Regularity for Double Phase Variational Problems

By Maria Colombo, Giuseppe Mingione. 2015

467 citations recorded by the source corpus.

AbstractWe prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero set of a modulating coefficient $${a(\cdot)}$$a(·). The model case is given by the functional $$ w \mapsto \int (|Dw|^p + a(x)|Dw|^q) \, {\rm d}x,$$w↦∫(|Dw|p+a(x)|Dw|q)dx,where q > p and $${a(\cdot) \geqq 0}$$a(·)≧0.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/7c368ef3187ef764561d6900b639efde82f8505e

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Constant sign solutions for double phase problems with superlinear nonlinearity (2019, methodology): “We refer to the papers of Baroni-Colombo-Mingione [2], [3], [5], Baroni-Kussi-Mingione [4], Colombo-Mingione [8], [9] and the references therein concerning the regularity.”
Multiplicity results for $(p,q)$ fractional elliptic equations involving critical nonlinearities (2018, background): “ocal case s1 = s2 = 1, the functional associated to the operator in (Pθ,λ) falls in the realm of general model functional P(w,Ω) := Z Ω |∇w|p +a(x)|∇w|q dx, where a(.) ≥ 0, was extensively studied in [4, 14, 15]. This kind of functional was ﬁrst introduced by Zhikov [21, 38] in order to produce mode”
Existence of solutions for singular double phase problems via the Nehari manifold method (2021, methodology): “We mention the papers of Baroni-ColomboMingione [3, 4, 5], Baroni-Kuusi-Mingione [6], Byun-Oh [7], Colombo-Mingione [10, 11], De Filippis-Palatucci [12], Marcellini [25, 26, 27], Ok [32, 33], RagusaTachikawa [42] and the references therein.”
Growth conditions and regularity for weak solutions to nonlinear elliptic pdes (2020, background): “The following results have been obtained by M. Colombo-Mingione; of course in the first one we need a more strict assumption than either (3.8) or (3.9).”
An existence result for singular Finsler double phase problems (2020, methodology): “We mention the papers of BaroniColombo-Mingione [3–5], Baroni-Kuusi-Mingione [6], Colombo-Mingione [12,13], Marcellini [32,31], Ok [35,36], Ragusa-Tachikawa [42] and the references therein.”
A new class of double phase variable exponent problems: Existence and uniqueness (2021, methodology): “We refer to the works of Baroni-Colombo-Mingione [6, 7, 8], Baroni-Kuusi-Mingione [9], Byun-Oh [16], Colombo-Mingione [20, 21], De Filippis [22], De Filippis-Palatucci [24], Harjulehto-Hästö-Toivanen [42], Marcellini [51, 52], Ok [54, 55], 2020 Mathematics Subject Classification.”
Existence and uniqueness results for double phase problems with convection term (2019, methodology): “We refer to the papers of Baroni-ColomboMingione [2], [3], [5], Baroni-Kussi-Mingione [4], Colombo-Mingione [9], [10] and the references therein concerning the regularity.”
Bounded Minimisers of Double Phase Variational Integrals (2015, background): “1 follows from the results of [14].”
The maximal operator on generalized Orlicz spaces (2015, methodology): “[18] regarding the optimality); (3) The double phase functional φ(x, t) = t + a(x)t of Mingione and collaborators [1, 2, 3], where the sharp condition for the regularity of minimizers is recovered, namely q p < 1 + α n with a ∈ C (Theorem 4.”
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.”
Lipschitz Bounds and Nonautonomous Integrals (2020, background): “tion 9], leads to the reverse Hölder inequality R−n/(1+δ)kH(·,Du)k L1+δ(B R/2) ≤ cR−nkH(·,Du)k L1(BR) +c, for c≡ c(data h),which in turn allows us to conclude. Let B⋐ Ω be a ball with r(B) ≤ 1. As is [23,24], we say that the p-phase occurs when (10.3) a i(B) := inf x∈B a(x) ≤ 4[r(B)]1−n/d[a] 0,1−n/d”
Existence and multiplicity of solutions for double‐phase Robin problems (2020, background): “P. Baroni, M. Colombo and G. Mingione, ‘Harnack inequalities for double phase functionals’, Nonlinear Anal.”
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].”