Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

Monticelli, Dario D.; Payne, Kevin R.

doi:10.1016/j.jde.2009.06.024

Cited by 24 publications

(40 citation statements)

References 27 publications

(68 reference statements)

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“…The hypothesis in Theorem 5.1 that (1.11) holds is required by the change-of-coordinates argument used in Step 1 of the proof. [60]. Although unattractive, Remark 5.3 indicates that this hypothesis is not unduly restrictive in applications since, typically, we can take h = n(x 0 ) for some x 0 ∈ ∂ 0 O.…”

Section: Weak Maximum Principle For Smooth Functionsmentioning

confidence: 99%

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

Feehan

2013

Communications in Partial Differential Equations

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Abstract. We prove weak and strong maximum principles, including a Hopf lemma, for C 2 subsolutions to equations defined by second-order, linear elliptic partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the C 2 subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol's vanishing locus. We obtain uniqueness and a priori maximum principle estimates for C 2 solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators with partial Dirichlet or Neumann boundary conditions. We also prove weak maximum principles and uniqueness for W 1,2 solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions. Contents List of

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Section: Weak Maximum Principle For Smooth Functionsmentioning

confidence: 99%

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

Feehan

2013

Communications in Partial Differential Equations

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“…(22) be the Kelvin transform of the solution u with respect to the origin, which is defined for z = (x, y) ∈ R d+k \ {0}. We recall that the Kelvin transform (22) is an element of the symmetry group of the Grushin operator (see [18]).…”

Section: Two Applications Of the Moving Planes To The Grushin Operatormentioning

confidence: 99%

“…We have addressed the problem of studying maximum principles for the class of operators considered here in a setting compatible with a suitable notion of weak solution in a work which will appear elsewhere (see [22]). …”

Section: Introductionmentioning

confidence: 99%

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Monticelli¹

2010

J. Eur. Math. Soc.

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Abstract. We deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for the linear equation. A good example of such an operator is the Grushin operator on R d+k , to which we devote particular attention. As an application of these tools in the degenerate elliptic setting, we prove a partial symmetry result for classical solutions of semilinear problems on bounded, symmetric and suitably convex domains, which is a generalization of the result of Gidas-Ni-Nirenberg [12], [13], and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space, which is a generalization of the result of Gidas-Spruck [14] and Chen-Li [6]. We use the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator.

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“…, w n (x) be a subunit vector field in . Assume that the global weak Poincaré inequality with gain ω > 1 holds, see (10). Then…”

Section: Lemma 21 Let Be a Bounded Open Set Such Thatmentioning

confidence: 99%

“…Section 4 contains a maximum principle for weak solutions of Xu ≤ 0 and in Section 5 we demonstrate a relationship between compact embeddings of Sobolev spaces and global Poincaré inequalities with gain; we refer the reader to Theorems 4.3 and 5.1 for these results. All of our results are developed in the spirit of [2] and [4] using ideas presented in [1,[9][10][11][13][14][15][16][17], and other related works.…”

Section: Introductionmentioning

confidence: 99%

Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients

Monticelli

Rodney

2015

Journal of Differential Equations

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In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated with second order linear degenerate elliptic partial differential operators X with rough coefficients, of the form X = −div(P ∇) + HR + S G + F , where the n × n matrix function P = P (x) is nonnegative definite and allowed to degenerate, R, S are families of subunit vector fields, G, H are vector valued functions and F is a scalar function. We operate in a geometric homogeneous space setting and we assume the validity of certain Sobolev and Poincaré inequalities related to a symmetric nonnegative definite matrix of weights Q = Q(x) that is comparable to P ; we do not assume that the underlying measure is doubling. We give a maximum principle for weak solutions of Xu ≤ 0, and we follow this with a result describing a relationship between compact projection of the degenerate Sobolev space QH 1,p , related to the matrix of weights Q, into L q and a Poincaré inequality with gain adapted to Q.

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Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

Cited by 24 publications

References 27 publications

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients

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