The Harnack inequality for a class of nonlocal parabolic equations (original) (raw)
Related papers
Weak Harnack inequality for a mixed local and nonlocal parabolic equation
2021
Abstract. This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.
SIAM Journal on Mathematical Analysis, 2017
We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator (∂t − ∆) s u(t, x) = f (t, x), for 0 < s < 1. This nonlocal equation of order s in time and 2s in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for spacetime nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for (∂t−∆) s u(t, x) and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. Hölder and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic Hölder spaces. Our methods involve the parabolic language of semigroups and the Cauchy Integral Theorem, which are original to define the fractional powers of ∂t − ∆. Though we mainly focus in the equation (∂t − ∆) s u = f , applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.
Journal of Functional Analysis, 2014
We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.
A Harnack inequality for a degenerate parabolic equation
Journal of Evolution Equations, 2006
We prove a Harnack inequality for a degenerate parabolic equation using proper estimates based on a suitable version of the Rayleigh quotient. Mathematics Subject Classification (2000): 35B05, 35K55, 35K65. Key words and phrases: Harnack Inequality, Degenerate Parabolic Equation, Rayleigh Quotient, p-Laplacian. This work has been partially supported by M.I.U.R. through F.A.R. funds and by I.M.A.T.I. -C.N.R. U. Gianazza and V. Vespri J.evol.equ.
Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations
2006
We establish the intrinsic Harnack inequality for nonnegative solutions of the parabolic p-Laplacian equation by a proof that uses neither the comparison principle nor explicit self-similar solutions. The significance is that the proof applies to quasilinear p-Laplacian-type equations, thereby solving a long-standing problem in the theory of degenerate parabolic equations.
Degenerate parabolic equations and Harnack inequality
Annali di Matematica Pura ed Applicata, 1984
(*). FILIP~o CmA~ENZ~ (Catania) (**)-I~AV~ SE~A~0~ (Trento) (**) Sunto.-Viene risolto il problema di Cauchy DirichIet relativo all'operatore parabolico dege~ere ~u/ St-~/ ~xi(aij(~, t) ~u/ ~xs), in opportune ipotesi di integrabilith per gti autovalori di ai~(x, t). Vengono inottre ]or~iti controese~npi circa l'impossibilit~ di risultati di regolarith per le soluzioni deboli q~ostrando i@ tal modo che operatori parabolici degeneri hanno ~n vomportamento radicalmente di//erentc c~a qucllo dei corrisponde~#i operatori cllittici degencri. Introduction. Degenerate elliptic and parabolic partial differential equations have been extensively studied in the last 10-15 years. In particular, for elliptic operators of the form: (0.1) ~/~aij(~) ~ fl , ~-1(2)(~)1~[2 < aij(~)~i~<= = ~(A}(X)I~[2 ~ it was clear, since I0 years ago, that some local assumptions on (~(x) (as the ones given in [T~] or [T2] and more or less implicitely assumed in [M-S] (see also [M-SIbyl)) were needed in order to get local ItSlder continuity of the solutions. More precisely these authors assume that:
Rendiconti Lincei - Matematica e Applicazioni, 2009
Non-negative solutions to quasi-linear, degenerate or singular parabolic partial differential equations, of p-Laplacian type for p > 2N Nþ1 , satisfy Harnack-type estimates in some intrinsic geometry ([2, 3]). Some equivalent alternative forms of these Harnack estimates are established, where the supremum and the infimum of the solutions play symmetric roles, within a properly redefined intrinsic geometry. Such equivalent forms hold for the non-degenerate case p ¼ 2 following the classical work of Moser ([5, 6]), and are shown to hold in the intrinsic geometry of these degenerate and/or parabolic p.d.e.'s. Some new forms of such an estimate are also established for 1 < p < 2.
A remark on a Harnack inequality for degenerate parabolic equations
1985
L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.