Higher integrability of the gradient for minimizers of the 2 d Mumford–Shah energy

Abstract

We prove the existence of an exponent p > 2 with the property that the approximate gradient of any local minimizer of the 2-dimensional Mumford-Shah energy belongs to L p loc .Nous démontrons l'éxistence d'un exposant p > 2 tel que le gradient approximé d'une fonction 2-dimensionelle quelconque qui minimise localement l'énergie de Mumford-Shah appartientà l'èspace L p loc .

Higher Integrability for Minimizers of the Mumford–Shah Functional

“…A positive answer to the above conjecture was given in [7] when n = 2. The proof there strongly relies on the two-dimensional assumption, since it uses the description of minimal Caccioppoli partitions.…”

Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture

“…The higher integrability of the gradient has been first established by De Lellis and Focardi [35] in dimension 2. Following a classical path, the key ingredient to establish Theorem 3.9 is a reverse Hölder inequality for the gradient, which we state independently (see [35,Theorem 1.3]).…”

“…We point out that Bucur and Luckhaus [16], independently from [35], have been able to improve the ideas in Theorem 2.10 carrying on the proof without the 2-dimensional limitation via a delicate induction argument. Their approach leads to a remarkable monotonicity formula for (a truncated version of) the energy valid for a broad class of approximate minimizers that shall be the topic of the next subsection 2.4.…”

“…So far only a first step into this direction has been established. A proof of Theorem 3.9 in 2-dimension has been given by De Lellis and Focardi in [35], shortly after De Philippis and Figalli established the result without any dimensional limitation in [37]. The exponent p in both papers is not explicitly computed despite some suggestions to do that are also proposed.…”

“…In particular, the quoted estimate on the Hausdorff dimension of the full singular set reduces to the higher integrability property of the gradient and a corresponding estimate on a special subset of singular points: those for which the scaled Dirichlet energy is infinitesimal. The latter topic is dealt with in full details in Section 4 in the setting of Caccioppoli partitions as done by De Lellis and Focardi in [35]. The analysis of Section 4 allowed the same Authors to prove the higher integrability property in 2-dimensions as explained in Section 5.…”