Regularity for general functionals with double phase

By Paolo Baroni, Maria Colombo, Giuseppe Mingione. 2017

552 citations recorded by the source corpus.

We prove sharp regularity results for a general class of functionals of the type $$\begin{aligned} w \mapsto \int F(x, w, Dw) \, dx, \end{aligned}$$w↦∫F(x,w,Dw)dx,featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$\begin{aligned} w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx,\quad 1<p < q, \quad a(x)\ge 0, \end{aligned}$$w↦∫b(x,w)(|Dw|p+a(x)|Dw|q)dx,1<p<q,a(x)≥0,with $$0<\nu \le b(\cdot )\le L $$0<ν≤b(·)≤L. This changes its ellipticity rate according to the geometry of the level set $$\{a(x)=0\}$${a(x)=0} of the modulating coefficient $$a(\cdot )$$a(·). We also present new methods and proofs that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b48a2423f185746778a03815cd9ca46ff9776fc8

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Citations in the corpus

A variational approach to parabolic equations under general and p,q-growth conditions (2020, background): “A motivation to consider a class of such energy functions f can be also easily found in the stationary case, where a large literature in the calculus of variations is devoted to the minimization of p, q-growth problems [45] and to double phase problems [23], [24], [4], [5], [6].”
New martingale inequalities and applications to Fourier analysis (2018, background): “The Musielak–Orlicz spaces not only have their own interest, but they are also very useful in partial differential equations [2, 5, 28, 26], in calculus of variations [13], in image restoration [27, 43] and in fluid dynamics [73, 51].”
Existence of solutions for double phase obstacle problems with multivalued convection term (2020, unknown): “We refer to Baroni-Colombo-Mingione [3], [4], [5], Baroni-Kussi-Mingione [6], Colombo-Mingione [10], [11], CupiniMarcellini-Mascolo [12] and Marcellini [21], [22] and the references therein.”
A borderline case of Calder\'on-Zygmund estimates for non-uniformly elliptic problems (2019, background): “In [2, 4, 9, 10] the authors have provided a basic regularity theory for minimizers of double phase functionals of the type P(v,Ω) := ∫”
Regularity for Double Phase Problems at Nearly Linear Growth (2023, methodology): “The techniques needed for Theorem 1.3 are entirely different from those used in the case of classical superlinear double phase functionals in (1.3) [5,18] and for pointwise uniformly elliptic problems with nonstandard growth conditions [43,44].”
Double phase problems with variable growth (2018, background): “A key role is played by the method developed by Esposito, Leonetti, and Mingione [24] in order to prove the equivalence between the absence of Lavrentiev phenomenon and the extra regularity of the minimizers for unconstrained, non-autonomous variational problems.”
Hölder regularity for weak solutions to nonlocal double phase problems (2021, background): “5) and minimizers of corresponding variational integral has been exhaustively studied, see [2, 4, 11, 14, 15, 37] and references therein.”
Double phase anisotropic variational problems and combined effects of reaction and absorption terms (2018, background): “(iv) As it has been kindly pointed out by one of the referees of this paper, an interesting double phase type operator considered in the papers of Baroni, Colombo and Mingione [6,7,14,15], addresses functionals of the type w → ∫ Ω (|∇w|p + a(x)|∇w|q)dx, (48) where a(x) ≥ 0.”
Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold (2020, result): “We refer to the works of Baroni-Colombo-Mingione [4], [5], [6], Baroni-Kuusi-Mingione [7], CupiniMarcellini-Mascolo [16], Colombo-Mingione [14], [15], Marcellini [27], [28], Ok [33], [34], Ragusa-Tachikawa [43] and the references therein.”
Multiple solutions for parametric double phase Dirichlet problems (2020, background): “The study of equations in which the energy density changes its ellipticity and growth properties according to the point of the domain, has been revived more recently by Mingione and coworkers, who in a series of remarkable papers produced local regularity results (see [1, 2, 5, 6]).”