Uniform bounds of minimizers of non-smooth constrained functionals on maps spaces
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Abstract
We consider the functional Φ(u) = ∫Ω |∇u|2dx - ∫ΩG(u)dx constrained to the set EF = {u ∈ W01,2(Ω,ℝk) : ∫ΩF(u)dx = 1}, where Ω is a bounded open subset of ℝn and F,G : ℝk → ℝ are continuous functions satisfying certain homogeneity conditions. We investigate the L∞ regularity of minimizers of Φ in EF. Moreover, we establish uniform L∞ bounds for such minimizers as well as concentration results on Ω̅. In the latter case, we prove that, up to dilations and translations, minimizers behave in a certain sense like a special type of vector bubble. The central difficulty in this study is the fact that the minimizers of Φ do not have an Euler–Lagrange equation associated.
Funding source: Capes
Funding source: CNPq
Funding source: Fapemig
The authors are indebted to the referee for his (her) valuable suggestions and comments pointed out concerning this work.
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Existence results for fractional p-Laplacian problems via Morse theory
- Uniform bounds of minimizers of non-smooth constrained functionals on maps spaces
- Allard-type boundary regularity for C1,α boundaries
- Nonsolvability of the asymptotic Dirichlet problem for the p-Laplacian on Cartan–Hadamard manifolds
- Gaussian ℬ𝒱 capacity
Articles in the same Issue
- Frontmatter
- Existence results for fractional p-Laplacian problems via Morse theory
- Uniform bounds of minimizers of non-smooth constrained functionals on maps spaces
- Allard-type boundary regularity for C1,α boundaries
- Nonsolvability of the asymptotic Dirichlet problem for the p-Laplacian on Cartan–Hadamard manifolds
- Gaussian ℬ𝒱 capacity
