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Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators
Full text linkIn this paper 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 results of Gidas-Ni-Nirenberg (1979, 1981) 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 (1981) and Chen-
Li (1991). The method we use to obtain these results is the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator
Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients
No full textOn the Dirichlet problem of mixed type for lower hybrid waves in axisymmetric cold plasmas
No full textOn the relation between conformally invariant operators and some geometric tensors
No full textIn this note we introduce and study some new tensors on general Riemannian manifolds which provide a link between the geometry of the underlying manifold and conformally invariant operators (up to order four). We study some of their properties and their relations with well-known geometric objects, such as the scalar curvature, the Q-curvature, the Paneitz operator and the Schouten tensor, and with the elementary conformal tensors recently constructed on the Euclidean space
On fully nonlinear CR invariant equations on the Heisenberg group
No full textAbstractIn this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author in Li and Li (2003) [21]. We also prove a comparison principle for solutions of second order fully nonlinear CR invariant equations defined on bounded domains of the Heisenberg group and a comparison principle for solutions of a family of second order fully nonlinear equations on a punctured ball
Nonexistence of Stable Solutions to Quasilinear Elliptic Equations on Riemannian Manifolds
No full textWe prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted growth conditions on geodesic balls
Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction
No full textAbstractFor second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L2-based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue
A variational characterization of flat spaces in dimension three
No full textWe prove that, in dimension three, flat metrics are the only complete metrics with nonnegative scalar curvature which are critical for the σ2-curvature functional
A note on curvature of Riemannian manifolds
No full textWith the aid of the weak maximum principle at infinity we give some sufficient conditions for Riemannian manifolds to be either Einstein or of constant sectional curvature
Harnack's inequality and Hölder continuity for weak solutions of degenerate quasilinear equations with rough coefficients
Full text linkWe continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form div(A(x,u,∇;u))=B(x,u,∇;u)for x ∈Ω as considered in our paper Monticelli etal. (2012). There we proved only local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local Hölder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin (1964) and N. Trudinger (1967) for quasilinear equations, as well as ones for subelliptic linear equations obtained in Sawyer and Wheeden (2006, 2010)
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