Regularity for the planar optimal p-compliance problem

In this paper we prove a partial C1,α regularity result in dimension N = 2 for the optimal p-compliance problem, extending for p≠2 some of the results obtained by Chambolle et al. (2017). Because of the lack of good monotonicity estimates for the p-energy when p≠2, we employ an alternative technique based on a compactness argument leading to a p-energy decay at any flat point. We finally obtain that every optimal set has no loop, is Ahlfors regular, and is C1,α at ℌ1-a.e. point for every p ∈ (1, +∞).

[1]  T. Bagby,et al.  Quasi topologies and rational approximation , 1972 .

[2]  G. David,et al.  Singular Sets of Minimizers for the Mumford-Shah Functional , 2005 .

[3]  Eugene Stepanov,et al.  Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem , 2003 .

[4]  Giuseppe M. Buttazzo,et al.  Asymptotical compliance optimization for connected networks , 2007, Networks Heterog. Media.

[5]  Antonin Chambolle,et al.  Regularity for the Optimal Compliance Problem with Length Penalization , 2015, SIAM J. Math. Anal..

[6]  W. Ziemer.  Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , 1989 .

[7]  É. Oudet,et al.  Optimal Transportation Problems with Free Dirichlet Regions , 2002 .

[8]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[9]  S. Semmes,et al.  Analysis of and on uniformly rectifiable sets , 1993 .

[10]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[11]  L. Hedberg.  Non-linear potentials and approximation in the mean by analytic functions , 1972 .

[12]  A. Bonnet,et al.  On the regularity of edges in image segmentation , 1996 .

[13]  J. Sikora,et al.  Optimal shape design , 1999 .

[14]  B. Dacorogna,et al.  Sur une généralisation de l’inégalité de Wirtinger , 1992 .

[15]  Topologie , 2020, Mathematische Begriffe in Beispielen und Bildern.

[16]  Emmanuele DiBenedetto,et al.  C1 + α local regularity of weak solutions of degenerate elliptic equations , 1983 .

[17]  T. O’Neil.  Geometric Measure Theory , 2002 .

[18]  P. Lindqvist.  Notes on the Stationary p-Laplace Equation , 2019, SpringerBriefs in Mathematics.

[19]  D. Bucur,et al.  Shape optimisation problems governed by nonlinear state equations , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[20]  Irene Fonseca,et al.  Regularity results for anisotropic image segmentation models , 1997 .

[21]  Ronald F. Gariepy.  FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs) , 2001 .

[22]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[23]  L. Hedberg,et al.  Function Spaces and Potential Theory , 1995 .

[24]  Antoine Lemenant,et al.  Approximation of Length Minimization Problems Among Compact Connected Sets , 2014, SIAM J. Math. Anal..

[25]  Antoine Lemenant,et al.  Partial regularity for the crack set minimizing the two-dimensional Griffith energy , 2019, Journal of the European Mathematical Society.

[26]  Gianni Dal Maso,et al.  Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators , 1997 .

[27]  O. Scherzer,et al.  Weakly Differentiable Functions , 2009 .

[28]  Al-hassem Nayam,et al.  Constant in two-dimensional $p$-compliance-network problem , 2014, Networks Heterog. Media.

[29]  D. Slepčev.  Counterexample to regularity in average-distance problem , 2014 .

[30]  Al-hassem Nayam.  Asymptotics of an optimal compliance-network problem , 2013, Networks Heterog. Media.

[31]  Eugene Stepanov,et al.  Existence and regularity results for the Steiner problem , 2012, Calculus of Variations and Partial Differential Equations.