Operatori ellittici estremanti

Pucci, Carlo

doi:10.1007/bf02414332

Cited by 88 publications

(57 citation statements)

References 3 publications

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“…The above conjecture, in this situation, is in fact the celebrated geometric maximum principle of Alexandrov-Bakelman-Pucci (ABP, henceforth); see [A1], [Ba1], [Pu2], and also [Ba2], [Ba3]. In this setting it is well-known that an estimate such as (1.2) can only hold with the L n norm of F in the right-hand side, in the sense that it is in general impossible to have it with any smaller norm; see Alexandrov [A2] and Pucci [Pu1].…”

Section: Conjecture Given a Connected Bounded Open Setmentioning

confidence: 99%

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On the best possible character of the 𝐿^{𝑄} norm in some a priori estimates for non-divergence form equations in Carnot groups

Danielli

Garofalo

Nhieu

2003

Proc. Amer. Math. Soc.

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Abstract. Let be a group of Heisenberg type with homogeneous dimension Q. For every 0 < < Q we construct a non-divergence form operator L and a non-trivial solution u ∈ L 2,Q− (Ω) ∩ C(Ω) to the Dirichlet problem: Lu = 0 in Ω, u = 0 on ∂Ω. This non-uniqueness result shows the impossibility of controlling the maximum of u with an L p norm of Lu when p < Q. Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such aswhere m is the dimension of the horizontal layer of the Lie algebra and (u ,ij ) is the symmetrized horizontal Hessian of u.

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Section: Conjecture Given a Connected Bounded Open Setmentioning

confidence: 99%

“…[Pu1] for elliptic equations. Our proof is inspired to Serrin's note [Se] on the existence of pathological solutions for divergence form operators.…”

Section: Conjecture Given a Connected Bounded Open Setmentioning

confidence: 99%

On the best possible character of the 𝐿^{𝑄} norm in some a priori estimates for non-divergence form equations in Carnot groups

Danielli

Garofalo

Nhieu

2003

Proc. Amer. Math. Soc.

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“…See [20] and [21]. The class of operators defined above includes operators constructed upon a general set of symmetric matrices.…”

Section: About Extremal Operators Dimension and Fundamental Solutionsmentioning

confidence: 99%

Fundamental solutions and two properties of elliptic maximal and minimal operators

Felmer

Quaas

2009

Trans. Amer. Math. Soc.

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Abstract. For a large class of nonlinear second order elliptic differential operators, we define a concept of dimension, upon which we construct a fundamental solution. This allows us to prove two properties associated to these operators, which are generalizations of properties for the Laplacian and Pucci's operators. If M denotes such an operator, the first property deals with the possibility of removing singularities of solutions to the equationwhere B is a ball in R N . The second property has to do with existence or nonexistence of solutions in R N to the inequalityIn both cases a common critical exponent defined upon the dimension number is obtained, which plays the role of N/(N − 2) for the Laplacian.

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“…Our proof includes the use of a modified version of Täcklind's bootstrapping argument. Our principal innovation, however, is the use of the recent theory of elliptic extremal operators (see [4]) to obtain a comparison function independent of coefficient regularity which is then used to derive the crucial estimates. This is in contrast to Täcklind and Zolotarev who depend upon a Green's function for this purpose, thus restricting themselves to regular coefficients.…”

mentioning

confidence: 99%

“…This leads us to seek our comparison function with the same boundary conditions but satisfying, instead, Lv < 0 in Q (I, t0) since existence of solutions of Lv = 0 is unknown in general for the operators L (2.1) which we consider. The theory of extremal operators (see [4]), which we will outline here, plays a major role in the construction of (3.1) and in the proof of Theorem II.…”

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confidence: 99%

Uniqueness in the Cauchy Problem for Parabolic Equations

Hayne¹

1978

Transactions of the American Mathematical Society

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Abstract. In a classical paper S. Täcklind (Nova Acta Soc. Sei. Upsaliensis (4) 10 (1936), 1-57) closed the uniqueness question for the Cauchy problem for the heat equation with a general growth hypothesis which was both necessary and sufficient. Täcklind's proof of the sufficiency involved an ingenious bootstrapping comparison technique employing the maximum principle and a comparison function constructed from the Green's function for a half cylinder. G. N. Zolotarev (Izv. Vyss. Ucebn. Zaved. Matematika 2 (1958), 118-135) has extended this result using essentially the same technique to show that Täcklind's uniqueness condition remains sufficient for a general second order parabolic equation provided the coefficients are regular enough to permit the existence and estimation of a Green's function. We have now shown, using a new approach which replaces the construction based upon a Green's function by an appropriate comparison solution of the maximizing equation (G Pucci, Ann. Mat. Pura Appl. 72 (1966), 141-170), that Täcklind's condition is sufficient without any regularity conditions on the coefficients.

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Operatori ellittici estremanti

Cited by 88 publications

References 3 publications

On the best possible character of the 𝐿^{𝑄} norm in some a priori estimates for non-divergence form equations in Carnot groups

On the best possible character of the 𝐿^{𝑄} norm in some a priori estimates for non-divergence form equations in Carnot groups

Fundamental solutions and two properties of elliptic maximal and minimal operators

Uniqueness in the Cauchy Problem for Parabolic Equations

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