A Harnack's inequality for mixed type evolution equations

Paronetto, Fabio

doi:10.1016/j.jde.2015.12.003

Cited by 14 publications

(19 citation statements)

References 27 publications

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“…while μ can change sign and also be zero. Denote by |μ| a suitable function (see [11] or [12] for this detail) such that |μ| > 0 a.e. (we choose |μ| = λ where μ ≡ 0) and…”

Section: If We Considermentioning

confidence: 99%

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Two generalized Tricomi equations

Paronetto

2021

Calc. Var.

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In this note we give existence results for the generalized Tricomi equations $${\mathcal {R}}u'' + {\mathcal {B}}u = f$$ R u ′ ′ + B u = f and $$({\mathcal {R}}u')' + {\mathcal {B}}u = f$$ ( R u ′ ) ′ + B u = f with suitable boundary data where $${\mathcal {R}}$$ R may be an operator (or a function) depending also on time assuming positive, null and negative sign, while $${\mathcal {B}}$$ B is an elliptic operator. To do that we also extend a result for equations like $$({\mathcal {R}}u')' + {\mathcal {A}}u' + {\mathcal {B}}u = f$$ ( R u ′ ) ′ + A u ′ + B u = f to equations like $${\mathcal {R}}u'' + {\mathcal {A}}u' + {\mathcal {B}}u = f$$ R u ′ ′ + A u ′ + B u = f and use these to derive the existence for the generalised Tricomi type equations mentioned above.

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“…while μ can change sign and also be zero. Denote by |μ| a suitable function (see [11] or [12] for this detail) such that |μ| > 0 a.e. (we choose |μ| = λ where μ ≡ 0) and…”

Section: If We Considermentioning

confidence: 99%

“…and the weighted Sobolev spaces (also for these details about this spaces we refer to [11] or to [12])…”

Section: If We Considermentioning

confidence: 99%

Two generalized Tricomi equations

Paronetto

2021

Calc. Var.

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Section: Introductionmentioning

confidence: 99%

“…The purpose of this paper is to announce and present in a very short version a result which is to appear (see [8]). Moreover the result we present here regards a simplified situation with respect to that considered in [8]. In [8] we consider equations like (2) satisfying ðAðx; t; u; DuÞ; DuÞ b clðxÞjDuj 2 ; jAðx; t; u; DuÞj a ClðxÞjDuj; jBðx; t; u; DuÞj a MlðxÞjDuj ð1Þ with m; l a L 1 loc ðWÞ, l > 0 almost everywhere, while sgnðmÞ can change and be zero also in some set of positive measure.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover the result we present here regards a simplified situation with respect to that considered in [8]. In [8] we consider equations like (2) satisfying ðAðx; t; u; DuÞ; DuÞ b clðxÞjDuj 2 ; jAðx; t; u; DuÞj a ClðxÞjDuj; jBðx; t; u; DuÞj a MlðxÞjDuj ð1Þ with m; l a L 1 loc ðWÞ, l > 0 almost everywhere, while sgnðmÞ can change and be zero also in some set of positive measure. One simple example (to stress that we are interested in the changing type equations) is mðxÞ qu qt À divðlðxÞDuÞ ¼ 0:…”

Section: Introductionmentioning

confidence: 99%

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A Harnack’s inequality and Hölder continuity for solutions of mixed type evolution equations

Paronetto¹

2015

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is mðxÞ qu qt À Du ¼ 0 where m can be positive, null and negative, so that elliptic-parabolic and forward-backward parabolic equations are included. For functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives Hö lder-continuity, in particular in the interface I where m change sign, and a maximum principle.

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Further existence results for elliptic–parabolic and forward–backward parabolic equations

Paronetto

2020

Calc. Var.

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We give an existence result for first order evolution equation of the type $$\mathcal {R}u' + \mathcal {A}u = f$$ R u ′ + A u = f where $$\mathcal {R}$$ R may be a function depending also on time assuming positive, null and negative sign, then the equation may be elliptic–parabolic, both forward and backward. The result is given in an abstract setting with Banach spaces depending on time (the functions u are defined in an interval [0, T] and $$u(t) \in X(t)$$ u ( t ) ∈ X ( t ) for a.e. t) and $$\mathcal {R}$$ R which is in fact a linear operator. We also extend a previous existence result for the equation $$(\mathcal {R}u)' + \mathcal {A}u = f$$ ( R u ) ′ + A u = f to the setting of moving Banach spaces. We also give a time regularity result in a particular case and give many examples of different possible choices of $$\mathcal {R}$$ R .

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A Harnack's inequality for mixed type evolution equations

Cited by 14 publications

References 27 publications

Two generalized Tricomi equations

Two generalized Tricomi equations

A Harnack’s inequality and Hölder continuity for solutions of mixed type evolution equations

Further existence results for elliptic–parabolic and forward–backward parabolic equations

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