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

Abstract

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.

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

“…At present, we do not have a suitable strong maximum principle for weak solutions on the entire domain Ω which would allow us to propagate the vanishing of u from one connected component to another. For classical solutions, a Hopf lemma has been developed for such a purpose in [33].…”

“…(5) As noted above, this class of operators has been studied in the context of classical solutions in [33], where weak and strong maximum principles as well as a Hopf lemma have been proved as well as generalized maximum principles and a priori estimates for the linear Dirichlet problem on bounded domains. To be precise, [33] uses only the continuity of a ij . In addition, (3) and (5) are used to obtain the weak maximum principle and the Hopf lemma, while the strong and generalized maximum principles exploit also a more elaborate condition (Σ) in place of (4) on the structure of the degeneracy set.…”

“…Finally we assume that L admits a uniformly elliptic direction ξ in the sense that ∃β > 0, ξ ∈ R N with |ξ | = 1 such that ξ, A(x)ξ β > 0 ∀x ∈ Ω. (5) As noted above, this class of operators has been studied in the context of classical solutions in [33], where weak and strong maximum principles as well as a Hopf lemma have been proved as well as generalized maximum principles and a priori estimates for the linear Dirichlet problem on bounded domains. To be precise, [33] uses only the continuity of a ij .…”

“…In [32,33] a general class of linear degenerate elliptic operators of second order was introduced whose minimal structural properties were chosen to allow for the use of maximum principles along the lines of classical approaches for uniformly elliptic equations and inequalities. In that work, attention was given to solutions with classical regularity, a request which often is too stringent for applications.…”

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

“…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.…”