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

Journal of the European Mathematical Society (2010)

- Volume: 012, Issue: 3, page 611-654
- ISSN: 1435-9855

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topMonticelli, Dario Daniele. "Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators." Journal of the European Mathematical Society 012.3 (2010): 611-654. <http://eudml.org/doc/277413>.

@article{Monticelli2010,

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 $\mathbb \{R\}^\{d+k\}$, 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.},

author = {Monticelli, Dario Daniele},

journal = {Journal of the European Mathematical Society},

keywords = {maximum principles; degenerate elliptic linear differential operators; Grushin operator; moving planes; maximum principles; elliptic differential operators with noncharacteristic degeneracy; Grushin operator},

language = {eng},

number = {3},

pages = {611-654},

publisher = {European Mathematical Society Publishing House},

title = {Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators},

url = {http://eudml.org/doc/277413},

volume = {012},

year = {2010},

}

TY - JOUR

AU - Monticelli, Dario Daniele

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

JO - Journal of the European Mathematical Society

PY - 2010

PB - European Mathematical Society Publishing House

VL - 012

IS - 3

SP - 611

EP - 654

AB - 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 $\mathbb {R}^{d+k}$, 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.

LA - eng

KW - maximum principles; degenerate elliptic linear differential operators; Grushin operator; moving planes; maximum principles; elliptic differential operators with noncharacteristic degeneracy; Grushin operator

UR - http://eudml.org/doc/277413

ER -

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