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 quasi-linear 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 in-trinsic 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 [DBGV] this gap was fullifed in the case of degenerate equations The proof is based on DeGiorgi techniques ([DG]) and on a theoretical measure lemma([DBGV1]). With this new approach we were able to give also a new proof by Moser only based on geometrical approaches and not on BMO properties or on Bombieri’s lemma. The gap concerning the singular case was fulfilled in this paper. The proof is different in some points, it requires a new lemma of expansion of positivity ([DBGV2]) and implies that a new form of Harnack holds. More precisely we prove that Let $u$ be a nonnegative weak solution of the $p$-Laplacean and let $2>p> \frac{2N}{N+1}$. 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 $$\gamma \sup_{B_\rho(x_0)\times[-\theta, + \theta]}u\leq u(x_0,t_0) \leq \gamma^{-1} \inf_{B_\rho(x_0)\times[-\theta, + \theta]}u$$ 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 $\Omega_T$. Therefore in the singular case, the Harnack inequality becomes more elliptic than parabolic. This fact confirms a previous result for the prototype equation ([DBKV]). This result was applied to investigate the case when $p\leq \frac{2N}{N+1}$ ([DBGV3]) and to study the asymptotic limit for a general class of singular parabolic equations in unbounded domains ([RVV]) From Google Scholar database it results that this paper was quoted 18 times, for ISI database 4 times, for Scopus database 3 times, for MathSciNet database 1. 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 differenziabilita' e l’analiticita' 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;Harnack estimates for quasi linear degenerate parabolic differential equations Acta Math 200 (2008), no. 2, 181-209. [DBGV1] E. DiBenedetto; U. Gianazza; V. Vespri; Local clustering of the non-zero set of func-tions 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. [DBGV2] E. DiBenedetto; U. Gianazza,; V. Vespri A new approach to the expansion of positivity set of non-negative solutions to certain singular parabolic equations. Proc. Amer. Math. Soc ., 138 (2010), pp 3521--3529. [DBGV3] E. DiBenedetto; U. Gianazza,; V. Vespri Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations. Manuscripta Math., 131 (2010), 231-245. [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. [DBK] E. DiBenedetto - Y.C. Kwong , V. Vespri Local space-analyticity of solutions of certain singular parabolic equations. Indiana Univ. Math. J. 40 (1991), no. 2, 741--765. [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, S. Vernier-Piro, V. Vespri; Pointwise estimates for solutions of singular parabolic problems in $R^N \times[0;+\infty)$ , submitted [TR] N.S. Trudinger, Pointwise Estimates and Quasi-Linear Parabolic Equations, Comm. Pure Appl. Math. 21, (1968), 205-226.

Titolo: | Forward, Backward and Elliptic Harnack Inequalities for Non-Negative Solutions to Certain Singular Parabolic Partial Differential Equations |

Autori di Ateneo: | |

Autori: | E. DiBenedetto; U. Gianazza; VESPRI, VINCENZO |

Data di pubblicazione: | 2010 |

Rivista: | |

Volume: | 9 |

Pagina iniziale: | 385 |

Pagina finale: | 422 |

Abstract: | 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 quasi-linear 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 in-trinsic 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 [DBGV] this gap was fullifed in the case of degenerate equations The proof is based on DeGiorgi techniques ([DG]) and on a theoretical measure lemma([DBGV1]). With this new approach we were able to give also a new proof by Moser only based on geometrical approaches and not on BMO properties or on Bombieri’s lemma. The gap concerning the singular case was fulfilled in this paper. The proof is different in some points, it requires a new lemma of expansion of positivity ([DBGV2]) and implies that a new form of Harnack holds. More precisely we prove that Let $u$ be a nonnegative weak solution of the $p$-Laplacean and let $2>p> \frac{2N}{N+1}$. 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 $$\gamma \sup_{B_\rho(x_0)\times[-\theta, + \theta]}u\leq u(x_0,t_0) \leq \gamma^{-1} \inf_{B_\rho(x_0)\times[-\theta, + \theta]}u$$ 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 $\Omega_T$. Therefore in the singular case, the Harnack inequality becomes more elliptic than parabolic. This fact confirms a previous result for the prototype equation ([DBKV]). This result was applied to investigate the case when $p\leq \frac{2N}{N+1}$ ([DBGV3]) and to study the asymptotic limit for a general class of singular parabolic equations in unbounded domains ([RVV]) From Google Scholar database it results that this paper was quoted 18 times, for ISI database 4 times, for Scopus database 3 times, for MathSciNet database 1. 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 differenziabilita' e l’analiticita' 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;Harnack estimates for quasi linear degenerate parabolic differential equations Acta Math 200 (2008), no. 2, 181-209. [DBGV1] E. DiBenedetto; U. Gianazza; V. Vespri; Local clustering of the non-zero set of func-tions 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. [DBGV2] E. DiBenedetto; U. Gianazza,; V. Vespri A new approach to the expansion of positivity set of non-negative solutions to certain singular parabolic equations. Proc. Amer. Math. Soc ., 138 (2010), pp 3521--3529. [DBGV3] E. DiBenedetto; U. Gianazza,; V. Vespri Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations. Manuscripta Math., 131 (2010), 231-245. [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. [DBK] E. DiBenedetto - Y.C. Kwong , V. Vespri Local space-analyticity of solutions of certain singular parabolic equations. Indiana Univ. Math. J. 40 (1991), no. 2, 741--765. [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, S. Vernier-Piro, V. Vespri; Pointwise estimates for solutions of singular parabolic problems in $R^N \times[0;+\infty)$ , submitted [TR] N.S. Trudinger, Pointwise Estimates and Quasi-Linear Parabolic Equations, Comm. Pure Appl. Math. 21, (1968), 205-226. |

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