On the Hölder regularity of all extrema in Hilbert’s 19th Problem

Friedrich Tomi; Anthony Tromba

doi:10.1515/acv-2021-0089

Article

On the Hölder regularity of all extrema in Hilbert’s 19th Problem

Friedrich Tomi and Anthony Tromba

Published/Copyright: June 29, 2022

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advances in Calculus of Variations Volume 17 Issue 1

Abstract

Let Ω ⊂ ℝ n be a C 1 smooth compact domain. Furthermore, let F : Ω × ℝ n ⁢ N → ℝ , F ⁢ ( x , p ) , be C 0 , differentiable with respect to p, and with F p := D p ⁢ F continuous on Ω × ℝ n ⁢ N and F strictly convex in p. Consider an n ⁢ N × n ⁢ N matrix A = ( A α ⁢ β i ⁢ j ) ∈ C 0 ⁢ ( Ω ) satisfying

(0.1) A α ⁢ β i ⁢ j ⁢ ( x ) ⁢ ξ α i ⁢ ξ β j = A β ⁢ α j ⁢ i ⁢ ( x ) ⁢ ξ α i ⁢ ξ β j ≥ λ ⁢ | ξ | 2 , λ > 0 .

Suppose that

(0.2) lim | p | → ∞ ⁡ 1 | p | ⁢ ( D p ⁢ F ⁢ ( x , p ) - A ⁢ ( x ) ⁢ p ) = 0 ,

(0.3) - C 0 + c 0 ⁢ | p | 2 ≤ F ⁢ ( x , p ) ≤ C 0 ⁢ ( 1 + | p | 2 ) ,

(0.4) | F p ⁢ ( x , p ) - F p ⁢ ( x , q ) | ≤ C 0 ⁢ | p - q | ,

(0.5) 〈 F p ⁢ ( x , p ) - F p ⁢ ( x , q ) , p - q 〉 ≥ c 0 ⁢ | p - q | 2

uniformly in x and with positive constants c 0 and C 0 . Consider the functional

(0.6) J ⁢ ( u ) := ∫ Ω F ⁢ ( x , D ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x + ∫ Ω G ⁢ ( x , u ) ⁢ 𝑑 x ,

where G ⁢ ( x , ⋅ ) ∈ C 1 ⁢ ( ℝ N ) for each x ∈ Ω , G ⁢ ( ⋅ , u ) is measurable for each u ∈ ℝ N , and

(0.7) | G u ⁢ ( x , u ) | ≤ C 0 ⁢ ( 1 + | u | s )

with s < n + 2 n - 2 . Under these conditions, we shall show that if n > 2 , then any weak solution u ∈ W 1 , 2 ⁢ ( Ω , ℝ N ) of the Euler equations of J, i.e.

∑ α ∂ ∂ ⁡ x α ⁢ F p α i ⁢ ( x , D ⁢ u ) = G u i ⁢ ( x , u ) , i = 1 , … , N ,

is Hölder continuous in the interior of Ω and under appropriate boundary conditions also Hölder continuous up to the boundary.

Keywords: Extrema of variational problems; Hölder continuity; flow of vector fields

MSC 2010: 35B65

Dedicated to Richard Palais on his 90th birthday

Communicated by Frank Duzaar

References

[1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Search in Google Scholar

[2] M. Carozza, J. Kristensen and A. Passarelli di Napoli A trace preserving operator and applications, J. Math. Anal. Appl. 501 (2021). 10.1016/j.jmaa.2020.124170Search in Google Scholar

[3] M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 3–4, 291–303. 10.1017/S0308210500026378Search in Google Scholar

[4] E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43. Search in Google Scholar

[5] F. Duzaar, A. Gastel and J. Grotowski, Partial regularity for almost minimizers of quasi-convex integrals, SIAM J. Math Anal. 32 (2000), no. 3, 665–687. 10.1137/S0036141099374536Search in Google Scholar

[6] M. Foss, A. Passarelli di Napoli and A. Verde, Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems, Ann. Mat. Pura Appl. (4) 189 (2010), no. 1, 127–162. 10.1007/s10231-009-0103-zSearch in Google Scholar

[7] M. Giaquinta, Introduction to regularity of non-linear systems, ETH Nachdiplom Vorlesungen, Birkhauser, 1993. Search in Google Scholar

[8] M. Giaquinta and L. Martinazzi, An Introduction to the Regularity of Elliptic Systems, Harmonic Maps and Minimal Graphs, Edizioni della Normale, Pisa, 2005. Search in Google Scholar

[9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, Berlin, 1977. 10.1007/978-3-642-96379-7Search in Google Scholar

[10] W. Hao, S. Leonardi and J. Nečas, An example of irregular solution to a nonlinear Euler–Lagrange elliptic system with real analytic coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 1, 57–67. Search in Google Scholar

[11] D. Kim, Trace theorems for Sobolev–Slobodeckij spaces with or without weights, J. Funct. Spaces Appl. 5 (2007), 243–268. 10.1155/2007/471535Search in Google Scholar

[12] A. Koshelev, Regularity Problem for Quasilinear Elliptic and Parabolic Systems, Lecture Notes in Math. 1614, Springer, Berlin, 1995. 10.1007/BFb0094482Search in Google Scholar

[13] J. Kristensen and G. Mingione, Sketches of regularity theory from the 20th century and the work of Jindřich Nečas, https://citeseerx.ist.psu.edu/viewoc/download?doi=10.1.1.306.8598&rep=rep1&type=pdf. Search in Google Scholar

[14] A. Kufner, O. John and S. Fučík, Function Spaces, Noordhoff, Leyden, 1977. Search in Google Scholar

[15] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Search in Google Scholar

[16] J. Marschall, The trace of Sobolev–Slobodeckij spaces on Lipschitz domains, Manuscripta Math. 58 (1987), no. 1–2, 47–65. 10.1007/BF01169082Search in Google Scholar

[17] C. Mooney and O. Savin, Some singular minimizers in low dimensions in the calculus of variations, Arch. Ration. Mech. Anal. 221 (2016), no. 1, 1–22. 10.1007/s00205-015-0955-xSearch in Google Scholar

[18] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Grundlehren Math. Wiss. 130, Springer, New York, 1966. 10.1007/978-3-540-69952-1Search in Google Scholar

[19] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. 10.2307/2372841Search in Google Scholar

[20] V. Šverák and X. Yan, A singular minimizer of a smooth strongly convex functional in three dimensions, Calc. Var. Partial Differential Equations 10 (2000), no. 3, 213–221. 10.1007/s005260050151Search in Google Scholar

[21] V. Sverák and X. Yan, Non-Lipschitz minimizers of smooth uniformly convex functionals, Proc. Natl. Acad. Sci. USA 99 (2002), no. 24, 15269–15276. 10.1073/pnas.222494699Search in Google Scholar PubMed PubMed Central

[22] F. Tomi and A. Tromba, Morse theory and Hilbert’s 19th Problem, Calc. Var. Partial Differential Equations 58 (2019), Paper No. 172. 10.1007/s00526-019-1614-0Search in Google Scholar

[23] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. 10.1007/978-3-0346-0416-1Search in Google Scholar

Received: 2021-11-03

Revised: 2022-03-27

Accepted: 2022-04-21

Published Online: 2022-06-29

Published in Print: 2024-01-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/acv-2021-0089

Keywords for this article

Extrema of variational problems; Hölder continuity; flow of vector fields