Continuous rearrangement and symmetry of solutions of elliptic problems

Brock, Friedemann

doi:10.1007/bf02829490

Cited by 73 publications

(124 citation statements)

References 23 publications

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“…Previously this result was known only for radial solutions in a ball, or general solutions again in a ball, once radial symmetry results are available (see e.g. [4], [5], [10]). …”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 93%

Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations

2005

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We consider the Dirichlet problem for positive solutions of the equation -Delta(m)(u) = f (u) in a bounded smooth domain Omega, with f positive and locally Lipschitz continuous. We prove a Harnack type inequality for the solutions of the linearized operator, a Harnack type comparison inequality for the solutions, and exploit them to prove a Strong Comparison Principle for solutions of the equation, as well as a Strong Maximum Principle for the solutions of the linearized operator. We then apply these results, together with monotonicity results recently obtained by the authors, to get regularity results for the solutions. In particular we prove that in convex and symmetric domains, the only point where the gradient of a solution u vanishes is the center of symmetry (i.e. Z equivalent to {x is an element of Omega vertical bar D(u)(x) = 0} = {0} assuming that 0 is the center of symmetry). This is crucial in the study of m-Laplace equations, since Z is exactly the set of points where the m-Laplace operator is degenerate elliptic. As a corollary u is an element of C-2(Omega \ {0})

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“…Previously this result was known only for radial solutions in a ball, or general solutions again in a ball, once radial symmetry results are available (see e.g. [4], [5], [10]). …”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 93%

Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations

2005

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“…Note that a 1 , u e in Proposition 3.2 of [9] are 0 here. We have from [20] that the problem (To obtain the symmetry results, in [20], the author used the results in [3] and [4].) Moreover, the proofs of Lemma 6.1 and (iii) of Theorem B imply that for any 0 < s g < m − r 1 , there exists C =C(s as e Q 0.…”

Section: Quasilinear Elliptic Eigenvalue Problemsmentioning

confidence: 98%

“…The following result, Theorem 2 from [4], was used in [20]. Theorem D was established by using a new rearrangement technique called continuous Steiner symmetrization (see [3,4]) together with the maximum principle for the p-Laplacian. Note that b(s)=Ms p − 1 ¥ A p .…”

Section: Quasilinear Elliptic Eigenvalue Problemsmentioning

confidence: 99%

Large and Small Solutions of a Class of Quasilinear Elliptic Eigenvalue Problems

Zhang

Webb

2002

Journal of Differential Equations

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Existence and uniqueness results for large positive solutions are obtained for a class of quasilinear elliptic eigenvalue problems in general bounded smooth domains via a generalization of a sweeping principle of Serrin. The nonlinear terms of the problems can be negative in some intervals. The existence and structure of a mountain pass solution are also discussed. We show that this solution develops to a spike layer solution. © 2002 Elsevier Science (USA)

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“…its domain is the union of annuli on which it is radial and of a set on which ∇u = 0 almost everywhere [3].…”

Section: Introductionmentioning

confidence: 99%

Symmetrization and Minimax Principles

Schaftingen

2005

Commun. Contemp. Math.

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We develop a method to prove that some critical levels for functionals invariant by symmetry obtained by minimax methods without any symmetry constraint are attained by symmetric critical points. It is used to investigate the symmetry properties of solutions of elliptic partial differential equations with Dirichlet or Neumann boundary conditions. It is also an alternative to concentration-compactness for some symmetric elliptic problems.

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Continuous rearrangement and symmetry of solutions of elliptic problems

Cited by 73 publications

References 23 publications

Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations

Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations

Large and Small Solutions of a Class of Quasilinear Elliptic Eigenvalue Problems

Symmetrization and Minimax Principles

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