math. Annalen 348 (2010) 779-795. [TR] N.S. Trudinger, Pointwise Estimates and Quasi-Linear Parabolic Equations, Comm. Pure Appl. Math. 21, (1968), 205-226.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/109520
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